Curves of Genus 2 With \sqrt{2} Multiplication

Peter Richard Bending Magdalen College, Oxford

Trinity Term, 1998

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Oxford.

Curves of Genus 2 With \sqrt{2} Multiplication

Peter Richard Bending Magdalen College, Oxford Trinity Term, 1998

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the University of Oxford.

Abstract

Fermat’s Last Theorem, one of the most celebrated results in mathematics, had been shown by K. Ribet to be a consequence of the conjecture that all semistable elliptic curves defined over ℚ are modular; this conjecture has been proved by A. Wiles and R. Taylor.

The principal aim of this thesis is to investigate the more general conjecture that all abelian varieties defined over ℚ which are ℚ-simple and of GL_{2}-type are modular, for jacobians of curves of genus 2 defined over ℚ which have \sqrt{2} ℚ-multiplication.

Our main result is a family of curves of genus 2 whose jacobians have \sqrt{2} multiplication, which contains all those satisfying the conjecture above. The family was derived using the classical Richelot isogeny between jacobians of curves of genus 2, in a modern context.

In general, the degree of an isogeny between the jacobians of two curves of genus 2 is a square. However, if the jacobians have \sqrt{2} multiplication, then its degree can be 2. We derive explicit isogenies of degree 2 between such jacobians.

We derive a family of curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication, and \overline{ℚ}-multiplication by the maximal order in the division quaternion algebra (\frac{2,−3}{ℚ}). If the jacobian of such a curve is modular, then a corresponding newform f has the property that the Galois conjugate of f is the twist of f by a Dirichlet character.

Finally, we list a table of curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor N^{2}, and hopefully modular. We do not prove modularity in general (a few of our curves are quotients of modular curves by Atkin-Lehner involutions, and so their jacobians are modular). Our investigation of the conjecture is fairly successful for N ≤ 500, less so for larger N.

Acknowledgements

I would like to thank Prof. Bryan Birch for his supervision over the last four years, and for introducing me to the exciting world of curves of genus 2.

I would like to thank EPSRC, and so in turn the British taxpayers, for providing my grant. I may well pay more than my grant by becoming one myself, so I hope they forgive me.

I would like to thank various people for useful discussions and general interest, especially my colleague John Wilson. Other people in this category are Dr. John Merriman and Dr. Frazer Jarvis.

I would like to thank X. Wang and M. Müller from the University of Essen for sending me via email useful information for this thesis.

I would like to thank Imin Chen, Steven Galbraith, Mike Richards, Tim Gardener (and probably other people who I’ve forgotten to mention) for enhancing my life within the Mathematical Institute.

I would like to thank the various societies in which I have been involved with during my time at Oxford, namely Magdalen Singers, Oxford University Chess Club, Taruithorn, and also the chapel at Magdalen, for providing much needed extracurricular activities.

Finally, I would like to thank God, and my family, for their continual support.

The LORD is my shepherd; I have everything I need.

Psalm 23 v. 1

Contents

1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 1

2 Background 3 2.1 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Abelian Varieties (Algebraic) . . . . . . . . . . . . . . . . . . . . 3 2.3 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 A sufficient condition for the

endomorphism ring to be commutative . . . . . . . . . . . . . . 16 2.5 L-Series and Conductors . . . . . . . . . . . . . . . . . . . . . . 19 2.6 The Richelot Isogeny . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7 Abelian Varieties (Complex-Analytic) . . . . . . . . . . . . . . . 35 2.8 The Abel-Jacobi Isomorphism . . . . . . . . . . . . . . . . . . . 42 2.9 The Action of a Θ-Symmetric Endomorphism on 2-Torsion … 48 2.10 Elliptic Curves with Complex Multiplication . . . . . . . . . . . 56 2.11 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.12 Modular Abelian Varieties . . . . . . . . . . . . . . . . . . . . . 62

3 Curves with \sqrt{2} Multiplication 65 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 The \sqrt{2} Multiplication Family of Curves . . . . . . . . . . . . . 66 3.3 A sufficient condition for K-simplicity . . . . . . . . . . . . . . . 68 3.4 The Galois Group . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 The Outline of the Proof of Theorem 3.2.1 . . . . . . . . . . . . 73 3.6 The Proofs of the Propositions . . . . . . . . . . . . . . . . . . . 75 3.7 A Likely Strengthening of Theorem 3.2.1 . . . . . . . . . . . . . 87 3.8 A Departure from Θ-Symmetricity . . . . . . . . . . . . . . . . 88

4 Isogenies of Degree 2 Between Curves 91 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.2 A restricted form of Theorem 3.2.1 . . . . . . . . . . . . . . . . 93 4.3 Stating Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . 95 4.4 The curves C_{1}, C_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.5 Discovering Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . 103 4.6 Proving Theorem 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . 106

5 Curves with (2, −3/ℚ) Multiplication 109 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 The curve

Y^{2} = X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3 . . . . . . . . . . 112 5.3 Discovering Theorem 5.1.1 . . . . . . . . . . . . . . . . . . . . . 116 5.4 The Outline of the Proof of Theorem 5.1.1 . . . . . . . . . . . . 118 5.5 The Proofs of the Propositions . . . . . . . . . . . . . . . . . . . 121

6 Conductors of Curves 134 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.2 Curves with Low Conductor and \sqrt{2} Multiplication . . . . . . . 134 6.3 Finding Curves with Low Conductor and \sqrt{2} Multiplication . . 137 6.4 Taking Isogenies of Degree 2 . . . . . . . . . . . . . . . . . . . . 138 6.5 Taking Richelot Isogenies . . . . . . . . . . . . . . . . . . . . . . 139 6.6 The Other Putative ℚ-Isogeny Classes . . . . . . . . . . . . . . 144 6.7 The ℚ-Endomorphism Rings of the Curves . . . . . . . . . . . . 144 6.8 Comments on Modularity . . . . . . . . . . . . . . . . . . . . . 146

7 Conclusion 153

A Tables 154 A.1 Curves With \sqrt{2} Multiplication and Low Conductor . . . . . . . 154 A.2 Discriminants and Galois Groups . . . . . . . . . . . . . . . . . 182 A.3 Parameterisations for Theorem 3.2.2 . . . . . . . . . . . . . . . 197 A.4 Parameterisations for Theorem 4.3.1 . . . . . . . . . . . . . . . 211

Chapter 1

Introduction

1.1 Motivation

Fermat’s Last Theorem, one of the most celebrated results in mathematics, had been shown by K. Ribet to be a consequence of the conjecture that all semistable elliptic curves defined over ℚ are modular; this conjecture has been proved by A. Wiles and R. Taylor. By a strengthening of their method, F. Diamond has shown that all elliptic curves which are semistable at 3 and 5 are modular (see [D-D-T] for further details). Part of the Shimura-Taniyama-Weil conjecture states the following:

Conjecture 1.1.1 All elliptic curves defined over ℚ are modular.

J. Cremona has computed all the modular elliptic curves of conductor ≤ 6000; the curves of conductor ≤ 1000 are listed in [Cre1]. Generalising from elliptic curves to abelian varieties, part of the generalised Shimura-Taniyama-Weil conjecture states the following:

Conjecture 1.1.2 All abelian varieties A defined over ℚ which are ℚ-simple and of GL_{2}-type (see Section 2.12 for the definition of GL_{2}-type) are modular.

Less is known about Conjecture 1.1.2 than about Conjecture 1.1.1; in particular, there are fewer explicitly known examples of modular abelian varieties of dimension ≥ 2 than of modular elliptic curves. The natural first generalisation is from elliptic curves defined over ℚ to jacobians of curves of genus 2 defined over ℚ which are simple over ℚ and of GL_{2}-type. In this thesis, we will investigate Conjecture 1.1.2 for jacobians of curves of genus 2 defined over ℚ which are simple over ℚ and have \sqrt{2} ℚ-multiplication.

1.2 Structure of the Thesis

In Chapter 2, we outline the mathematics which will be used in this thesis, establishing the notation to be used.

In Chapter 3, we derive a family of curves of genus 2 defined over a subfield K of ℂ whose jacobians have \sqrt{2} K-multiplication (i.e. a \sqrt{2} multiplication defined over K), and show that, if the K-endomorphism algebra of the jacobian of a curve C of genus 2 defined over K is a field containing a \sqrt{2} multiplication, then C is K-isomorphic to a curve in our family. The proof will use the classical Richelot isogeny between jacobians of curves of genus 2, in a modern context. Moreover, we will see that if the jacobian a curve C of genus 2 defined over ℚ has a \sqrt{2} ℚ-multiplication, is ℚ-simple and is modular, then C is ℚ-isomorphic to a curve in our \sqrt{2} multiplication family, with K = ℚ; this fact is useful for our investigation of Conjecture 1.1.2.

In Chapter 4, we derive explicit K-isogenies of degree 2 between the jacobians of two curves of genus 2 defined over K, provided that they have \sqrt{2} K-multiplication. Note that the property of having \sqrt{2} K-multiplication is essential, since, in general, the degree of a K-isogeny between the jacobians of two curves of genus 2 defined over K is a square. Sometimes, given a curve of genus 2 defined over ℚ whose jacobian has \sqrt{2} ℚ-multiplication and conductor N^{2}, we can obtain other such curves via ℚ-isogenies of degree 2.

In Chapter 5, we derive a family of curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication, and \overline{ℚ}-multiplication by the maximal order in the division quaternion algebra (\frac{2,−3}{ℚ}). If the jacobian of such a curve is modular, then a corresponding newform f has the property that the Galois conjugate of f is the twist of f by the Dirichlet character (5.3). In fact, our derivation of the family involves showing that the \overline{ℚ}-endomorphism algebra of a certain curve of genus 2 is isomorphic to (\frac{2,−3}{ℚ}), assuming that its jacobian is modular, finding another curve of genus 2 with similar properties, and then finding the family from these two curves. The method of proof will tie together algebraic and complex-analytic techniques, rather than use explicit computations which are potentially very tedious.

In Chapter 6, we list a table of curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor N^{2}, and hopefully modular. Some of the curves are obtained from our \sqrt{2} multiplication family, and the rest are obtained by taking Richelot isogenies and ℚ-isogenies of degree 2, mostly the latter. We do not prove modularity in general (a few of our curves are quotients of modular curves by Atkin-Lehner involutions, and so their jacobians are modular). Our investigation of Conjecture 1.1.2 is fairly successful for N ≤ 500, less so for larger N.

Finally, in Chapter 7, we discuss problems arising from this thesis.

Background

2.1 Basic Notation

Throughout this thesis, we fix once and for all a field ℂ of complex numbers, and let ℤ, ℚ, \overline{ℚ}, ℝ denote the appropriate subsets of ℂ. For all n ∈ ℂ, we choose once and for all an element in ℂ whose square is n, and denote it by \sqrt{n}. We also fix once and for all finite fields 𝔽_{p}^{n} (n ranging across the positive integers) such that 𝔽_{p}^{m} is a subfield of 𝔽_{p}^{n} if m is the greatest proper divisor of n, an algebraic closure \overline{𝔽_{p}} of 𝔽_{p}, a field of p-adic numbers ℚ_{p} containing ℚ, and a ring of p-adic integers ℤ_{p} containing ℤ and contained in ℚ_{p}. We fix these sets in order to reduce the number of homomorphisms used in our proofs.

2.2 Abelian Varieties (Algebraic)

In this section, we define the basic notions of abelian varieties and polarised abelian varieties, which are of fundamental importance in this thesis, from an algebraic viewpoint. Further details can be found in [Mil1] and [Mum1]. We also look at differential forms on abelian varieties, which can be used to establish important information about the endomorphism rings. To start with, we establish some notation, which will be used in this section and in later sections.

• k is a perfect field;

• \overline{k} is an algebraic closure of k;

• G_{k} is the absolute Galois group of k;

• X is a projective variety defined over k;

• X(k) is the set of points on X defined over k; • O_{X,k} is the sheaf of regular functions defined over k on X;

• k(X) is the field of functions defined over k on X;

and, in the case where X is non-singular and defined over k,

• Div_{k}(X) is the abelian group of divisors on X defined over k;

• [Div]_{k}(X) is the abelian group of linear equivalence classes of divisors on X defined over k;

• Pic_{k}(X) is the abelian group of isomorphism classes of invertible sheaves on X defined over k, which is isomorphic (as abelian groups) to [Div]_{k}(X);

• Ω_{k}^{1}(X) is the k(X)-vector space of differential 1-forms on X defined over k;

• Ω_{k}^{1}[X] is the k-vector space of regular differential 1-forms on X defined over k.

A point on X will denote a point on X defined over \overline{k}, with X denoting X(\overline{k}) where appropriate, and a rational point on X will denote a point on X defined over k (note that the term ‘rational’ does not mean that k = ℚ).

Whatever category we are working in, if A, B are objects on which G_{k} acts, a morphism ϕ: A → B will be called a k-morphism if σ^{−1}ϕσ = ϕ for all σ ∈ G_{k}.

More notation concerning abelian varieties will be defined later on.

An abelian variety A defined over k is a projective variety A defined over k with k-morphisms

ϕ_{GL}: A × A → A, ϕ_{I}: A → A

and a rational point O on A such that the morphism ϕ_{GL} induces a group law on the points on A with the identity being the point O and the inverse of P ∈ A being the point ϕ_{I}(P). Two fundamental and important facts are that A is non-singular and that the group law on A is commutative. A k-morphism ϕ: A → B of abelian varieties is a k-morphism of projective varieties which is a homomorphism of abelian groups; throughout this thesis, the multiplicationby-n morphism n: A → A, which simply sends a point P on A to nP (n an integer), is denoted by [n]_{A} (or sometimes just by [n] when only one A is involved).

From now on in this section, A is an abelian variety defined over k unless otherwise specified.

For P_{0} ∈ A, define t_{P_{0}}, the translation by P_{0}-morphism, to be the kmorphism

t_{P_{0}}: A → A, P ↦ ϕ_{GL}(P, P_{0}).

We have the following theorem, which leads to the concept of polarisations on abelian varieties, as will be discussed later.

Theorem 2.2.1 (Theorem of the Square): Let L be an invertible sheaf on A defined over k, and let P_{1}, P_{2} ∈ A(k). Then,

[t_{P_{1}+P_{2}}^{∗}L ⊗_{O_{A,k}} L] = [t_{P_{1}}^{∗}L ⊗_{O_{A,k}} t_{P_{2}}^{∗}L.]

Proof: See [Mil1], Section 6 Theorem 6.7.□

Let A be an abelian variety defined over k, and let Pic_{k}^{0}(A) be the subgroup of Pic_{k}(A) of isomorphism classes of invertible sheaves L on A defined over k such that

[t_{P}^{∗} L] = [L] ∀P ∈ A(k).  (2.1)

There is an abelian variety Â defined over k, whose dimension is that of A, called the dual abelian variety of A, with a k-isomorphism of abelian groups

φ: Pic0 ; \overline{k}(A) →^{∼} Â;

moreover, the pair (Â, φ) is uniquely determined up to unique k-isomorphism, and ˆ Â is k-isomorphic to A. See [Mil1], Sections 9 and 10 for more details. If A, B are abelian varieties defined over k and ϕ: A → B is a k-morphism of abelian varieties, then

ϕ̂: B̂ → Â, [L] ↦ [ϕ^{∗}L]

is a k-morphism of abelian varieties. (Note that this definition is better stated using invertible sheaves than using divisors, as the pull-back of an invertible sheaf is easier to define than the pull-back of a divisor; see [Sha], p. 279).

We now establish notation and concepts which are crucial for the notion of a polarisation on an abelian variety. Throughout, we fix (Â, φ).

Let L be an invertible sheaf on A defined over \overline{k}, and consider the map

ϕ_{L}: A → Â, P → φ([t_{P}^{∗} (L) ⊗_{O} L^{−1}]).  (2.2)

This definition is possible, since, for all P_{0} ∈ A,

[t_{P_{0}}^{∗}(t_{P}^{∗} L ⊗_{O_{A,\overline{k}}} L^{−1})] = [t_{P_{0}+P}^{∗} L ⊗_{O_{A,\overline{k}}} (t_{P_{0}}^{∗}L)^{−1}] = [t_{P}^{∗} L ⊗_{O_{A,\overline{k}}} L^{−1}]

(the second equality following from Theorem 2.2.1); by 2.1, [t_{P}^{∗} L⊗_{O_{A,\overline{k}}} L^{−1}] lies in Pic0 ; \overline{k}(A). ϕ_{L} is a homomorphism of abelian groups, since, for all P_{1}, P_{2} ∈ A,

[t_{P_{1}+P_{2}}^{∗}L ⊗_{O_{A,\overline{k}}} L^{−1}] = [(t_{P_{1}}^{∗}L ⊗_{O_{A,\overline{k}}} L^{−1}) ⊗_{O_{A,\overline{k}}} (t_{P_{2}}^{∗}L ⊗_{O_{A,\overline{k}}} L^{−1})]

by Theorem 2.2.1 (tensor both sides with L^{2}). Also, ϕ_{L} is a \overline{k}-morphism of varieties (see [Mil1], Section 10). So ϕ_{L} is a \overline{k}-morphism of abelian varieties. If D is a divisor on A defined over \overline{k}, define ϕ_{D}: = ϕ_{L(D)}, where L(D) is the invertible sheaf on A defined over \overline{k} associated to D. If L_{1}, L_{2} are invertible sheaves on A defined over \overline{k}, then ϕ_{L_{1}} = ϕ_{L_{2}} ⇔ L_{1} ⊗_{O_{A,\overline{k}}} L_{2}^{−1} ∈ Pic0 ; \overline{k}(A), by (2.2) and the fact that φ is an isomorphism; in particular, if D_{1}, D_{2} are divisors on A defined over \overline{k}, then ϕ_{D_{1}} = ϕ_{D_{2}} if and only if D_{1}, D_{2} are algebraically equivalent.

Let D be a divisor on A defined over \overline{k}. Then, for all σ ∈ G_{k},

σ^{−1}ϕ_{D}σ = σ^{−1}ϕ_{L(D)}σ = ϕ_{σ^{−1}L(D)} = ϕ_{σ^{−1}D}.

If, for all σ ∈ G_{k}, there exists P_{σ} ∈ A such that σ^{−1}D = {P ∈ A|P + P_{σ} ∈ D}, which implies that L(σ^{−1}D) = t_{P_{σ}}^{∗}L(D), then ϕ_{D} is defined over k; indeed, for all σ ∈ G_{k},

σ^{−1}ϕ_{D}σ = ϕ_{σ^{−1}D} = ϕ_{L(σ^{−1}D)} = ϕ_{t_{Pσ}^{∗} L(D)} = ϕ_{L(D)} = ϕ_{D};

the penultimate equality comes from the fact that [t_{P_{σ}}^{∗}L(D) ⊗_{O_{A,\overline{k}}} L(D)^{−1}] lies

in Pic0 ; \overline{k}(A).

A k-polarisation on A is a k-morphism λ: A → Â of abelian varieties such that

λ = ϕ_{D}

for some ample divisor D on A defined over \overline{k}. λ is a k-isogeny whose degree is a square (see [Mil1], Section 10 Theorem 10.1 and Section 13 Theorem 13.3). A principal k-polarisation is a k-polarisation which is an isomorphism. It is a standard fact that A has a k-polarisation (see [Mil1], Section 7 Theorem 7.1). If λ_{A} is a k-polarisation/principal k-polarisation on A, then the pair (A, λ_{A}) is called a k-polarised/principally k-polarised abelian variety defined over k. If (A, λ_{A}), (B, λ_{B}) are k-polarised abelian varieties defined over k, then a kmorphism

ϕ: (A, λ_{A}) → (B, λ_{B})

of polarised abelian varieties is a k-morphism ϕ: A → B of (unpolarised) abelian varieties such that λ_{A} = ϕ̂λ_{B}ϕ.

We now move on to the concept of endomorphisms of abelian varieties. We define End_{k}(A) to be the k-endomorphism ring of A; this notation will be used throughout this thesis. Also, End_{k}(A) ⊗ ℚ is the k-endomorphism algebra of A, which is a ℚ-algebra. Let D be an ample divisor on A defined over \overline{k} such that ϕ_{D} is a k-polarisation. There is a map †_{D} on the k-endomorphism algebra of A, defined by

†_{D}: End_{k}(A) ⊗ ℚ → End_{k}(A) ⊗ ℚ, ε ↦ \frac{1}{e(ϕ_{D})}ψ_{D} ∘ ε̂∘ ϕ_{D},

where ψ_{D} is the dual isogeny of ϕ_{D} and e(ϕ_{D}) is the exponent of the kernel of ϕ_{D}; we define ε^{†_{D}}: = †_{D}(ε). †_{D} has the following properties (see [Mil1], Section • (r_{1}ε_{1} + r_{2}ε_{2})^{†_{D}} = r_{1}ε_{1}^{†_{D}} + r_{2}ε_{2}^{†_{D}} for all r_{1}, r_{2} ∈ ℚ, ε_{1}, ε_{2} ∈ End_{k}(A) ⊗ ℚ;

• (ε_{1}ε_{2})^{†_{D}} = ε_{2}^{†_{D}}ε_{1}^{†_{D}} for all ε_{1}, ε_{2} ∈ End_{k}(A) ⊗ ℚ;

• ε^{†_{D}}^{†_{D}} = ε for all ε ∈ End_{k}(A) ⊗ ℚ;

so †_{D} is an anti-involution on the k-endomorphism algebra of A. It is called the Rosati involution w.r.t. D (or w.r.t. ϕ_{D}). We call a k-endomorphism ε D-symmetric if ε^{†_{D}} = ε, and D-antisymmetric if ε^{†_{D}} = −ε.

Define NS_{k}(A) to be the abelian group of the linear equivalence classes of divisors on A defined over k modulo the abelian group of the algebraic equivalence classes. We have the following theorem, which will be useful in this thesis:

Theorem 2.2.2 For a principal k-polarisation ϕ_{D_{0}} on A, there is an isomorphism of abelian groups defined by

φ_{D_{0}}: NS_{k}(A) →^{∼} End_{k}(A)^{†_{D}=1},

[D] ↦ ϕ_{D_{0}}^{−1}ϕ_{D},

defining End_{k}(A)^{†_{D}=1} to be the abelian group of D-symmetric k-endomorphisms of A.

Proof: See [Mil1], Section 17 Proposition 17.2.□

An abelian variety A defined over k is said to be k-simple if A does not contain any abelian subvariety defined over k apart from A and 0. If A is k-simple, then the k-endomorphism algebra of A is a ℚ-algebra which is a skew field. We have the following theorem, which says that the decomposition of an abelian variety A defined over k into a product of k-simple abelian varieties defined over k is essentially unique, and a corollary, which gives information on the k-endomorphism ring of A.

Theorem 2.2.3 Let A be an abelian variety defined over k. Then, there is a k-isogeny

A → A_{1}^{n_{1}} × … × A_{r}^{n_{r}},

where the A_{i}’s are abelian varieties which are simple over k and not isogenous to each other, and the n_{i}’s are positive integers. Moreover, the A_{i}’s and the n_{i}’s are uniquely determined up to isogenies and permutations.

Proof: See [Mil1], Chapter 12, p. 122.□

Corollary 2.2.4

End_{k}(A) ⊗ ℚ ≅ M_{n_{1}}(End_{k}(A_{1}) ⊗ ℚ) ⊕ … ⊕ M_{n_{r}}(End_{k}(A_{r}) ⊗ ℚ)

(in particular, the k-endomorphism algebra of A is a semisimple ℚ-algebra).

Proof: See [Mil1], Chapter 12 p. 122.□

An important proposition about the k-endomorphism ring of an abelian variety defined over k is as follows:

Proposition 2.2.5 Let A be an abelian variety defined over k. Then, the kendomorphism A is a free abelian group of rank ≤ 4 dim(A)^{2}.

Proof: See [Mil1], Section 12 Proposition 12.5.□

We now discuss how the action of the k-endomorphism ring of an abelian variety defined over k on the space of regular differential 1-forms defined over k influences the former. First, we recall relevant notation concerning differential 1-forms on non-singular projective varieties. Let X be a non-singular projective variety defined over k. Recall that Ω_{k}^{1}(X) is the k(X)-space of differential 1forms on X defined over k, and that Ω_{k}^{1}[X] is the k-vector space of regular differential 1-forms on X defined over k.

Letting A be an abelian variety defined over k, there is an action of the kendomorphism ring of A on the space of differential 1-forms defined over k on A, defined by sending a form ω to the form ε^{∗}ω, where ε is a k-endomorphism of A. As we will see below, it turns out that the k-endomorphism ring of A acts on the space of regular differential 1-forms defined over k on A in the same way, and that, if the characteristic of k is zero and A is k-simple, the resulting representation

End_{k}(A) → M_{dim(A)}(k)

(we will see that the dimension of Ω_{k}^{1}[A] as a k-vector space is that of A) is faithful, giving a ring embedding

End_{k}(A) ↪ M_{dim(A)}(k).  (2.3)

If A is not k-simple, then (2.3) is also true, by Corollary 2.2.4. In the context of this thesis, (2.3) will be useful, as will be explained later. We now state the crucial theorems, referring the proofs to established texts on algebraic geometry.

Theorem 2.2.6 Let X, Y be non-singular projective varieties defined over k. If ϕ: X → Y is a k-morphism and k(X) has a separable transcendence basis over ϕ^{∗}k(Y), then the map of k-vector spaces

ϕ^{∗}: Ω_{k}^{1}(Y) → Ω_{k}^{1}(X)  (2.4)

is an embedding.

Proof: See [Sha], Chapter 3 Section 5 Theorem 1.□

Theorem 2.2.7 Let X, Y be non-singular projective varieties defined over k. If ϕ: X → Y is a k-morphism and ϕ(X) is dense in Y, then the map of kvector spaces (2.4) sends regular differential 1-forms on Y to regular differential 1-forms on X.

Proof: See [Sha], Chapter 3 Section 5 Theorem 2.□

The map ω ↦ t_{a}^{∗}ω, where ω ∈ Ω_{k}^{1}(A) and a ∈ A(k), induces an action of A(k) on Ω_{k}^{1}(A), since t_{a}: A → A is a k-morphism. Consider H^{0}(A(k), Ω_{k}^{1}(A)), the k-vector subspace of Ω_{k}^{1}(A) invariant under the action of A(k). We have

Theorem 2.2.8

H^{0}(A(k), Ω_{k}^{1}(A)) = Ω_{k}^{1}[A].

Proof: See [Mum1], p. 42.□

Theorem 2.2.9

dim Ω_{k}^{1}[A] = dim A.

Proof: See Theorem 2.2.8, [Sha], Chapter 2 Section 1 Theorem 3 and [Sha], Chapter 3 Section 5 p. 169.□

Given these theorems, we are in a position to prove the following corollary, giving (2.3).

Corollary 2.2.10 Let A be an abelian variety defined over k, and suppose that the characteristic of k is zero. Then, (2.3) is true.

Proof: To prove the corollary, it suffices to assume that A is k-simple, by Corollary 2.2.4.

Let ε be a non-zero endomorphism in the k-endomorphism ring of A. Since A is k-simple, ε is a k-isogeny, so k(A) is a finite extension of ε^{∗}k(A), equal to the degree of ε (recall that k(A) is the field of functions defined over k on A). Since the characteristic of k is zero, this extension is also separable. By Theorem 2.2.6 and Theorem 2.2.7, the map of k-vector spaces

ε^{∗}: Ω_{k}^{1}[A] → Ω_{k}^{1}[A]

is an embedding (note that ε(A) is dense in A, since ε is surjective). This is true for all non-zero endomorphisms in the k-endomorphism ring of A and dim Ω_{k}^{1}[A] = dim A by Theorem 2.2.9, so (2.3) is true, as required.□

In the context of this thesis, we have an easy but important corollary of Corollary 2.2.10 for abelian surfaces, which will be used for D = 2:

Corollary 2.2.11 Let A be an abelian surface defined over ℚ, suppose that A has a primitive \sqrt{D} (resp. \frac{1+\sqrt{D}}{2}) ℚ-multiplication ε, where D is an integer which is not a square, (primitive means that, for all integers n ≥ 2, [\frac{1}{n}]ε, which lies in the ℚ-endomorphism algebra of A, does not lie in the ℚ-endomorphism ring of A) and that A is ℚ-simple. Then, there is a ring isomorphism from the ℚ-endomorphism ring of A to ℤ(\sqrt{D}) (resp. ℤ(\frac{1+\sqrt{D}}{2})).

Proof: Since A is ℚ-simple, the ℚ-endomorphism ring of A is not isomorphic as rings to M_{2}(ℚ). Since A has a \sqrt{D} (resp. \frac{1+\sqrt{D}}{2}) ℚ-multiplication, there is a ring embedding from ℤ(\sqrt{D}) (resp. ℤ(\frac{1+\sqrt{D}}{2})) to the ℚ-endomorphism ring of A. By Corollary 2.2.10 and the fact that ε is primitive, there is a ring isomorphism from the ℚ-endomorphism ring of A to ℤ(\sqrt{D}) (resp. ℤ(\frac{1+\sqrt{D}}{2})), as required.□

Note that, if A has a \sqrt{2} ℚ-multiplication ε and A is ℚ-simple, then there is a ring isomorphism from the ℚ-endomorphism ring of A to ℤ(\sqrt{2}); indeed, ε is primitive, since the kernel of ε is of order 4 and so does not contain the kernel of [n] for any non-zero integer n ≥ 2.

2.3 Curves

A curve is a non-singular projective variety of dimension one. Let C be a curve defined over k. In this section, we look at divisors and differentials on curves, the Riemann-Roch theorem, jacobian of curves, Torelli’s theorem, and, finally, curves of genus 2.

We now look at divisors on a curve. Suppose that k is algebraically closed. Let P be a point on C. Define O_{P} to be the ring of functions in k(C) which are regular at P, a local ring, and define 𝔪_{P} to be the maximal ideal of O_{P}. There is a valuation

v_{P}: k(C)^{∗} → ℤ

with v_{P} (f) = 1 if f ∈ 𝔪_{P} \𝔪_{P}^{2} . The divisor of f on C is defined to be the sum

(f): = ∑_{P ∈C} v_{P} (f)P.

For f ∈ k(C)^{∗}, the degree of (f) is zero (see [Har], Chapter 2 Section 6 Corollary 6.10), so the divisors of degree zero modulo linear equivalence form an abelian group, denoted by [Div^{0}]_{k}(C).

We now look at differentials on a curve. Let P ∈ C, and let t_{P} ∈ k(C) such that v_{P} (t_{P}) = 1. For all ω ∈ Ω_{k}^{1}(C), there is a unique f_{P} ∈ k(C) (depending on ω and t_{P}) such that ω = f_{P} dt_{P}, and v_{P} (f_{P}) depends only on ω and P, not on t_{P} (see [Sil1], Chapter 2 Section 4 Proposition 4.3). We define

v_{P} (ω): = v_{P} (f_{P}), div(ω) = ∑_{P ∈C} v_{P} (f_{P})P.

A divisor K is called a canonical divisor if K = div(ω) for some ω ∈ Ω_{k}^{1}(C). A divisor K′ is a canonical divisor if and only if K′ is linearly equivalent to K, since Ω_{k}^{1}(C) is a one-dimensional k(C)-vector space. Define the k-vector space V_{D} as follows:

V_{D}: = {f ∈ k(C)^{∗}: div(f) ≥ −D} ∪ {0}.

The following theorem is of fundamental importance in the theory of curves; for example, it implies that dim V_{K} = g (putting D = 0) and deg K = 2g − 2 (putting D = K).

Theorem 2.3.1 (Riemann-Roch): Let C be a curve defined over an algebraically closed field k, and let K be a canonical divisor on C. Then, there is an integer g ≥ 0, called the genus of C, such that for every divisor D ∈ C,

dim_{k} V_{D} − dim_{k} V_{K−D} = deg D − g + 1.

Proof: See [Har], Chapter 4 Section 1 Theorem 1.3. The proof involves the use of Serre duality, an aspect of the cohomological theory of algebraic geometry.□

We now relax the condition that k is algebraically closed, and study jacobians of curves. Suppose that g ≥ 1. Then, there is an abelian variety of dimension g defined over k, called a Jacobian of C and denoted by Jac(C), with a kisomorphism of abelian groups

φ: [Div^{0}]\overline{k}(C) → Jac(C).

The pair (Jac(C), φ) is uniquely determined up to unique k-isomorphism. In this thesis, as is standard practice, we will identify two abelian varieties defined over k which are jacobians of C, and simply refer to ‘the jacobian Jac(C) of C’; the issues concerning the jacobian will be independent of the abelian variety chosen. Also, using a slight but standard abuse of language, we will denote a point on the jacobian by a divisor class in [Div^{0}]\overline{k}(C) rather than by the image of a divisor class in [Div^{0}]\overline{k}(C).

Let P be a point on C. A translate by a point on the jacobian of the subset

∪_{P_{1},...,P_{g−1}∈C}{[P_{1} + … + P_{g−1} − (g − 1)P]}

of the jacobian is a divisor on the jacobian (see [Mil2], Section 5), called a theta divisor. Let Θ be a theta divisor; then, the map

φ_{Θ}: Jac(C) → Ĵac(C)

is a principal k-polarisation (see [Mil2], Section 6 Theorem 6.7), called the canonical polarisation.

A hyperelliptic curve is a curve C which is a double cover of the projective line (i.e. there is a k-morphism of degree 2 from C to the projective line). For example, as we will see, every curve of genus 2 is hyperelliptic.

The following theorem gives a useful condition for deducing when two curves defined over k are k-isomorphic, given that their jacobians are k-isomorphic, especially when they are hyperelliptic:

Theorem 2.3.2 (Torelli’s Theorem): Let C_{1}, C_{2} be curves defined over k, let P_{i} be a point on C_{i}, and define a \overline{k}-morphism f_{i}: C_{i} → Jac(C_{i}), P ↦ [P − P_{i}].

Suppose that there is a \overline{k}-isomorphism ι_{J} from the canonically polarised jacobian of C_{1} to that of C_{2}. Then, there is a \overline{k}-isomorphism ι_{C} from C_{1} to C_{2} such that f_{2} ∘ ι_{C} = ±ι_{J} ∘ f_{1} + c for some constant c on the jacobian of C_{2}.

Suppose that the genus of C_{1} is ≥ 2. Then, ι_{C} is defined over k if ι_{J} is. Moreover, if C_{1} is not hyperelliptic, then ι_{C}, the sign ± and c are uniquely determined by ι_{J}, P_{1}, P_{2}, and if C_{1} is hyperelliptic, then the sign ± can be chosen, and with this choice ι_{C} and c are uniquely determined by ι_{J}, P_{1}, P_{2}.

Proof: See [Mil2], Section 12 and Section 13. The proof is fairly elementary, but also fairly involved.□

Torelli’s Theorem will be used a lot in this thesis for curves C of genus 2 defined over a subfield K of ℂ such that the K-endomorphism algebra of the jacobian of C is a field; an important proposition using it will be proved in Section 2.8. One needs to include polarisations in Torelli’s Theorem, for an abelian variety can have several isomorphism classes of principal polarisations; in particular, an abelian variety can be the jacobian of two non-isomorphic curves simultaneously. From the complex-analytic point of view (see Section 2.7), H. Lange has proved that if X is an abelian variety of dimension g which has a principal polarisation and whose endomorphism ring is an order in a totally real field F of degree g over ℚ, then the number of isomorphism classes of principal polarisations of X is less than or equal to \frac{h^{+}}{h} (equal if the order is maximal), where h^{+} is the narrow class number of F and h is the class number of F (see [Lan2], Theorem 3.1 p. 624). Looked at from an algebraic viewpoint rather than a complex-analytic viewpoint, the fact that \frac{h^{+}}{h} is 1 when F is isomorphic to ℚ(\sqrt{2}) will turn out to be useful in this thesis.

We now recall the basic concepts of curves of genus 2, which are the subject of this thesis. Suppose that the characteristic of k is not 2, which will usually be the case in this thesis. Using the Riemann-Roch theorem, it can be shown that a curve of genus 2 defined over k is isomorphic over k to the curve of genus 2

C: Y^{2} = f_{6}X_{3}^{2} + f_{5}X_{2}X_{3} + f_{4}X_{2}^{2} + f_{3}X_{1}X_{2} + f_{2}X_{1}^{2} + f_{1}X_{0}X_{1} + f_{0}X_{0}^{2}, (2.5)

X_{1}^{2} = X_{0}X_{2}, X_{2}^{2} = X_{1}X_{3}, X_{0}X_{3} = X_{1}X_{2}

for some f_{0}, …, f_{6} ∈ k (see [C-F], Chapter 1 Sections 1 and 3). Conversely, if f_{0}, …, f_{6} ∈ k such that the discriminant of ∑_{i=0}^{6} f_{i}X^{i} is non-zero, then C is a curve of genus 2.

Using a slight but standard abuse of language (the model is a singular affine variety when the point(s) at ∞ are included rather than a non-singular projective variety) for notational convenience, the model for C which we will use in this thesis will be

C: Y^{2} = ∑_{i=0}^{6} f_{i}X^{i}.  (2.6)

A point on C of the form (x, y) (x, y ∈ \overline{k}) represents the point (1, x, x^{2}, x^{3}, y) for the model (2.5), and there are either one or two points at infinity on C according to whether f_{6} = 0 or f_{6} ≠ 0 respectively, denoted by ∞ in the former case and ∞^{+}, ∞^{−} in the latter case; a point at ∞ represents a point of the form (0, 0, 0, 1, ±\sqrt{f_{6}}) for the model (2.5). There is an involution ι on C, defined simply by (x, y) ↦ (x, −y). A Weierstrass point on C is a point fixed by ι. If f_{6} ≠ 0, there are six Weierstrass points of the form (x, 0). If f_{6} = 0, there are five Weierstrass points of the form (x, 0), and ∞ is the other Weierstrass point. To summarise, there are six Weierstrass points on C whether f_{6} is zero or not. C is a double cover of the projective line ramified at the Weierstrass points, the cover being simply (x, y) ↦ x, so is a hyperelliptic curve.

We now study isomorphisms between curves of genus 2, using, w.l.o.g., models of the form (2.6). Let M be a Möbius k-transformation of the form

x ↦ \frac{M_{11}x + M_{12}}{M_{21}x + M_{22}} , (M_{11}, M_{12}, M_{21}, M_{22} ∈ k),

(in this thesis, all Möbius k-transformations will require M_{11}, M_{12}, M_{21}, M_{22} ∈ k rather than merely having the ratios of the M_{ij}’s in k; a reason for this convention will be apparent later), let N ∈ k^{∗}, define

x|_{M}: = \frac{M_{11}x + M_{12}}{M_{21}x + M_{22}} , y|_{M,N,x}: = \frac{Ny}{(M_{21}x + M_{22})^{3}} , (x, y)|_{M,N}: = (x|_{M}, y|_{M,N,x}),

(2.7)

and define C|_{M,N} to be the curve

C|_{M,N}: (M_{11}M_{22}−M_{12}M_{21})^{6}Y^{2} = (−M_{21}x+M_{11})^{6}N^{6} ∑_{i=0}^{6} f_{i} (\frac{M_{22}x − M_{12}}{−M_{21}x + M_{11}} )^{i} .

Then, C is k-isomorphic to C|_{M,N}, the k-isomorphism C →^{∼} C|_{M,N} being given by (x, y) ↦ (x, y)|_{M,N} and the inverse k-isomorphism C|_{M,N} →^{∼} C being given by (x, y) ↦ (x, y)|_{M^{−1},det(M)^{3}N^{−1}}, where we define

det(M): = M_{11}M_{22} − M_{21}M_{12}  (2.8)

(as expected); moreover, P is a Weierstrass point on C if and only if P |_{M,N} is a Weierstrass point on C|_{M,N}. Two curves

C: Y^{6} = F_{6}(X), F_{6}(X) ∈ k[X], 5 ≤ ∂F_{6} ≤ 6,
C̃: Y^{6} = F̃_{6}(X), F̃_{6}(X) ∈ k[X], 5 ≤ ∂F̃_{6} ≤ 6

are k-isomorphic if and only if C̃ = C|_{M,N} for some Möbius k-transformation M and N ∈ k^{∗} (we need the convention that M_{11}, M_{12}, M_{21}, M_{22} ∈ k, to ensure that \frac{Ny}{(M_{21}x+M_{22})^{3}} is defined over k).

We now discuss integral invariants for a curve of genus 2 defined over ℂ such that ∞ is not a Weierstrass point on C (for simplicity), and state a lemma, easy to apply, which says that two such curves of genus 2 defined over ℂ are ℂ-isomorphic if and only if their integral invariants are equal, provided one of them is non-zero for both curves. This lemma will be used in Chapter 5, where we will link up two families of curves of genus 2.

Let C be a curve of genus 2 defined over ℂ such that ∞ is not a Weierstrass point on C, let x_{0}, …, x_{5} ∈ ℂ such that (x_{i}, 0) are the Weierstrass points on C. We define integral invariants I_{2}, I_{4}, I_{6}, I_{10} for C as follows:

I_{2} = f_{6}^{2} ∑_{15}(i_{0}i_{1})^{2}(j_{0}j_{1})^{2}(k_{0}k_{1})^{2},
I_{4} = f_{6}^{4} ∑_{10}(i_{0}j_{0})^{2}(j_{0}k_{0})^{2}(k_{0}j_{0})^{2}(i_{1}j_{1})^{2}(j_{1}k_{1})^{2}(k_{1}i_{1})^{2},
I_{6} = f_{6}^{6} ∑_{60}(i_{0}j_{0})^{2}(j_{0}k_{0})^{2}(k_{0}j_{0})^{2}(i_{1}j_{1})^{2}(j_{1}k_{1})^{2}(k_{1}i_{1})^{2}(i_{0}i_{1})^{2}(j_{0}j_{1})^{2}(k_{0}k_{1})^{2},
I_{10} = f_{6}^{10} ∏_{0≤i<j≤5}(ij)^{2},

writing (ij): = x_{i} −x_{j}; i_{0}, i_{1}, j_{0}, j_{1}, k_{0}, k_{1} are distinct integers from 0 to 5 in each term of the first three invariants. The invariants are symmetric in x_{0}, …, x_{5} and, for all n ∈ {2, 4, 6, 10}, each x_{m} appears n times in each term of I_{n}. The invariants I_{2}, I_{4}, I_{6} have 15, 10, 60 terms respectively. I_{10} is the discriminant of F_{6}. In [Wan1], X. Wang has derived expressions for I_{2}, I_{4}, I_{6}, I_{10} in terms of the coefficients of F_{6}, which we do not list here (because the expressions for I_{6}, I_{10} are lengthy).

Lemma 2.3.3 Let C and C̃ be curves of genus 2 as above, and let I_{2}, I_{4}, I_{6}, I_{10} and Ĩ_{2}, Ĩ_{4}, Ĩ_{6}, Ĩ_{10} be the integral invariants of C and C̃ respectively. Suppose that I_{2} ≠ 0 ≠ Ĩ_{2}. Then, C and C̃ are ℂ-isomorphic if and only if

\frac{I_{2}^{5}}{I_{10}} = \frac{Ĩ_{2}^{5}}{Ĩ_{10}} , \frac{I_{2}^{3}I_{4}}{I_{10}} = \frac{Ĩ_{2}^{3}Ĩ_{4}}{Ĩ_{10}} , \frac{I_{2}^{2}I_{6}}{I_{10}} = \frac{Ĩ_{2}^{2}Ĩ_{6}}{Ĩ_{10}} .

Proof: See [Wan1], Lemma 7. □

The theory of curves of genus 2 is more complicated for characteristic 2 than for characteristic not 2, and we will just scratch its surface. We will discuss curves of genus 2 defined over 𝔽_{2} coming from the reduction at 2 of curves of genus 2 defined over ℚ, and apply the discussion in Chapter 6 of this thesis.

Let G_{3}^{2}(X), F_{6}(X) ∈ ℤ[X], ∂G_{3} ≤ 3, ∂F_{6} ≤ 6 with the properties that G_{3}^{2}(X) is congruent modulo 4 to one of the polynomials

X^{6},
X^{6} + 2X^{3} + 1,
X^{6} + 2X^{4} + X^{2},
X^{6} + 2X^{4} + 2X^{3} + X^{2} + 2X + 1,
X^{6} + 2X^{5} + X^{4},
X^{6} + 2X^{5} + X^{4} + 2X^{3} + 2X^{2} + 1,
X^{6} + 2X^{5} + 3X^{4} + 2X^{3} + X^{2},
X^{6} + 2X^{5} + 3X^{4} + 3X^{2} + 2X + 1,

and that 2^{12} exactly divides the discriminant of G_{3}^{2}(X) + 4F_{6}(X) w.r.t. X. Write F_{6}(X): = ∑_{i=0}^{6} f_{i}X^{i}, G_{3}(X): = ∑_{i=0}^{3} g_{i}X^{i} (f_{i}, g_{i} ∈ ℤ) as before, and let \overline{f}_{i}, \overline{g}_{i} ∈ 𝔽_{2} be the reduction of f_{i}, g_{i} respectively modulo 2. Then, it turns out that

Y^{2}+(∑_{i=0}^{3} g_{i}X_{i}) Y = f_{6}X_{3}^{2}+f_{5}X_{2}X_{3}+f_{4}X_{2}^{2}+f_{3}X_{1}X_{2}+f_{2}X_{1}^{2}+f_{1}X_{0}X_{1}+f_{0}X_{0}^{2},

(2.9)

X_{1}^{2} = X_{0}X_{2}, X_{2}^{2} = X_{1}X_{3}, X_{0}X_{3} = X_{1}X_{2}

is a curve of genus 2 defined over ℚ which is ℚ-isomorphic to the curve Y^{2} + G_{3}(X)Y = F_{6}(X) (most of the curves in Chapter 6 will be of the latter form), and that

Y^{2}+(∑_{i=0}^{3} \overline{g}_{i}X_{i}) Y = \overline{f}_{6}X_{3}^{2}+\overline{f}_{5}X_{2}X_{3}+\overline{f}_{4}X_{2}^{2}+\overline{f}_{3}X_{1}X_{2}+\overline{f}_{2}X_{1}^{2}+\overline{f}_{1}X_{0}X_{1}+\overline{f}_{0}X_{0}^{2},

(2.10)

X_{1}^{2} = X_{0}X_{2}, X_{2}^{2} = X_{1}X_{3}, X_{0}X_{3} = X_{1}X_{2}

is a curve of genus 2 defined over 𝔽_{2}; the curve (2.10) is the reduction at 2 of the curve (2.9) (see Section 2.4 for the definition of the reduction of a non-singular projective variety).

The following discussion is independent of the characteristic of k. A point on the jacobian of a curve C of genus 2 is of the form [P_{0} − P_{1}] where P_{0}, P_{1} ∈ C, and, using the Riemann-Roch theorem, it can be shown that [P_{0}−P_{1}] = [P_{2}−P_{3}] if and only if P_{0} = P_{2}, P_{1} = P_{3} or P_{0} = ι(P_{3}), P_{1} = ι(P_{2}), and that [P_{0} −P_{1}] is a point of order dividing 2 if and only if P_{0}, P_{1} are both Weierstrass points on C. For the case where the characteristic of k is not 2, 3 or 5, E. V. Flynn has found a projective embedding in ℙ^{15} of the jacobian, the resulting projective variety being given by 72 quadratic forms, in [Fly], but we will not be concerned with his particular model, or any other, in this thesis.

2.4 A sufficient condition for the endomorphism ring to be commutative

In this section, we present a proposition which gives a sufficient (but not necessary) condition for the ℚ-endomorphism ring of a given abelian surface defined over ℚ to be commutative, easy to apply for the jacobian of a curve of genus 2 defined over ℚ. It uses a small part of the theory about abelian varieties defined over finite fields described in [Tat]. First, we need to discuss briefly the reduction modulo 𝔭 of a non-singular projective variety defined over a number field K, where 𝔭 is a prime ideal of K, and the Frobenius endomorphism of an abelian variety defined over 𝔽_{p}, where p is a prime.

Let K be a number field, let 𝔭 be a prime ideal of K, and let k_{𝔭} be the residue field of K w.r.t. 𝔭. Let X be a non-singular projective variety defined over K. Then, X is said to have good reduction at 𝔭 if there is a model Y of X and polynomials f_{1}, …, f_{r} defined over O_{K}, the ring of integers of K, such that the projective variety given by f_{1}, …, f_{r} is Y and the projective variety defined over k_{𝔭} given by \overline{f_{1}}, …, \overline{f_{r}} is non-singular (here, \overline{x} ∈ 𝔭 is the reduction of x ∈ O_{K} modulo 𝔭). This variety is called a reduction of X at 𝔭; in practice, we will usually identify two reductions, and simply refer to ‘the reduction of X at 𝔭’. X is said to have bad reduction at 𝔭 if X does not have good reduction at 𝔭. X is said to have stable reduction at 𝔭 if there is no finite extension K′ of K and no prime 𝔭′ of K′ above 𝔭 such that X has good reduction at 𝔭′. X is said to have semistable reduction if X has either good or stable reduction.

It is well-known that X has bad reduction at finitely many primes. It is also well-known that if C is a curve defined over K which has good reduction at 𝔭, then the jacobian of C has good reduction at 𝔭; the converse is false. A deep theorem, proved to help establish the irreducibility of the moduli space of curves of given genus, and useful for this thesis, is the following:

Theorem 2.4.1 Let C be a curve defined over K. Then, C has semistable reduction at 𝔭 if and only if its jacobian has semistable reduction at 𝔭.

Proof: See [D-M], Theorem 2.4. Note that the notion of stability in [D-M] is our notion of semistability. □

For a curve C of genus 2, we have an easy sufficient (but not necessary) criterion for establishing semistable reduction of C and so for establishing semistable reduction of the jacobian of C at a prime whose residue field is not of characteristic 2.

Lemma 2.4.2 Let C be a curve of genus 2 defined over K, of the form

Y^{2} = F_{6}(X), F_{6}(X) ∈ O_{K}[X], 5 ≤ ∂F_{6} ≤ 6.

Suppose that the residue field of 𝔭 is not of characteristic 2, and that the zeros of the reduction of F_{6} modulo 𝔭 (regarded as a sextic) all have multiplicity ≤ 2. Then, C has semistable reduction at 𝔭.

Proof: See [Mes2], p. 327. □

A well-known result, which will be used in this section and in Chapter 5, is as follows:

Proposition 2.4.3 If A is an abelian variety defined over K and 𝔭 is a prime ideal of good reduction of A, then the K-endomorphism ring of A embeds (via ϕ, say) into the k_{𝔭}-endomorphism ring of the reduction of A at 𝔭; if ε is a K-endomorphism of A, then the explicit definition of ϕ(ε) is simply the explicit definition of ε reduced modulo 𝔭.

From now on in this section, we will take K = ℚ, which will be the most important case in this thesis. We will refer to ‘reduction at pℤ’ by ‘reduction at p’, p being a prime.

Let A_{p} ⊆ ℙ_{𝔽_{p}}^{n} be an abelian variety defined over 𝔽_{p} (the reason why we use the notation A_{p} instead of the notation A will become apparent). The Frobenius endomorphism of A_{p} is defined to be the 𝔽_{p}-endomorphism which sends (x_{0}, …, x_{n}) ∈ A_{p} to (x_{0}^{p}, …, x_{n}^{p}). Define f to be the characteristic polynomial of this endomorphism.

If A_{p} is an elliptic curve, the it is well-known that

f(X) = 1 + (|A_{p}(𝔽_{p})| − p − 1)X + pX^{2}.  (2.11)

If A_{p} is an abelian surface, which is the case of interest, then it is well-known that

f(X) = 1 − a_{p}X + b_{p}X^{2} − a_{p}pX^{3} + p^{2}X^{4}  (2.12)

for some a_{p}, b_{p} ∈ ℚ; moreover, if C is a curve of genus 2 defined over ℚ which has good reduction at p, C_{p} is the reduction of C at p, and A_{p} is the reduction of the jacobian of C at p, then

a_{p} = p + 1 − |C_{p}(𝔽_{p})|  (2.13)
b_{p} = \frac{1}{2}|C_{p}(𝔽_{p^{2}})| + \frac{1}{2}|C_{p}(𝔽_{p})|^{2} − (p + 1)|C_{p}(𝔽_{p})| + p,  (2.14)

(see [M-S], Lemma 3). Due to these formulae, a_{p}, b_{p} are easy to compute, particularly if p is odd (if p is odd, one can always use the model Y^{2} = F_{6}(X), F_{6}(X) ∈ 𝔽_{p}[X], 5 ≤ F_{6} ≤ 6 for C_{p}, taking care to count points at ∞ the correct number of times, as described in [C-F], p. 80; if p = 2, one cannot).

We will now discuss what happens to the a_{p}’s and the b_{p}’s when C is twisted, this discussion being applied in Chapter 5. More precisely, we have the following lemma:

Lemma 2.4.4 Let C, C̃ be the curves

C: Y^{2} = F_{6}(X), C̃: Y^{2} = ΔF_{6}(X), F_{6} ∈ ℤ[X], ∂F_{6} = 6, 0 ≠ Δ ∈ ℤ,

let p be an odd prime of good reduction of both C and C̃, with the degree of F_{6} modulo p equal to 6 for simplicity, and define a_{p}, b_{p} for C, ã_{p}, b̃_{p} for C̃ as in (2.13), (2.14) respectively.

Then, ã_{p} = a_{p} if Δ is a quadratic residue modulo p, ã_{p} = −a_{p} if Δ is not a quadratic residue modulo p, and b̃_{p} = b_{p} whether p is a quadratic residue modulo p or not.

Proof: If Δ is a quadratic residue modulo p (let δ be an element of 𝔽_{p} such that δ^{2} ≡ Δ mod p), then there is a one-to-one correspondence between the points in C_{p}(𝔽_{p}) and the points in C̃_{p}(𝔽_{p}) given by

(x, y) ∈ C_{p}(𝔽_{p}) ↔ (δx, y) ∈ C̃_{p}(𝔽_{p}),

which implies that |C̃_{p}(𝔽_{p})| = |C_{p}(𝔽_{p})| and ã_{p} = a_{p}. If Δ is not a quadratic residue modulo p, then, for all x ∈ 𝔽_{p} ∪ ∞, either (x, 0) ∈ C_{p}(𝔽_{p}) ∩ C̃_{p}(𝔽_{p}) (in which case (x, 0) is a Weierstrass point of the reduction of both C_{p} and C̃_{p} modulo p) or there are precisely two y_{1}, y_{2} ∈ 𝔽_{p} ∪ {∞^{+}, ∞^{−}} such that (x, y_{1}), (x, y_{2}) ∈ C_{p}(𝔽_{p}) or (x, y_{1}), (x, y_{2}) ∈ C̃_{p}(𝔽_{p}) but (x, y_{1}), (x, y_{2}) ∉ C_{p}(𝔽_{p})∩ C̃_{p}(𝔽_{p}), which implies that |C_{p}(𝔽_{p})| + |C̃_{p}(𝔽_{p})| = 2(p + 1) and ã_{p} = −a_{p}. Whether Δ is a quadratic residue modulo p or not, |C̃_{p}(𝔽_{p}^{2})| = |C_{p}(𝔽_{p}^{2})|, since there is an element δ of 𝔽_{p}^{2} such that δ^{2} ≡ Δ mod p. So b̃_{p} = b_{p}.□

The following lemma is the basis for our proposition:

Lemma 2.4.5 Let A_{p} be an abelian variety defined over 𝔽_{p}, and let f be the characteristic polynomial of the Frobenius endomorphism of A_{p}. Then, f has no multiple root if and only if the 𝔽_{p}-endomorphism ring of A_{p} is commutative, and, if B_{p} is an abelian variety defined over 𝔽_{p} which is 𝔽_{p}-isogenous to an abelian subvariety of A_{p}, then the characteristic polynomial of the Frobenius endomorphism of B_{p} divides f.

Proof: See [Tat], Theorem 1 part (b) and Theorem 2 part (c).□

Our proposition is:

Proposition 2.4.6 Let A be an abelian surface defined over ℚ, let p be a prime of good reduction of A, and let A_{p} be the reduction of A at p (this is the reason why we used the notation A_{p} instead of the notation A for an abelian variety defined over 𝔽_{p}). Suppose that the 𝔽_{p}-endomorphism ring of A_{p} is not commutative. Then, we have

a_{p}^{2} = 4(b_{p} − 2p),  (2.15)

where a_{p}, b_{p} ∈ ℚ such that the characteristic polynomial of the Frobenius endomorphism of A_{p} is (2.12).

Proof: Since the 𝔽_{p}-endomorphism ring of A is not commutative, f has a multiple root, by Lemma 2.4.5. So the discriminant of f, which is

p^{2}((b_{p} + 2p)^{2} − p(2a_{p})^{2})(a_{p}^{2} − 4(b_{p} − 2p)),

is zero. If a_{p}^{2} − 4(b_{p} − 2p) is zero, then (2.15) holds, and so we are done. If (b_{p} + 2p)^{2} − p(2a_{p})^{2} is zero, then a_{p} = 0 and b_{p} = −2p. From (2.12), we see that f(X) = (pX^{2} − 1)^{2}. By a remark in [Tat], p. 141, the 𝔽_{p}-endomorphism algebra is isomorphic to the algebra of 2 by 2 matrices over ℚ(\sqrt{p}), and so Jac(C)_{p} is 𝔽_{p}-isogenous to the square of an elliptic curve E_{p} defined over 𝔽_{p}. By Lemma 2.4.5, pX^{2} − 1 is the characteristic polynomial of the Frobenius endomorphism of E_{p}, which is impossible from (2.11).□

We may use this proposition to prove that the ℚ-endomorphism ring of a given abelian surface A defined over ℚ is commutative, by simply finding a prime p of good reduction of A such that a_{p}^{2} ≠ 4(b_{p} − 2p), by Proposition 2.4.3. On the other hand, it is not possible to prove that the ℚ-endomorphism ring is not commutative using this proposition; one can verify that a_{p}^{2} = 4(b_{p} − 2p) for finitely many primes p of good reduction of A, but not for all such primes. If A is the jacobian of a curve of genus 2 defined over ℚ, then this proposition is easy to use, since, as we have remarked, a_{p}, b_{p} are easy to compute.

2.5 L-Series and Conductors

In this section, we discuss the L-series and the conductor of an abelian variety defined over ℚ.

Let A be an abelian variety defined over ℚ, and let p be a prime of good reduction of A. The L-series of A at p, L_{p}(A, s), is defined to be

L_{p}(A, s): = f(p^{−s})^{−1},  (2.16)

where f is the characteristic polynomial of the Frobenius endomorphism of the reduction of A at p, discussed in the previous section. L-series are very useful, due to the following deep theorem based on a theorem by G. Faltings:

Theorem 2.5.1 Let A_{1}, A_{2} be abelian varieties defined over ℚ. Then, the following statements are equivalent:

1. A_{1} and A_{2} are ℚ-isogenous.

2. If l is a prime, then T_{l}(A_{1}) ⊗_{ℤ_{l}} ℚ_{l} and T_{l}(A_{2}) ⊗_{ℤ_{l}} ℚ_{l} are isomorphic as G_{ℚ}-modules.

3. L_{p}(A_{1}, s) = L_{p}(A_{2}, s) for all but finitely many primes p.

4. L_{p}(A_{1}, s) = L_{p}(A_{2}, s) for all primes p.

Proof: See [Fal], Section 5 Korollar 2.□

This theorem can be used to easily prove that the jacobians of two given curves defined over ℚ are not ℚ-isogenous, using (2.12), (2.13), (2.14), (2.16). However, it cannot be used to prove that the jacobians of two given curves defined over ℚ are ℚ-isogenous, although it can provide empirical evidence that they are, as we will demonstrate in this thesis.

We now move on to the conductor of an abelian variety defined over ℚ. Let p be a prime, K be a finite extension of ℚ_{p}, \overline{K} be an algebraic closure of K, A be an abelian variety defined over K, l be a prime other than p, L be the smallest field containing K such that the points in A[l] are defined over L. Define I(L/K) to be the inertia group associated to L/K, and G_{i}(L/K) to be the i^{th} inertia group associated to L/K (in particular, G_{0}(L/K) = I(L/K)).

Define

ε(A/K): = dim_{ℚ_{l}} (\frac{T_{l}(A) ⊗_{ℤ_{l}} ℚ_{l}}{H^{0}(I(\overline{K}/K), T_{l}(A) ⊗_{ℤ_{l}} ℚ_{l})}) ,

δ(A/K): = ∑_{i>0} \frac{|G_{i}(L/K)|}{|I(L/K)|} dim_{𝔽_{l}} (\frac{A[l]}{H^{0}(G_{i}(L/K), A[l])}) .  (2.17)

It is well-known that ε(A/K), δ(A/K) are independent of l, and that, if l ≠ 2, then

ε(A/K) = dim_{𝔽_{l}} (\frac{A[l]}{H^{0}(I(L/K), A[l])}) .  (2.18)

Let A be an abelian variety defined over ℚ. The conductor of A is defined to be the product

∏_{p} p^{ε(A/ℚ_{p})+δ(A/ℚ_{p})}

(taken over all primes p); the product

∏_{p} p^{ε(A/ℚ_{p})}

is called the tame part of the conductor, and the product

∏_{p} p^{δ(A/ℚ_{p})}

is called the wild part of the conductor. See [L-R-S] for more details.

The exponent of the conductor of A at a prime p measures how bad the reduction of A is at p, the larger the exponent the worse the reduction; in particular, p divides the conductor of A if and only if A has bad reduction at p. Computing the conductor of an abelian variety defined over ℚ is no trivial matter in general; it depends on the reduction of the minimal complete Néron model, a geometrical object which will not be discussed in this thesis, of the variety at each prime p. J. Tate has devised an algorithm which gives the conductor of an elliptic curve defined over ℚ (see [Sil2]). In [N-U], Y. Namikawa and K. Ueno have classified the possible types of reductions for a jacobian of a curve of genus 2 defined over ℚ, listing over 120 types! Q. Liu has devised an algorithm which gives the odd part of the conductor of a jacobian of a curve of genus 2 defined over ℚ, which will be used in this thesis.

If A and B are abelian varieties which are ℚ-isogenous, an isogeny from A to B being ϕ, then they have the same conductor. Indeed, letting p be a prime and l be be a prime other than p and not dividing the degree of ϕ, we see that A[l] = B[l], and so that δ(A/ℚ_{p}) = δ(B/ℚ_{p}), ε(A/ℚ_{p}) = ε(B/ℚ_{p}) from (2.18), (2.17) respectively.

Finally in this section, we have the following lemma, which will be used in Chapter 6 when conjecturing conductors of jacobians of curves of genus 2 defined over ℚ with \sqrt{2} ℚ-multiplication:

Lemma 2.5.2 Suppose that Δ is an integer congruent to 1 modulo 4. Then, the same power of 2 exactly divides the conductors of the jacobians of the curves

Y^{2} = F (X), Y^{2} = ΔF (X), F (X) ∈ ℚ[X], ∂F ≥ 3.

Proof: Directly from the definitions (2.17) and (2.18). Since ∂F ≥ 3, the genus of both of the curves is ≥ 1, so that their jacobians are definable. □

2.6 The Richelot Isogeny

In this section, we look at the Richelot isogeny on the jacobian of a curve of genus 2, which is of great importance in Theorem 3.2.1. The Richelot isogeny, originally discovered by F. J. Richelot (1808-1875) and used to calculate elliptic integrals with great rapidity and precision, has been used to study the Mordell-Weil group and height functions of the jacobians of certain curves of genus 2, but we will not go into any detail about these applications here. For more details, see [C-F], Chapters 9 to 12. Throughout this section, the characteristic of k will not be 2.

First, we will introduce obvious but useful conventions concerning discriminants and zeros of polynomials, which will be applied in this section and throughout this thesis. Let F be a non-zero polynomial defined over k, and let n be an integer greater than or equal to the degree of F, chosen as convenient (usually, n will be the degree of F); we will say that F is regarded as an n-tic. The discriminant of F will simply be the discriminant calculated as an n-tic (the coefficients of X^{∂F +1}, …, X^{n} being zero). We will regard x ∈ k, ∞ as a zero of F (X) with multiplicity m if and only if (x : 1), (1 : 0) respectively is a zero of Y^{n}F (X/Y) with multiplicity m. Applying these conventions to hyperelliptic curves, we can say, letting g be a non-negative integer, that the zeros of a polynomial F regarded as a 2g + 2-tic whose discriminant is non-zero are the x-coordinates of the Weierstrass points on the hyperelliptic curve Y^{2} = F (X) of genus g.

Let C be a curve defined over k of the form

C: Y^{2} = Δ ∏_{i=0}^{2} G_{i}(X), (G_{i}: = ∑_{j=0}^{2} g_{ij}X^{j}, Δ ∈ k^{∗}, g_{ij} ∈ \overline{k}) ,  (2.19)

For the majority of this section, we will look at the case when det(g_{uv}) ≠ 0. Suppose that this is so, and let C_{G_{0},G_{1},G_{2}} be the curve

C_{G_{0},G_{1},G_{2}}: det(g_{uv})T^{2} = Δ ∏_{i=0}^{2} H_{i}(Z),  (2.20)

where, for all i ∈ {0, 1, 2} (regarding, as we will throughout this thesis, addition on this set to be addition modulo 3),

H_{i}(Z) = G′_{i+1}(Z)G_{i+2}(Z) − G′_{i+2}(Z)G_{i+1}(Z)  (2.21)

(here, ′ denotes differentiation); we need det(g_{uv}) ≠ 0 for this definition to make sense. C_{G_{0},G_{1},G_{2}} is called the Richelot dual w.r.t. the G_{i}’s of C; this makes sense since the Richelot dual curve w.r.t. the H_{i}’s of C_{G_{0},G_{1},G_{2}} is

Y^{2} = 4Δ ∏_{i=0}^{2} G_{i}(X);

indeed, det(h_{uv}) = −2 det(g_{uv})^{2}, and

H′_{i+1}(X)H_{i+2}(X) − H′_{i+2}(X)H_{i+1}(X) = −2 det(g_{uv})G_{i}(X).

So the Richelot dual w.r.t. the H_{i}’s of the Richelot dual w.r.t. the G_{i}’s of C is k-isomorphic to C, the isomorphism from the former curve to the latter curve being (x, y) ↦ (x, \frac{y}{2}). Usually, when the G_{i}’s are known, we will use the more succinct terminology ‘the Richelot dual’. Note that if we replace one of the G_{i}’s by nG_{i} and Δ by \frac{Δ}{n} (n ∈ k^{∗}), leaving C unaffected, then the its Richelot dual is also unaffected. When looking at curves C of genus 2 defined over a subfield K of ℂ such that the jacobian of C has a Θ-symmetric \sqrt{2} K-multiplication, where Θ is a theta divisor on the jacobian, Δ will be a non-zero rational number and the G_{i}’s will be defined over K as a set (not necessarily individually).

We now define explicitly the Richelot isogeny and the Richelot dual isogeny between the jacobians of C and its Richelot dual as two-to-two correspondences on the curves. Indeed, there exist \overline{k}-isogenies

ρ_{G_{0},G_{1},G_{2}} : Jac(C) → Jac(C_{G_{0},G_{1},G_{2}}),
ρ_{G_{0},G_{1},G_{2}}^{∨} : Jac(C_{G_{0},G_{1},G_{2}}) → Jac(C),

called the Richelot isogeny w.r.t. the G_{i}’s and the Richelot dual isogeny w.r.t. the G_{i}’s respectively (the terminology is justified, since ρ_{G_{0},G_{1},G_{2}}^{∨}ρ_{G_{0},G_{1},G_{2}} is the multiplication by 2 endomorphism on the jacobian of C, as we will see) given explicitly as follows:

ρ_{G_{0},G_{1},G_{2}}([(x, y) − (x′, y′)])  (2.22)
: = [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})],
ρ_{G_{0},G_{1},G_{2}}^{∨}([(z, t) − (z′, t′)])  (2.23)
: = [(x_{1}, y_{1}) + (x_{2}, y_{2}) − (x′_{1}, y′_{1}) − (x′_{2}, y′_{2})],

where z_{1}, z_{2} (resp. x_{1}, x_{2}) are the zeros w.r.t. z (resp. x) of

G_{1}(x)H_{1}(z) + G_{2}(x)H_{2}(z) = 0  (2.24)

(this polynomial is regarded as a quadratic for both isogenies), and t_{j} (resp. y_{j}) satisfies

yt_{j} = ΔG_{1}(x)H_{1}(z_{j})(x − z_{j}),  (2.25)
ty_{j} = ΔG_{1}(x_{j})H_{1}(z)(x_{j} − z),  (2.26)

respectively, z′_{j}, t′_{j} (resp. x′_{j}, y′_{j}) having analogous definitions. Usually, when the G_{i}’s are known, we will use the more succinct terminology ‘the Richelot isogeny’ and ‘the Richelot dual isogeny’. Note that if we replace one of the G_{i}’s by nG_{i} and Δ by \frac{Δ}{n}, n ∈ k^{∗}, leaving C and its Richelot dual unaffected, then the Richelot isogeny and the Richelot dual isogeny are also unaffected, as is reasonable to expect.

To ensure that our definitions (2.22) and (2.23) for the Richelot isogeny and the Richelot dual isogeny make sense (we shall see later that they do indeed give \overline{k}-isogenies), we need the following lemma:

Lemma 2.6.1 (z_{j}, t_{j}), (z′_{j}, t′_{j}) are both points on the Richelot dual of C and (x_{j}, y_{j}), (x′_{j}, y′_{j}) are both points on C.

Proof: We will just show that (z_{j}, t_{j}), (z′_{j}, t′_{j}) are both points on the Richelot dual of C; then, a similar argument will establish that (x_{j}, y_{j}), (x′_{j}, y′_{j}) are both points on C.

After some computation, we see that

G_{0}(x)H_{0}(z_{j}) + (x − z_{j})^{2} det(g_{uv}) = 0;  (2.27)

then, taking (2.19), (2.24), (2.25) and (2.27) into account, we see that

Δ (∏_{i=0}^{2} G_{i}(x)) t_{j}^{2} = (ΔG_{1}(x)H_{1}(z_{j})(x − z_{j}))^{2},

= −Δ^{2} (∏_{i=1}^{2} G_{i}(x)H_{i}(z_{i})) (x − z_{j})^{2}

= \frac{1}{det(g_{uv})}Δ^{2} ∏_{i=0}^{2} G_{i}(x)H_{i}(z_{j}),

det(g_{uv})t_{j}^{2} = Δ ∏_{i=0}^{2} H_{i}(z_{j}).

i.e. (z_{j}, t_{j}) is a point on the Richelot dual of C. Similarly, (z′_{j}, t′_{j}) is a point on the Richelot dual of C, establishing the lemma.□

We will now show that ρ_{G_{0},G_{1},G_{2}}^{∨}ρ_{G_{0},G_{1},G_{2}} is the multiplication by 2 endomorphism on the jacobian on C, therefore showing that our definitions (2.22) and (2.23) for the Richelot isogeny and the Richelot dual isogeny do indeed give \overline{k}- isogenies, and justifying our terminology ‘Richelot isogeny’ and ‘Richelot dual isogeny’, in the following lemma:

Lemma 2.6.2 ρ_{G_{0},G_{1},G_{2}}^{∨}ρ_{G_{0},G_{1},G_{2}} is the multiplication by 2 endomorphism on the jacobian on C.

Proof: To start with, we define

M_{11}: = g_{10}g_{22} − g_{12}g_{20}, M_{12}: = g_{10}g_{21} − g_{11}g_{20}, M_{21}: = g_{12}g_{21} − g_{11}g_{22},
M_{22}: = −M_{11},

and M to be the Möbius \overline{k}-transformation as in Section 2.3; the application of this Möbius \overline{k}-transformation will soon become apparent. det(M) is non-zero, since, after some computation, we see that det(M) divides the discriminant of Δ ∏_{i=0}^{2} G_{i}(x), computed as a sextic, which is non-zero.

Let z_{1}, z_{2}, z′_{1}, z′_{2}, t_{1}, t_{2}, t′_{1}, t′_{2} be as defined in (2.24) and (2.25), so that (2.22) holds. After some computation, using (2.24), we see that

ρ_{G_{0},G_{1},G_{2}}^{∨}([(z_{1}, t_{1}) − (z′_{1}, t′_{1})])  (2.28)
= [(u_{1}, v_{1}) + (u_{3}, v_{3}) − (u′_{1}, v′_{1}) − (u′_{3}, v′_{3})],
ρ_{G_{0},G_{1},G_{2}}^{∨}([(z_{2}, t_{2}) − (z′_{2}, t′_{2})])  (2.29)
= [(u_{2}, v_{2}) + (u_{4}, v_{4}) − (u′_{2}, v′_{2}) − (u′_{4}, v′_{4})],

where u_{1} = u_{2} = x, u′_{1} = u′_{2} = x′ and u_{3} = u_{4} = x|_{M}, u′_{3} = u′_{4} = x′|_{M}, which implies that

ρ_{G_{0},G_{1},G_{2}}^{∨}ρ_{G_{0},G_{1},G_{2}}([(x, y) − (x′, y′)])
= [(u_{1}, v_{1}) + (u_{2}, v_{2}) + (u_{3}, v_{3}) + (u_{4}, v_{4}) − (u′_{1}, v′_{1}), (u′_{2}, v′_{2}), (u′_{3}, v′_{3}), (u′_{4}, v′_{4})].

To finish the proof, we just need to show that v_{1} = v_{2} = y, v′_{1} = v′_{2} = y′, v_{3} = −v_{4}, v′_{3} = −v′_{4}.

Consider v_{1}, v_{2}, v′_{1}, v′_{2}; we require v_{1} = v_{2} = y, v′_{1} = v′_{2} = y′. We will concentrate on showing that v_{1} = y; the other three equalities will then follow from similar arguments. From (2.25), (2.26), (2.28) and (2.29), we have

yt_{1} = ΔG_{1}(x)H_{1}(z_{1})(x − z_{1}), t_{1}v_{1} = ΔG_{1}(u_{1})H_{1}(z_{1})(u_{1} − z_{1});

so, since u_{1} = x, we have v_{1} = y, as required.

Consider v_{3}, v_{4}, v′_{3}, v′_{4}; we require v_{3} = −v_{4}, v′_{3} = −v′_{4}. We will concentrate on showing that v_{3} = −v_{4}; the other equality will then follow from a similar argument. From (2.25), (2.26), (2.28) and (2.29), we have

yt_{1} = ΔG_{1}(x)H_{1}(z_{1})(x − z_{1}), t_{1}v_{3} = ΔG_{1}(u_{3})H_{1}(z_{1})(u_{3} − z_{1}),
yt_{2} = ΔG_{1}(x)H_{1}(z_{2})(x − z_{2}), t_{2}v_{4} = ΔG_{1}(u_{4})H_{1}(z_{2})(u_{4} − z_{2}),

and, after some computation, we have

(x − z_{1})(x|_{M} − z_{2}) + (x − z_{2})(x|_{M} − z_{1}) = 0;

so, since u_{3} = u_{4} = x|_{M}, we have v_{3} = −v_{4}, as required.□

For use throughout this section, we will define P_{ij}, Q_{ij}, i ∈ {0, 1, 2}, j ∈ {0, 1} to be labellings of the Weierstrass points on C, its Richelot dual respectively such that the x-coordinates, z-coordinates of the Weierstrass points P_{ij}, Q_{ij} are the zeros of G_{i}, H_{i} respectively (throughout this thesis, G_{i}, H_{i} will be regarded as quadratics). Permuting the G_{i}’s affects z_{j}, t_{j}, z′_{j}, t′_{j}, x_{j}, y_{j}, x′_{j}, y′_{j} in general, but, as is reasonable to expect and desirable, does not affect either the Richelot isogeny or the Richelot dual isogeny, as we will show in the following lemma (the proof of which will use the Q_{ij}’s).

Lemma 2.6.3 Permuting the G_{i}’s does not affect either the Richelot isogeny or the Richelot dual isogeny.

Proof: We will just show that permuting the G_{i}’s does not affect the Richelot isogeny; then, a similar argument will establish that permuting the G_{i}’s does not affect the Richelot dual isogeny.

It suffices to show that swapping G_{1}, G_{2} and swapping G_{0}, G_{2} do not affect the Richelot isogeny; then, since (12) and (02) generate the symmetric group S_{3}, permuting the G_{i}’s will not affect it.

It is easy to show, by considering the definition (2.22) of the Richelot isogeny, that swapping G_{1}, G_{2} does not affect it. It is a bit harder to show that swapping G_{0}, G_{2} does not affect it, since we need to consider divisors of functions on the Richelot dual of C. We will concentrate on the latter.

Define

G̃_{0}: = G_{2}, G̃_{1}: = G_{1}, G̃_{2}: = G_{0},  (2.30)
⇒ H̃_{0} = −H_{2}, H̃_{1} = −H_{1}, H̃_{2} = −H_{0}, det(g̃_{uv}) = − det(g_{uv}),

so that the G̃_{i}’s are the G_{i}’s with G_{0}, G_{2} swapped; then, C_{G̃_{0},G̃_{1},G̃_{2}} = C_{G_{0},G_{1},G_{2}}.

To show that swapping G_{0}, G_{2} does not affect the Richelot isogeny, it suffices to show that ρ_{G̃_{0},G̃_{1},G̃_{2}}([(x, y)−P_{10}]) = ρ_{G_{0},G_{1},G_{2}}([(x, y)−P_{10}]) for all (x, y) ∈ C, since the divisor classes [(x, y) − P_{10}], (x, y) ∈ C generate the points on the jacobian of C. So it suffices to show, for all (x, y) ∈ C, that

(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z̃_{1}, t̃_{1}) − (z̃_{2}, t̃_{2}) + Q_{00} + Q_{01} − Q_{20} − Q_{21}

is the divisor of a function on the Richelot dual of C, where z_{i}, t_{i} are as in the definition of the Richelot isogeny, i.e.

G_{1}(x)H_{1}(z_{i}) + G_{2}(x)H_{2}(z_{i}) = 0, yt_{i} = ΔG_{1}(x)H_{1}(z_{i})(x − z_{i}),  (2.31)

and z̃_{i}, t̃_{i} are as in the definition of ρ_{G̃_{0},G̃_{1},G̃_{2}}, i.e.

G̃_{1}(x)H̃_{1}(z̃_{i}) + G̃_{2}(x)H̃_{2}(z̃_{i}) = 0, yt̃_{i} = ΔG̃_{1}(x)H̃_{1}(z̃_{i})(x − z̃_{i}),

i.e.

G_{1}(x)H_{1}(z̃_{i}) + G_{0}(x)H_{0}(z̃_{i}) = 0, yt̃_{i} = ΔG_{1}(x)H_{1}(z̃_{i})(z̃_{i} − x).  (2.32)

Since the divisor Q_{00} + Q_{01} − Q_{20} − Q_{21} is linearly equivalent to the divisor Q_{10} − Q_{11}, it suffices to show, for all (x, y) ∈ C, that the divisor

(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z̃_{1}, t̃_{1}) − (z̃_{2}, t̃_{2}) + Q_{10} − Q_{11}

is the divisor of a function on the Richelot dual of C. In fact, it is the divisor of the function

\frac{yT + ΔG_{1}(x)(h_{12}Z^{3} + (h_{11} − h_{12}x)Z^{2} + (h_{10} − h_{11}x)Z − h_{10}x)}{(Z − z̃_{1})(Z − z̃_{2})(Z − z_{11})}  (2.33)

where the h_{1j} satisfy H_{1}(Z): = ∑_{j=0}^{2} h_{1j}Z^{j}, i.e.

h_{10} = g_{00}g_{21} − g_{01}g_{20}, h_{11} = 2(g_{00}g_{22} − g_{02}g_{20}), h_{12} = g_{01}g_{22} − g_{02}g_{21};

indeed, after some computation and using (2.30), we see that

\frac{Δ ∏_{i=0}^{2} H_{i}(Z)}{det(g_{uv})}  (2.34)
− (\frac{ΔG_{1}(x)(h_{12}Z^{3} + (h_{11} − h_{12}x)Z^{2} + (h_{10} − h_{11}x)Z − h_{10}x)}{−y})^{2}
= \frac{Δ^{2}G_{1}(x)(G_{1}(x)H_{1}(Z) + G_{2}(x)H_{2}(Z))(G_{1}(x)H_{1}(Z) + G_{0}(x)H_{0}(Z))H_{1}(Z)}{det(g_{uv})y^{2}}
= \frac{Δ^{2}G_{1}(x)(G_{1}(x)H_{1}(Z) + G_{2}(x)H_{2}(Z))(G̃_{1}(x)H̃_{1}(Z) + G̃_{2}(x)H̃_{2}(Z))H_{1}(Z)}{− det(g_{uv})y^{2}},

and the zeros of (2.34) in X, regarded as a sextic, are z_{1}, z_{2}, z̃_{1}, z̃_{2}, z_{10}, z_{11}, where z_{10}, z_{11} are the z-coordinates of the points Q_{10}, Q_{11} respectively (from the definition of H_{1} and (2.31), (2.32)), and the values of T killing (2.33) for Z = z_{1}, z_{2}, z̃_{1}, z̃_{2}, z_{10}, z_{11} are t_{1}, t_{2}, −t̃_{1}, −t̃_{2}, t_{10}, t_{11} respectively, where t_{1j} satisfies Q_{1j} = (z_{1j}, t_{1j}) (either equal to 0 or to ∞).□

We have the following simple corollary, which is important for our thesis:

Corollary 2.6.4 Suppose that the G_{i}’s are defined over k as a set (not necessarily individually). Then, the Richelot isogeny and the Richelot dual isogeny are both k-isogenies.

Proof: σ ∈ G_{k} permutes the G_{i}’s, and, by Lemma 2.6.3, permuting the G_{i}’s does not affect either the Richelot isogeny or the Richelot dual isogeny.□

The importance of this corollary is due to the fact that the hypotheses will turn out to be satisfied when C is defined over a subfield K of ℂ and the jacobian of C has a Θ-symmetric \sqrt{2} K-multiplication ε, where Θ is a theta divisor on the jacobian (with k = K).

We will now use Lemma 2.6.2 and Lemma 2.6.3 to study the actions of the Richelot isogeny and the Richelot dual isogeny on the 2-torsion of the jacobians of C and its Richelot dual; we will consider these actions when looking at curves C of genus 2 defined over a subfield K of ℂ such that the jacobian of C has a Θ-symmetric \sqrt{2} K-multiplication ε, where Θ is a theta divisor on the jacobian. To start with, we have

ker ρ_{G_{0},G_{1},G_{2}} = {0, [P_{00} − P_{01}], [P_{10} − P_{11}], [P_{20} − P_{21}]},  (2.35)
ker ρ_{G_{0},G_{1},G_{2}}^{∨} = {0, [Q_{00} − Q_{01}], [Q_{10} − Q_{11}], [Q_{20} − Q_{21}]};  (2.36)

indeed, in both cases, the LHS contains the RHS from the definition of the isogeny, and equality is implied by Lemma 2.6.2.

From the definitions (2.22) and (2.23) for the Richelot isogeny and the Richelot dual isogeny respectively, using Lemma 2.6.3 to simplify the computations, we see that

ρ_{G_{0},G_{1},G_{2}}([P_{00} − P_{10}]) = [Q_{20} − Q_{21}],  (2.37)
ρ_{G_{0},G_{1},G_{2}}([P_{00} − P_{20}]) = [Q_{10} − Q_{11}],
ρ_{G_{0},G_{1},G_{2}}^{∨}([Q_{00} − Q_{10}]) = [P_{20} − P_{21}],  (2.38)
ρ_{G_{0},G_{1},G_{2}}^{∨}([Q_{00} − Q_{20}]) = [P_{10} − P_{11}].

The actions of the Richelot isogeny and the Richelot dual isogeny on the 2torsion of the jacobians of C and its Richelot dual are completely determined by (2.35), (2.36), (2.37) and (2.38).

Suppose that k = ℚ, the G_{i}’s are defined over ℚ as a set (not necessarily individually), and ∏_{i=0}^{2} G_{i}(X), \frac{1}{det(g_{uv})} ∏_{i=0}^{2} H_{i}(Z) both have integer coefficients. Recall that the Richelot isogeny is defined over ℚ. It is of interest to compare the set of primes dividing the discriminant of ∏_{i=0}^{2} G_{i}(X) to the set of primes dividing the discriminant of \frac{1}{det(g_{uv})} ∏_{i=0}^{2} H_{i}(Z), regarding the polynomials as sextics; we expect the two sets of primes to be similar, as we now explain.

Two abelian varieties defined over ℚ which are ℚ-isogenous have the same conductor; in particular, the jacobians of C and its Richelot dual have the same conductor, since the Richelot isogeny is defined over ℚ.

Now the primes which divide the discriminant of ∏_{i=0}^{2} G_{i}(X) (regarded as a sextic) roughly correspond to the primes of bad reduction of C; more precisely, if p is an odd prime, then p divides the discriminant if and only if p is a prime of bad reduction, but 2 can divide the discriminant and be a prime of good reduction (see Section 2.3). Also the primes of bad reduction of C roughly correspond to the primes of bad reduction of the jacobian of C, which are the primes dividing the conductor of it; more precisely, the former set contains the latter set, and in fact is often equal to it. Similar observations can be made for \frac{1}{det(g_{uv})} ∏_{i=0}^{2} H_{i}(Z) and the Richelot dual of C.

For these reasons, we expect the two sets of primes as defined above to be similar.

Consider the discriminants of ∏_{i=0}^{2} G_{i}(X), \frac{1}{det(g_{uv})} ∏_{i=0}^{2} H_{i}(Z), supposing that the G_{i}’s are defined by

G_{i} = X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R, i ∈ {0, 1, 2},

∏_{i=0}^{2}(X − α_{i}) = X^{3} + BX + C,

P, Q, R, B, C ∈ ℤ.

After some computation, we see that the discriminant of ∏_{i=0}^{2} G_{i}(X) is

−(27C^{2} + 4B^{3})^{2}Φ_{1}Φ_{2}^{2},

and that the discriminant of \frac{1}{det(g_{uv})} ∏_{i=0}^{2} H_{i}(Z) is

64P^{2}(27C^{2} + 4B^{3})^{2}Φ_{2}Φ_{1}^{2},

where

Φ_{1} = C^{2}(4P − 1)^{3} − 4CQ(B(4P − 1)^{2} + 16Q^{2} − 12R(4P − 1))
+ 4R(B^{2}(4P − 1)^{2} + 8B(2Q^{2} − R(4P − 1)) + 16R^{2}),
Φ_{2} = C^{2}P^{3}(P − 1)^{3} + 2CQP^{2}(BP (P − 1)(P + 2) + Q^{2}(P + 3) − 3R(P − 1))
− (BP − R − Q^{2})(B^{2}P^{4} + 2BP^{2}(Q^{2} − R) + (Q^{2} + R)^{2}).

From these expressions, we see that the two sets of primes as defined above are similar when the G_{i}’s are defined as above.

We now state and prove a lemma concerning fundamental relationships between the Richelot duals of isomorphic curves of genus 2 which will be used in the proof of Proposition 3.5.3 and the proof of Proposition 3.5.5, a corollary which will be used in the proof of Proposition 3.5.2, and a corollary which will be used in the proof of Proposition 3.5.3. We will prove the lemma using fairly general models for the curves, since some of the equations derived in the proof will be used individually in the proof of Proposition 3.5.4. The lemma will allow us to specialise the models for the curves used, and so simplify the computations involved, in the proofs of the corollaries.

Lemma 2.6.5 Let C, C̃ be two curves defined over an algebraically closed field k of characteristic not 2, given by

C: Y^{2} = Δ ∏_{i=0}^{2} G_{i}(X), (G_{i}: = ∑_{j=0}^{2} g_{ij}X^{j}, Δ ∈ k^{∗}, g_{ij} ∈ k) ,

C̃: Y^{2} = Δ̃ ∏_{i=0}^{2} G̃_{i}(X), (G̃_{i}: = ∑_{j=0}^{2} g̃_{ij}X^{j}, Δ̃ ∈ k^{∗}, g̃_{ij} ∈ k) ,

with the properties that det(g_{uv}) ≠ 0 ≠ det(g̃_{uv}). Suppose that there is a kisomorphism ι_{C} from C to C̃, given by (x, y) ↦ (x, y)|_{M,N}, where M is a Möbius k-transformation which sends the zeros of G_{i} to the zeros of G̃_{i}, and N ∈ k^{∗}.

Then, there is a k-isomorphism ι′_{C} from the Richelot dual of C to the Richelot dual of C̃, given by (z, t) ↦ (z, t)|_{M,N}, and M sends the zeros of H_{i} to the zeros of H̃_{i}. Moreover, defining ι_{J} to be the k-isomorphism from the jacobian of C to that of C̃ given by

[(x, y) − (x′, y′)] ↦ [ι_{C}((x, y)) − ι_{C}((x′, y′))], (x, y), (x′, y′) ∈ C,

and ι′_{J} to be the k-isomorphism from the jacobian of the Richelot dual of C to that of the Richelot dual of C̃ given by

[(z, t) − (z′, t′)] ↦ [ι′_{C}((z, t)) − ι′_{C}((z′, t′))], (z, t), (z′, t′) ∈ C_{G_{0},G_{1},G_{2}},

we have

ρ_{G̃_{0},G̃_{1},G̃_{2}} = ι′_{J}ρ_{G_{0},G_{1},G_{2}}ι_{J}^{−1}.  (2.39)

Proof: Let M_{11}, M_{12}, M_{21}, M_{22} ∈ k such that

X|_{M} = \frac{M_{11}X + M_{12}}{M_{21}X + M_{22}} .

Since M sends the zeros of G_{i} to the zeros of G̃_{i},

G_{i}(X) = C_{i}(M_{21}X + M_{22})^{2}G̃_{i}(X|_{M})  (2.40)

for some constant C_{i} ∈ k^{∗}. Moreover, (x, y) ∈ C implies that (x, y)|_{M,N} ∈ C̃; from this and (2.40) with X = x, we see that

N^{2} ∏_{i=0}^{2} C_{i} = \frac{Δ̃}{Δ}.  (2.41)

After some computation, we see, from (2.40) and (2.41), that

\frac{N^{2}}{det(M)^{3}} = \frac{Δ̃ det(g̃_{uv})}{Δ det(g_{uv})},  (2.42)

det(M) being as in (2.8). Also, after some computation, we see, from (2.21) and (2.40), that

H_{i}(Z) = G′_{i+1}(Z)G_{i+2}(Z) − G′_{i+2}(Z)G_{i+1}(Z)  (2.43)
= C_{i+1}C_{i+2} det(M)(M_{21}Z + M_{22})^{2}
× (G̃′_{i+1}(Z|_{M})G̃_{i+2}(Z|_{M}) − G̃′_{i+2}(Z|_{M})G̃_{i+1}(Z|_{M}))
= C_{i+1}C_{i+2} det(M)(M_{21}Z + M_{22})^{2}H̃_{i}(Z|_{M}),

so M sends the zeros of H_{i} to the zeros of H̃_{i}. We see, from (2.41), (2.42) and (2.43), that (z, t) ∈ C_{G_{0},G_{1},G_{2}} implies that (z, t)|_{M,N} ∈ C̃_{G̃_{0},G̃_{1},G̃_{2}}.

Finally, using the definition (2.22) for the Richelot isogeny, we see that

ι_{J}^{′−1}ρ_{G̃_{0},G̃_{1},G̃_{2}}ι_{J}([(x, y) − (x′, y′)]) = [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})], where z_{1}, z_{2} are the zeros w.r.t. z of

G̃_{1}(x|_{M})H̃_{1}(z|_{M}) + G̃_{2}(x|_{M})H̃_{2}(z|_{M}) = 0

and t_{i} satisfies

y|_{M,N,x}t_{i}|_{M,N,z_{i}} = Δ̃G̃_{1}(x|_{M})H̃_{1}(z|_{M})(x|_{M} − z_{i}|_{M}),

z′_{i}, t′_{i} having analogous definitions. From (2.40), (2.41) and (2.43), again using the definition (2.22) for the Richelot isogeny, we see that (2.39) holds, as required.□

Corollary 2.6.6 Let C be a curve defined over an algebraically closed field k of characteristic not 2, given by the same equation as in Lemma 2.6.5, and suppose that there is a k-isomorphism from C to its Richelot dual, given by (x, y) ↦ (x, Ny) for some N ∈ k^{∗} such that x is a zero of G_{i} if and only if x is a zero of H_{i}. Let ι be the k-isomorphism from the jacobian of C to that of its Richelot dual given by

[(x, y) − (x′, y′)] ↦ [(x, Ny) − (x′, Ny′)], (x, y), (x′, y′) ∈ C.

Then, ι^{−1}ρ_{G_{0},G_{1},G_{2}} is a \sqrt{−2} k-multiplication.

Proof: By Lemma 2.6.5, we may assume, w.l.o.g., that C is a curve of the form

C: Y^{2} = X(X − 1)(X − λ)(X − μ)(X − ν)  (2.44)

for some λ, μ, ν ∈ k, and that

Δ = 1, G_{0}(X) = X, G_{1}(X) = (X −1)(X −λ), G_{2}(X) = (X −μ)(X −ν). (2.45)

Define H_{i}(Z) as in (2.21), so that

H_{0}(Z) = (λ + 1 − μ − ν)Z^{2} + 2(μν − λ)Z + λ(μ + ν − μν) − μν, (2.46)
H_{1}(Z) = Z^{2} − μν,  (2.47)
H_{2}(Z) = −Z^{2} + λ.  (2.48)

Since x is a zero of G_{i} if and only if x is a zero of H_{i} (by hypothesis), we have

λ = −1, μ = ±\sqrt{−1}, ν = −μ.  (2.49)

Looking at (2.19), (2.20), (2.45), (2.46) and (2.49), we see that N = ±\sqrt{−2}; w.l.o.g., we may assume that N = \sqrt{−2}. Using the definition (2.22) for the Richelot isogeny, we see that

ι^{−1}ρ_{G_{0},G_{1},G_{2}}([(x, y) − (x′, y′)]) = [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})],

where z_{1}, z_{2} are the zeros w.r.t. z of

G_{1}(x)H_{1}(z) + G_{2}(x)H_{2}(z) = 0

and t_{i} satisfies

yt_{i} = \frac{\sqrt{−2}}{2}G_{1}(x)H_{1}(z)(x − z_{i}),

z′_{i}, t′_{i} having analogous definitions. Taking (2.45), (2.46) and (2.49) into account, z_{1} = \sqrt{−1}x, z_{2} = −\sqrt{−1}x (w.l.o.g.) and yt_{i} = −\frac{\sqrt{−2}}{2}(x^{2} − 1)(z_{i}^{2} − 1)(x − z_{i}), i.e. t_{1} = y \frac{\sqrt{−2}}{2}(1 − \sqrt{−1}), t_{2} = y \frac{\sqrt{−2}}{2}(1 + \sqrt{−1}); z′_{i}, t′_{i} have analogous values. We deduce that, under ι^{−1}ρ_{G_{0},G_{1},G_{2}},

[(x, y) − (x′, y′)]
↦ [(\sqrt{−1}x, y \frac{\sqrt{−2}}{2}(1 − \sqrt{−1})) + (−\sqrt{−1}x, y \frac{\sqrt{−2}}{2}(1 − \sqrt{−1}))
− (\sqrt{−1}x′, y′ \frac{\sqrt{−2}}{2}(1 − \sqrt{−1})) − (−\sqrt{−1}x′, y′ \frac{\sqrt{−2}}{2}(1 − \sqrt{−1}))]
↦ [(−x, \sqrt{−1}y) + (x, −y) + (−x, −\sqrt{−1}y) + (x, −y)
− (−x′, \sqrt{−1}y′) − (x′, −y′) − (−x′, −\sqrt{−1}y′) − (x′, −y′)]
= −2[(x, y) − (x′, y′)],

so that ι^{−1}ρ_{G_{0},G_{1},G_{2}} is a \sqrt{−2} k-multiplication, as required.□

From the proof of Corollary 2.6.6, we see that the curves with the properties described in the corollary are very special in the sense that they are mutually k-isomorphic and that their jacobians have k-multiplication by an eighth root of unity (if k is a subfield of ℂ, their jacobians are k-isogenous to the square of an elliptic curve defined over k whose k-endomorphism ring is isomorphic to ℤ[\sqrt{−2}]).

Corollary 2.6.7 Let C be a curve defined over an algebraically closed field k of characteristic not 2, given by the same equation as in Lemma 2.6.5, and suppose that there is a k-isomorphism from C to its Richelot dual, given by (x, y) ↦ (x, y)|_{M,N}, where M is a Möbius k-transformation which sends the zeros of G_{i} to the zeros of H_{i}, and N ∈ k^{∗}.

Then, M is self-inverse.

Proof: By Lemma 2.6.5, there is a k-automorphism of C which is given by (x, y) ↦ (x, y)|_{M^{2},N^{2}/2}, and M^{2} sends the zeros of G_{i} to the zeros of G_{i} for all i. If M^{2} is the identity Möbius k-transformation, then we are done; otherwise, permuting the G_{i}’s if necessary, we have two cases (bearing in mind that if a Möbius k-transformation fixes four points, then it is the identity):

1. M^{2} sends any zero of G_{i} to the other zero of G_{i} for all i.

2. M^{2} sends any zero of G_{0} to itself, any zero of G_{1} to the other zero of G_{1}, and any zero of G_{2} to the other zero of G_{2}.

If the first case holds, then it is not too hard to see that det(g_{uv}) is zero, which is a contradiction. So consider the second case; we will also derive a contradiction, by showing that M^{2} is the identity Möbius k-transformation, and then the corollary will be established.

By Lemma 2.6.5, we may assume, w.l.o.g., that C is a curve of the form (2.44) for some λ, μ, ν ∈ k, and that (2.45) holds.

Firstly, we will use the hypothesis for the second case to show that λ = −1 and ν = −μ. By this hypothesis and (2.45), M^{2} sends 0 to 0, ∞ to ∞, swaps 1, λ, and swaps μ, ν. So λ = −1 and ν = −μ.

Define H_{i}(Z) as in (2.21), so that

H_{0}(Z) = −2(μ − 1)(μ + 1)Z, H_{1}(Z) = Z^{2} + μ^{2}, H_{2}(Z) = −Z^{2} − 1.

Now M sends the zeros of G_{0} to the zeros of H_{0}, i.e. 0, ∞ to 0, ∞ (not necessarily in that order), so X|_{M} = κX or X|_{M} = \frac{κ}{X} for some κ ∈ k^{∗}. If M is of the latter form, then M^{2} is the identity Möbius k-transformation, and so we are done. It remains to deal with the possibility that M is of the former form, i.e. that X|_{M} = κX. Now M sends the zeros of G_{1} to the zeros of H_{1}, i.e. 1, −1 to \sqrt{−1}μ, −\sqrt{−1}μ (not necessarily in that order), so κ = ±\sqrt{−1}μ. Also M sends the zeros of G_{2} to the zeros of H_{2}, i.e. μ, −μ to \sqrt{−1}, −\sqrt{−1} (not necessarily in that order), so μ^{4} = 1, and so μ = ±\sqrt{−1} to ensure that μ is not equal to 1 or −1 (a requirement for the definition of C). But this implies that κ = ±1, i.e. that X|_{M} = ±X, which implies that M^{2} is the identity Möbius k-transformation, and so we are done.□

This section would not be complete without a discussion of what happens when det(g_{uv}) = 0, particularly as we will need to consider this case when looking at curves of genus 2 whose jacobians have \sqrt{2} multiplication.

Proposition 2.6.8 Let C be a curve of the form (2.19), but now with the property that det(g_{uv}) = 0. Then, there is an abelian surface A defined over k with the following properties:

• There is a k-isogeny ρ_{G_{0},G_{1},G_{2}}, called the Richelot isogeny, from the jacobian of Cto A satisfying (2.35).

• A is the direct sum of two abelian subvarieties E_{1}, E_{2} of A, which are elliptic curves and defined over k as a pair (not necessarily individually), so that E_{1} ∪ E_{2} is a divisor on A defined over k and ϕ_{E_{1}∪E_{2}} is a principal k-polarisation on A, called the canonical polarisation.

Proof: The first property comes from a well-known result for abelian varieties in general (see [Mum1], p. 66).

Since det(g_{uv}) = 0 and the G_{i}’s are defined over k as a set, there is a selfinverse Möbius k-transformation M which sends any zero of G_{i} to the other zero of G_{i}. So there exists N ∈ \overline{k}^{∗}, lying in a quadratic extension k_{0} of k, such that (x, y) ∈ C implies that (x, y)|_{M,N} ∈ C. So there is a k_{0}-automorphism ι of of the jacobian given by

[(x, y) − (x′, y′)] ↦ [(x, y)|_{M,N} − (x′, y′)|_{M,N}], (x, y), (x′, y′) ∈ C.

We will now show that im ([1] + ι), im ([1] − ι) are elliptic curves (call them E_{+}, E_{−} respectively) which are defined over k_{0} individually and over k as a pair; it suffices to show that the k_{0}-endomorphisms [1] ± ι are neither zero nor surjective. They are not zero, since [1] ≠ ±ι, and they are not surjective, since either both or none are surjective, being defined over k as a pair, and ([1] + ι)([1] − ι) = [0].

We will now derive a k_{0}-isomorphism ψ from E_{+} × E_{−} to A such that ψϕ = ρ_{G_{0},G_{1},G_{2}}, where ϕ is the k_{0}-isogeny from the jacobian to E_{+} × E_{−} given by

[(x, y) − (x′, y′)] ↦ (([1] + ι)([(x, y) − (x′, y′)]), ([1] − ι)([(x, y) − (x′, y′)]));

it suffices to show that ker ϕ = ker ρ_{G_{0},G_{1},G_{2}}. Now ker ϕ consists precisely of the points on the jacobian of C which are sent by [1], ι, −ι to the same point. Since ι is an automorphism, the order of a point in ker ϕ divides 2. Then, we see that ker ϕ = ker ρ_{G_{0},G_{1},G_{2}}, since M sends any zero of G_{i} to the other zero of G_{i}.

Let E_{1}, E_{2} be the images under ρ_{G_{0},G_{1},G_{2}} of E_{+}, E_{−} respectively, which are also the images under ψ of ϕ(E_{+}), ϕ(E_{−}) respectively; then, E_{1}, E_{2} are subvarieties of A which are elliptic curves, and A is the direct sum of E_{1}, E_{2}, since the analogous statements with E_{1}, E_{2}, A replaced by ϕ(E_{+}), ϕ(E_{−}), E_{+} × E_{−} respectively are true. Moreover, E_{1}, E_{2} are defined over k as a pair, since E_{+}, E_{−} are defined over k as a pair (as established earlier) and ρ_{G_{0},G_{1},G_{2}} is defined over k. So the proposition is established.□

The nature of the abelian surface which is the image under the Richelot isogeny corresponding to a curve of the form (2.19) does not depend on whether det(g_{uv}) is zero or non-zero; if the curve is altered in a continuous manner, the abelian surface is altered in a continuous manner too, whether det(g_{uv}) is zero at any stage or not. However, there is a curve of genus 2 or a union of two elliptic curves on the abelian surface according to whether det(g_{uv}) is zero or non-zero; this fact is important for Chapter 3.

2.7 Abelian Varieties (Complex-Analytic)

In this section, we define the basic notions of abelian varieties and polarised abelian varieties, which are of fundamental importance in this thesis, from a complex-analytic viewpoint. Further details can be found in [L-B].

First, we define complex tori and the associated concepts, bearing in mind the fact that an abelian variety is a complex torus with special properties.

Let g be a positive integer. A complex torus X of dimension g is a quotient of abelian groups

X: = V/Λ_{V} ,

where V is a g-dimensional ℂ-vector space and Λ_{V} is a lattice in V.

Define complex tori X: = V/Λ_{V} , Y: = W/Λ_{W}. A homomorphism of complex tori

ϕ: X → Y

is a holomorphic map which is a homomorphism of abelian groups. For an integer n, let [n]_{X} denote the homomorphism X → X, x ↦ nx. For x_{0} ∈ X, the translation by x_{0} map t_{x_{0}}: X → X is the holomorphic map x ↦ x+x_{0}. For each holomorphic map ψ: X → Y of complex tori, there is a homomorphism ϕ: X → Y of complex tori such that ψ = t_{ψ(Λ_{V} )}ϕ (see [L-B], Chapter 1 Section 2 Proposition 2.1). Under addition, the set of homomorphisms ϕ: X → Y form an abelian group, denoted by Hom(X, Y). There is a unique ℂ-linear map Φ: V → W with Φ(Λ_{V}) ⊆ Λ_{W} inducing ϕ (see [L-B], Chapter 1 Section 2 Proposition 2.1), so we have injective homomorphisms

ρ_{a}: Hom(X, Y) → Hom_{ℂ}(V, W), ϕ ↦ Φ,
ρ_{r}: Hom(X, Y) → Hom_{ℤ}(Λ_{V} , Λ_{W}), ϕ ↦ Φ|_{Λ},

which induce injective homomorphisms

ρ_{a}: Hom(X, Y) ⊗ ℚ → Hom_{ℂ}(V, W) ⊗ ℚ,
ρ_{r}: Hom(X, Y) ⊗ ℚ → Hom_{ℤ}(Λ_{V} , Λ_{W}) ⊗ ℚ,

which are called the analytic,rational representations of Hom(X, Y)⊗ℚ respectively. With respect to bases of V and W, ρ_{a}(ϕ) is given by a matrix over ℂ. With respect to bases of Λ_{V} and Λ_{W}, ρ_{r}(ϕ) is given by a matrix over ℚ.

For this paragraph, suppose that X = Y. Define

P_{a}(ϕ, t): = det(t1_{g} − ρ_{a}(ϕ)), P_{r}(ϕ, t): = det(t1_{2g} − ρ_{r}(ϕ)),

the characteristic polynomials of ρ_{a}(ϕ), ρ_{r}(ϕ) respectively, which are monic with complex,rational coefficients respectively. Define tr_{a}(ϕ), tr_{r}(ϕ) to be the t^{g−1}- coefficient of P_{a}(ϕ, t), the t^{2g−1}-coefficient of P_{r}(ϕ, t) respectively.

Choose bases e_{1}, …, e_{g} of V and λ_{1}, …, λ_{2g} of Λ_{V}; then, the matrix Π ∈ M_{g,2g}(ℂ) (g rows, 2g columns) such that

(λ_{1} … λ_{2g})^{t} = Π^{t}(e_{1} … e_{g})^{t}

is a period matrix for X (w.r.t. the chosen bases).

We now move on to the concept of hermitian forms on complex tori, motivated by the fact that an abelian variety will be a complex torus with a special type of hermitian form on it.

A hermitian form on V is a map

H: V × V → ℂ

which is ℂ-linear in the first component and satisfies

H(v_{1}, v_{2}) = \overline{H(v_{2}, v_{1})} ∀v_{1}, v_{2} ∈ V.

A real valued alternating form on V is a map

E: V × V → ℝ

which is ℝ-linear in the first component and satisfies

E(v_{1}, v_{2}) = −E(v_{2}, v_{1}) ∀v_{1}, v_{2} ∈ V.

There is a one-to-one correspondence between the set of hermitian forms H on V and the real-valued alternating forms on V satisfying E(iv_{1}, iv_{2}) = E(v_{1}, v_{2}) for all v_{1}, v_{2} ∈ V, given by

E(v_{1}, v_{2}) = ℑH(v_{1}, v_{2}), H(v_{1}, v_{2}) = E(iv_{1}, v_{2}) + iE(v_{1}, v_{2})

for all v_{1}, v_{2} ∈ V (see [L-B], Chapter 2 Section 1 Lemma 1.7). The group of hermitian forms H on V satisfying H(Λ_{V} , Λ_{V}) ⊆ ℤ (which is isomorphic to the group of real-valued alternating forms E on V satisfying E(iv_{1}, iv_{2}) = E(v_{1}, v_{2}) for all v_{1}, v_{2} ∈ V and E(Λ_{V} , Λ_{V}) ⊆ ℤ by the above) is called the Néron-Severi group of X and denoted by NS(X).

There is a basis λ_{1}, …, λ_{2g} of Λ_{V} such that

E(λ_{i}, λ_{j}) = ( \begin{matrix}0_{g} & D \\ −D & 0_{g} \end{matrix} )_{ij} ,

where D is the diagonal g × g matrix such that the element in the i^{th} row and the i^{th} column of D is d_{i}, where 0 ≤ d_{i} ∈ ℤ (1 ≤ i ≤ g) and d_{i} | d_{i+1} (1 ≤ i ≤ g − 1); moreover, the vector (d_{1}, …, d_{g}) is uniquely determined by E and Λ_{V} (see [L-B], Chapter 3 Section 1). The vector (d_{1}, …, d_{g}) is called the type of H, and the basis λ_{1}, …, λ_{2g} is called a symplectic basis of Λ_{V} for H.

Define

V̂: = Hom\overline{ℂ}(V, ℂ),

the g-dimensional ℂ-vector space of ℂ-antilinear forms f: V → ℂ. The canonical ℝ-bilinear form

⟨ , ⟩: V̂ × V → ℝ, ⟨f, v⟩ = ℑf(v)

is nondegenerate, which implies that

Λ̂_{V}: = {f ∈ V̂ | ⟨f, v⟩ ⊆ ℤ}

is a lattice in V̂. The quotient of abelian groups

X̂: = V̂ /Λ̂_{V}

is a complex torus of dimension g, called the dual complex torus of X. V is isomorphic as ℂ-vector spaces to

ˆV̂: = Hom\overline{ℂ}(V̂ , ℂ),

the g-dimensional ℂ-vector space of ℂ-antilinear forms V̂ → ℂ, by double antiduality, so Λ_{V} is isomorphic as ℤ-vector spaces to ˆ Λ̂_{V} and X is isomorphic as complex tori to ˆ X̂. If ϕ: X → Y is a homomorphism of complex tori, then there is a homomorphism ϕ̂: Ŷ → X̂ of complex tori which sends fΛ̂_{W} ∈ Ŷ to (f ∘ ϕ)Λ̂_{V} ∈ X̂, f: W → ℂ being a ℂ-antilinear form.

To H ∈ NS(X), we associate a homomorphism

ϕ_{H}: X → X̂

of complex tori whose analytic representation in Hom_{ℂ}(V, V̂) sends v ∈ V to H(v, ·).

A polarisation on X is a homomorphism

λ: X → X̂

of complex tori such that λ = ϕ_{H} for some H ∈ NS(X) which is positive definite. The type of ϕ_{H} is defined to be the type of H, and a symplectic basis of Λ for ϕ_{H} is defined to be a symplectic basis of Λ for H. λ is an isogeny of complex tori whose degree is ∏_{i=1}^{g} d_{i}^{2}, where (d_{1}, …, d_{g}) is the type of ϕ_{H}. If ϕ_{H} is an isomorphism of complex tori (i.e. if d_{1} = … = d_{g} = 1), then λ is called a principal polarisation.

An abelian variety is a complex torus X having a polarisation λ. If g = 1, then an abelian variety is called an elliptic curve, and if g = 2, then an abelian variety is called an abelian surface. A polarised/principally polarised abelian variety is a pair (X, λ), where X is an abelian variety and λ is a polarisation/principal polarisation. A homomorphism of polarised abelian varieties (X, λ_{X}) → (Y, λ_{Y}) is a homomorphism ϕ: X → Y of complex tori such that λ_{X} = ϕ̂λ_{Y} ϕ.

We now move on to the concept of endomorphisms of abelian varieties. We define End(X) to be the endomorphism ring of X; this notation will be used throughout this thesis. Also, End(X) ⊗ ℚ is the endomorphism algebra of X, which is a ℚ-algebra. There is a map †_{H} on End(X) ⊗ ℚ defined by

†_{H}: End(X) ⊗ ℚ → End(X) ⊗ ℚ, ε ↦ \frac{1}{e(ϕ_{H})}ψ_{H} ∘ ε̂∘ ϕ_{H},

where ψ_{H} is the dual isogeny of ϕ_{H} and e(ϕ_{H}) is the exponent of the kernel of ϕ_{H}; we define ε^{†_{H}}: = †_{H}(ε). †_{H} has the following properties (see [L-B], Chapter 5 Section 1):

• (r_{1}ε_{1} + r_{2}ε_{2})^{†_{H}} = r_{1}ε_{1}^{†_{H}} + r_{2}ε_{2}^{†_{H}} for all r_{1}, r_{2} ∈ ℚ, ε_{1}, ε_{2} ∈ End(X) ⊗ ℚ;

• (ε_{1}ε_{2})^{†_{H}} = ε_{2}^{†_{H}}ε_{1}^{†_{H}} for all ε_{1}, ε_{2} ∈ End(X) ⊗ ℚ;

• ε^{†_{H}}^{†_{H}} = ε for all ε ∈ End(X) ⊗ ℚ;

so †_{H} is an anti-involution on the endomorphism algebra of X. It is called the Rosati involution w.r.t. H (or w.r.t. ϕ_{H}). We call an endomorphism ε H-symmetric if ε^{†_{H}} = ε, and H-antisymmetric if ε^{†_{H}} = −ε.

The following theorem is of great importance in determining the possible endomorphism rings of an abelian variety, as explained in [L-B], Chapter 5. It will be used to prove an important proposition in Section 2.8.

Theorem 2.7.1 †_{H} is a positive anti-involution , i.e. the form

(ε_{1}, ε_{2}) ↦ tr_{r}(ε_{1}^{†_{H}}ε_{2})

is a positive definite symmetric bilinear form on the ℚ-vector space End(X)⊗ℚ.

Proof: See [L-B], Chapter 5 Section 2 Theorem 1.8.□

Another important theorem, which will be used to prove an important proposition in Section 2.8, is

Theorem 2.7.2 For a principal polarisation ϕ_{H_{0}} on X, there is an isomorphism of abelian groups defined by

φ_{H_{0}}: NS(A) →^{∼} End(X)^{†_{H_{0}}=1}

H ↦ ϕ_{H_{0}}^{−1}ϕ_{H},

which induces a bijection between the polarisations ϕ_{H} on X of degree d and the H_{0}-symmetric endomorphisms α of X of degree d such that all of the zeros of P_{a}(α, t) are positive, defining End(X)^{†_{H_{0}}=1} to be the abelian group of H_{0}- symmetric endomorphisms of X.

Proof: See [L-B], Chapter 5 Section 2 Theorem 2.4).□

We now look at the problem of classifying polarised abelian varieties up to isomorphism. Two key notions here are the Siegel upper half space and the symplectic group.

Suppose that (X, λ) is a polarised abelian variety of dimension g. Let H ∈ NS(X) such that λ = ϕ_{H}, let (d_{1}, …, d_{g}) be the type of ϕ_{H} and D be the corresponding diagonal g × g matrix, and let λ_{1}, …, λ_{2g} be a symplectic basis of Λ_{V} for ϕ_{H}. Define the Siegel upper half space 𝔥_{g} as follows:

𝔥_{g}: = {τ ∈ M_{2}(ℂ) | τ^{t} = τ, ℑτ > 0}.

Define e_{i} = \frac{1}{d_{i}} λ_{i+g} for 1 ≤ i ≤ g; e_{1}, …, e_{g} is a ℂ-basis for V (see [L-B], Chapter 8 Section 1). Then, w.r.t. the basis e_{1}, …, e_{g} of V and the basis λ_{1}, …, λ_{2g} of Λ_{V}, the period matrix of X is Π = (τ, D) for some τ ∈ 𝔥_{g}, and H(e_{i}, e_{j}) = ((ℑτ)^{−1})_{ij} (see [L-B], Chapter 8 Section 1).

Define the lattice Λ_{τ,D} in ℂ^{g} by Λ_{τ,D}: = τℤ^{g} ⊕Dℤ^{g}, and define the hermitian form H_{τ} on ℂ^{g} by H_{τ}(e_{i}, e_{j}) = ((ℑτ)^{−1})_{ij}, where e_{i} = (0, …, 0, 1, 0 …, 0) (1 in the i^{th} position); e_{1}, …, e_{g} is the standard basis of ℂ^{g}. We deduce that (X, λ) and (ℂ^{g}/Λ_{τ,D}, ϕ_{H_{τ}}) are isomorphic as polarised abelian varieties, with the symplectic basis of Λ_{τ,D} for ϕ_{H_{τ}} being the columns of Π.

Having shown that (X, λ) is isomorphic to (ℂ^{g}/Λ_{τ,D}, ϕ_{H_{τ}}) for some τ, D as above, we need to discuss when two polarised abelian varieties of the latter form are isomorphic in order to complete the classification problem. The notion of the symplectic group is useful here.

For a commutative ring R with identity, the symplectic group Sp_{2g}(R) is defined to be

Sp_{2g}(R): = \{M ∈ M_{2g}(R)|M^{t} ( \begin{matrix}0_{g} & 1_{g} \\ −1_{g} & 0_{g} \end{matrix} ) M = ( \begin{matrix}0_{g} & 1_{g} \\ −1_{g} & 0_{g} \end{matrix} ) .\} (2.50)

Consider (d_{1}, …, d_{g}), where 0 ≤ d_{i} ∈ ℤ (1 ≤ i ≤ g) and d_{i} | d_{i+1} (1 ≤ i ≤ g−1), and let D be the corresponding g × g matrix. Define

G_{D}: = \{M ∈ Sp_{2g}(ℚ)|M^{t} ( \begin{matrix}1_{g} & 0_{g} \\ 0_{g} & D \end{matrix} ) ℤ^{2g} ⊆ ( \begin{matrix}1_{g} & 0_{g} \\ 0_{g} & D \end{matrix} ) ℤ^{2g}\} .  (2.51)

G_{D} acts on 𝔥_{g} as follows: let

M = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) , τ ∈ 𝔥_{g}

(α, β, γ, δ are 2 × 2 matrices), and define

M(τ): = (ατ + β)(γτ + δ)^{−1}.

Then, we have:

Proposition 2.7.3 For τ_{1}, τ_{2} ∈ 𝔥_{g}, (ℂ^{g}/Λ_{τ_{1},D}, ϕ_{H_{τ_{1}}}) and (ℂ^{g}/Λ_{τ_{2},D}, ϕ_{H_{τ_{2}}}) are isomorphic as polarised abelian varieties if and only if τ_{2} = M(τ_{1}) for some M ∈ G_{D}; if this is the case, then the rational representation of an isomorphism from the former variety to the latter variety, with the symplectic bases being the columns of (τ_{1}D) and (τ_{2}D), is

( \begin{matrix}1_{g} & 0 \\ 0 & D \end{matrix} )^{−1} M^{t} ( \begin{matrix}1_{g} & 0 \\ 0 & D \end{matrix} ) .  (2.52)

Proof: See [L-B], Chapter 8 Section 1 Proposition 1.3. The proof involves elementary manipulations of the definitions.□

Proposition 2.7.3 means that 𝔥_{g}/G_{D} can be loosely regarded as a moduli space for polarised abelian varieties of dimension g. Note that, if D = 1_{g}, then G_{D} = Sp_{2g}(ℤ), as can be easily seen by comparing (2.50) for R = ℤ and (2.51); so 𝔥_{g}/Sp_{2g}(ℤ) can be loosely regarded as a moduli space for principally polarised abelian varieties of dimension g.

We now look at the endomorphism ring of an abelian variety in the complexanalytic sense; it is in general easier to determine the endomorphism ring complex-analytically than algebraically, which is the chief motivation for studying abelian varieties in the complex-analytic sense.

An abelian variety X is said to be simple if X does not contain any abelian subvariety apart from X and 0; if X is simple, then the endomorphism algebra of X is a skew field. We have the following theorem and corollary, giving a description of End(X) for general abelian varieties X:

Theorem 2.7.4 (Poincare’s Complete Reducibility Theorem) Let X be an abelian variety. Then, there is an isogeny

X → X_{1}^{n_{1}} × … × X_{r}^{n_{r}},

where the X_{i}’s are simple abelian varieties not isogenous to each other, and the n_{i}’s are positive integers. Moreover, the X_{i}’s and the n_{i}’s are uniquely determined up to isogenies and permutations.

Proof: See [L-B], Chapter 5 Section 3 Theorem 3.7.□ Corollary 2.7.5

End(X) ⊗ ℚ ≅ M_{n_{1}}(End(X_{1}) ⊗ ℚ) ⊕ … ⊕ M_{n_{r}}(End(X_{r}) ⊗ ℚ)

(in particular, the endomorphism algebra of X is a semisimple ℚ-algebra).

Proof: See [L-B], Chapter 5 Section 3 Corollary 3.8.□

Suppose that (X, λ) is a polarised abelian variety of dimension g such that X is simple, and let H ∈ NS(X) such that λ = ϕ_{H}; then, as we have seen, the Rosati involution †_{H} acts on F: = End(X) ⊗ ℚ. Let K be the centre of F; then, †_{H} acts on K. Let K_{0} be the fixed field w.r.t. this action on K. Define

d^{2} = [F: K], e = [K: ℚ], e_{0} = [K_{0}: ℚ].

We have

Theorem 2.7.6 F is one of the following:

F  d e_{0} Restriction
Totally real number field 1 e e|g
Totally indefinite quaternion algebra over ℚ 2 e  2e|g
Totally definite quaternion algebra over ℚ 2 e  2e|g
(F, †_{H}) of the second kind  d \frac{e}{2}  e_{0}d^{2}|g

Proof: See [L-B], Chapter 4 Section 5 Proposition 5.7.□

To explain the description of the fourth line in the table above, the pair (F, †_{H}) is said to be of the first kind if the fixed field with respect to the action of †_{H} on K is K, and of the second kind otherwise. Note that there are further restrictions for the third and fourth possibilities (see [L-B], Chapter 9 Section 9 Theorem 9.1): for example, if X is an abelian surface, it is not possible for F to be an imaginary quadratic extension of ℚ or a definite quaternion algebra of degree 4 over ℚ. We have the following standard and important theorems:

Theorem 2.7.7 If X is an elliptic curve, then F is either ℚ or an imaginary quadratic extension of ℚ.

Proof: Theorem 2.7.6.□

Theorem 2.7.8 If X is a simple abelian surface, then F is one of the following:

• a real extension of degree 1 or 2 over ℚ;

• a totally indefinite quaternion algebra of degree 4 over ℚ;

• a CM field (i.e. an imaginary quadratic extension of a real quadratic extension of ℚ).

Proof: Theorem 2.7.6 and [Shi1], Theorem 5.□

2.8 The Abel-Jacobi Isomorphism

Many of the definitions in Sections 2.2 have an analogous definition in Section 2.7, and vice-versa. In view of this, it is reasonable to guess that algebraic abelian varieties and complex-analytic abelian varieties are inextricably linked, despite their apparent diversity. In fact, they are, by virtue of the Lefschetz Embedding Theorem (see [Lan1], Theorem 6.1 p. 202), which, given a complexanalytic abelian variety X and a polarisation on X, gives a complex-analytic embedding from X to a projective space over ℂ, the image of which is an algebraic abelian variety (defined over ℂ). This linkage is useful, as it allows algebraic theorems and complex-analytic theorems to work together, as we will demonstrate in this section. In this section, we mainly discuss the situation for the jacobian of a curve of genus g defined over ℂ, which is simpler than for a general abelian variety, although the map constructed here will go from an algebraic abelian variety to a complex-analytic abelian variety, rather than the other way round as constructed by the Lefschetz Embedding Theorem. Although we will use the map when g = 2 in this thesis, it is just as simple to describe when g is any positive integer.

Let C be a curve of genus g ≥ 1 defined over ℂ, let Jac(C) be the jacobian of C (for the theory below, it does not matter which model for Jac(C) is chosen), and let P_{0} be a point on C. Let ω_{1}, …, ω_{g} be a basis over ℂ for the vector space of holomorphic differentials on C, and let \underline{ω}: = (ω_{1}, …, ω_{g}). Define

Λ\underline{ω}: = \{∫_{c} \underline{ω} | c ∈ H_{1}(C, ℤ)\} ,

and consider the map

C → ℂ^{g}/Λ\underline{ω}, P ↦ ∫_{P_{0}}^{P} \underline{ω} + Λ\underline{ω},

which is independent of the choice of path from P_{0} to P by the definition of Λ\underline{ω}. The map extends by linearity to a surjective complex-analytic group homomorphism Div^{0}(C) → ℂ^{g}/Λ\underline{ω} (independent of the choice of P_{0} since

P_{1} − P_{2} ↦ ∫_{P_{2}}^{P_{1}} \underline{ω} + Λ\underline{ω}

for all P_{1}, P_{2} ∈ C) whose kernel consists precisely of the principal divisors (by well-known theorems of Abel and Jacobi), so there is a complex-analytic group isomorphism

Φ\underline{ω}: Jac(C) →^{∼} ℂ^{g}/Λ\underline{ω},  (2.53)

depending on \underline{ω}, called the Abel-Jacobi isomorphism. It can be shown that there is a basis ω_{1}, …, ω_{g} for the vector space of holomorphic differentials on C such that

Λ\underline{ω} = Λ_{τ}: = Λ_{τ,1_{g}}  (2.54)

for some τ ∈ 𝔥_{2}, as expected since (Jac(C), Θ) (defining Θ to be a theta divisor on the jacobian) and (ℂ^{g}/Λ_{τ}, ϕ_{H_{τ}}) are both principally polarised abelian varieties. There is a complex-analytic ring isomorphism

I\underline{ω}: End_{ℂ}(Jac(C)) →^{∼} End(ℂ^{g}/Λ\underline{ω}), ε ↦ ε\underline{ω}: = Φ\underline{ω}εΦ−1 ; \underline{ω},  (2.55)

and it is a well-known fact that

I\underline{ω}(ε^{†_{Θ}}) = ε†_{Hτ} ; \underline{ω} .  (2.56)

The complex-analytic group isomorphism Φ\underline{ω}, the principal ℂ-polarisation ϕ_{Θ} on the jacobian and the principal polarisation ϕ_{H_{τ}} on ℂ^{g}/Λ_{τ} induce a complexanalytic group isomorphism

Φ̂\underline{ω}: Ĵac(C) →^{∼} ℂ̂^{g}/Λ\underline{ω}  (2.57)

(it does not matter which model for Ĵac(C) is chosen); more precisely, we have Φ̂\underline{ω} = ϕ_{H_{τ}} Φ\underline{ω}ϕ_{Θ}^{−1}. It is a well-known fact that there is a one-to-one correspondence between algebraic equivalence classes of ample divisors [D] on the jacobian and positive definite hermitian forms H in NS(ℂ^{g}/Λ\underline{ω}) such that, if [D] corresponds to H, then

ϕ_{D} = Φ̂\underline{ω}^{−1}ϕ_{H}Φ\underline{ω},  (2.58)

(recall that ϕ_{D} is independent of the divisor D chosen within the algebraic equivalence class), i.e. the isomorphisms (2.53) and (2.57) induce a one-to-one correspondence between the ℂ-polarisations on the jacobian and the polarisations on ℂ^{g}/Λ\underline{ω}. For example, [Θ] corresponds to H_{τ}, since Φ̂\underline{ω} = ϕ_{H_{τ}} Φ\underline{ω}ϕ_{Θ}^{−1}.

Our first application of the linkage between algebraic varieties and complexanalytic abelian varieties is the following proposition based on Theorem 2.7.1, which we will use a lot in this thesis for \sqrt{2} multiplications. It can be extended from jacobians of curves of genus 2 to abelian surfaces, but will be proved for jacobians as we have an explicit complex-analytic isomorphism from algebraic jacobians to complex-analytic jacobians.

Proposition 2.8.1 Let K be a subfield of ℂ, let C be a curve of genus 2 defined over K, and let Θ be a theta divisor on the jacobian of C. Suppose that D is an integer that is not a square. If there are precisely two \sqrt{D} K-multiplications ±ε of the jacobian, then ε^{†_{Θ}} = ±ε. If ε^{†_{Θ}} = ±ε (whether there are precisely two \sqrt{D} K-multiplications or not), then ε is Θ-symmetric or Θ-antisymmetric according to whether D is positive or negative respectively.

Proof: Suppose that there are precisely two \sqrt{D} K-multiplications of the jacobian. Since ε and ϕ_{Θ} are defined over K, ε̂ is defined over K, so ε^{†_{Θ}} = ϕ_{Θ}^{−1}ε̂ϕ_{Θ} is defined over K as well. Moreover, the square of ε^{†_{Θ}} is [D] since the square of ε is [D], so, since there are precisely two \sqrt{D} K-multiplications, ε^{†_{Θ}} = ±ε.

Suppose that ε^{†_{Θ}} = ±ε. Applying the complex-analytic theory, let τ ∈ 𝔥_{2} and ω_{1}, ω_{2} be a basis over ℂ for the holomorphic differentials on C such that there is an isomorphism as in (2.53), and that (2.54) holds. Defining I\underline{ω} as in (2.55), we have ε†_{Hτ} ; \underline{ω} = ±ε\underline{ω} by (2.56), since ε^{†_{Θ}} = ±ε. If D is positive, then ε†_{Hτ} ; \underline{ω} = ε\underline{ω} by Theorem 2.7.1, so ε^{†_{Θ}} = ε by (2.56), i.e. ε is Θ-symmetric, as required. Similarly, if D is negative, then ε is Θ-antisymmetric, as required. □

We state and prove a corollary, which uses Theorem 2.7.2 as well as Proposition 2.8.1. We will use it a lot in this thesis in the case where F = ℚ(\sqrt{2}), but it is just as easy to prove for more general F.

Corollary 2.8.2 Let K be a subfield of ℂ, let C be a curve of genus 2 defined over K, and let Θ be a theta divisor on the jacobian of C. Suppose that there is a sub-ℚ-algebra F of the algebra of K-endomorphisms of the jacobian which is a real quadratic extension of ℚ, and that the K-endomorphism ring of the jacobian is either the ring of integers of F or an order in an imaginary quadratic extension of F which contains the ring of integers of F, and that the real Kautomorphisms of the jacobian are generated by [−1] and an automorphism α_{0} such that N_{F/ℚ}(α_{0}) = [−1] (since the K-endomorphism ring of the jacobian embeds into ℂ, it makes sense to talk about ‘real K-endomorphisms’ here). Then, any principal K-polarisation on the jacobian is ϕ_{Θ}α_{0}^{2n} for some integer n.

Proof: Firstly, we will show that the real K-endomorphisms of the jacobian are precisely the Θ-symmetric K-endomorphisms. This simply follows from Proposition 2.8.1, using the fact that the K-endomorphism ring of the jacobian is either the ring of integers of a real quadratic extension of ℚ or the ring of integers of an imaginary quadratic extension of a real quadratic extension of ℚ (a hypothesis in the corollary).

Now we will show that any principal K-polarisation on the jacobian is ϕ_{Θ}α_{0}^{2n} for some integer n, to establish the corollary. By Theorem 2.7.2, taking d = 1, and the correspondence (2.58) between algebraic polarisations and complex-analytic polarisations, we see that there is a bijection between the principal ℂ-polarisations ϕ_{D} on the jacobian and the Θ-symmetric ℂautomorphisms α of the jacobian such that all of the zeros of P_{a}(I\underline{ω}(α)) are positive, the automorphism corresponding to ϕ_{D} being ϕ_{Θ}^{−1}ϕ_{D}; also, since ϕ_{Θ} is defined over K, K-polarisations correspond to K-automorphisms. We have established that the real K-endomorphisms are precisely the Θ-symmetric Kendomorphisms, so the Θ-symmetric K-automorphisms are generated by [±1] and α_{0}. Since N_{F/ℚ}(α_{0}) = [−1] (a hypothesis in the corollary), the Θ-symmetric K-automorphisms α such that all of the zeros of P_{a}(I\underline{ω}(α)) are positive are generated by α_{0}^{2}, so any principal K-polarisation on the jacobian is ϕ_{Θ}α_{0}^{2n} for some integer n, as required.□

We state and prove a corollary which also uses Theorem 2.3.2, which will be the main application of Corollary 2.8.2 used in this thesis.

Corollary 2.8.3 Let K be a subfield of ℂ, and let C_{1}, C_{2} be curves of genus 2 defined over K such that there is a K-isomorphism ι from the jacobian of C_{1} to that of C_{2} as (unpolarised) abelian varieties. Suppose that C_{1} satisfies the conditions in Corollary 2.8.2. Then, for some integer n, ι_{J}: = ια_{0}^{n} is a Kisomorphism from the canonically polarised jacobian of C_{1} to that of C_{2}, and ι_{C} (as defined in Theorem 2.3.2) is a K-isomorphism from C_{1} to C_{2}.

Proof: Before starting the proof, we fix appropriate notation; let Θ_{1}, Θ_{2} be theta divisors on the jacobians of C_{1}, C_{2} respectively. By Theorem 2.3.2, it suffices to establish the existence of ι_{J} with the required properties.

Firstly, we will show that, if C is a curve of genus 2 defined over K which satisfies the conditions in Corollary 2.8.2 and λ is a principal K-polarisation on the jacobian of C, then (Jac(C), ϕ_{Θ}), (Jac(C), λ) are K-isomorphic as polarised abelian varieties. By Corollary 2.8.2, λ = ϕ_{Θ}α_{0}^{2n} for some integer n, which implies that ϕ_{Θ}^{−1}α̂_{0}^{n}ϕ_{Θ}α_{0}^{n} = (α_{0}^{n})^{†_{Θ}}α_{0}^{n} = α_{0}^{2n}, which implies that λ = α̂_{0}^{n}ϕ_{Θ}α_{0}^{n}. We deduce that α_{0}^{n} is a K-isomorphism from (Jac(C), ϕ_{Θ}) to (Jac(C), λ), as required.

Returning to the curves C_{1} and C_{2}, we see that ι is a K-isomorphism from (Jac(C_{1}), ι̂ϕ_{Θ_{2}}ι) to (Jac(C_{2}), ϕ_{Θ_{2}}) as polarised abelian varieties, by the definition of morphisms between polarised abelian varieties. The corollary follows from putting λ: = ι̂ϕ_{Θ_{2}}ι, a principal K-polarisation on the jacobian of C_{1}; indeed, for some integer n, ια_{0}^{n} is a K-isomorphism from the canonically polarised jacobian of C_{1} to that of C_{2} (by the previous paragraph, putting C = C_{1}), as required. □

Our second application of the linkage between algebraic varieties and complexanalytic abelian varieties is the following proposition based on Theorem 2.7.1, which we will use in Chapter 5. Like Proposition 2.8.1, it can be extended from jacobians of curves of genus 2 to abelian surfaces, but will be proved for jacobians as we have an explicit complex-analytic isomorphism from algebraic jacobians to complex-analytic jacobians.

Proposition 2.8.4 Let C be a curve of genus 2 defined over ℂ, let Θ be a divisor on C, and let ε, ε′ be Θ-symmetric ℂ-endomorphisms of the jacobian of C such that ε^{2} = [m], ε^{′2} = [m′] for some positive integers m, m′ such that m, m′, mm′ are not squares. Then,

ε′ε + εε′ = [n]

for some integer n such that −m − m′ ≤ n ≤ m + m′.

Proof: Applying the complex-analytic theory, let τ ∈ 𝔥_{2} and ω_{1}, ω_{2} be a basis over ℂ for the holomorphic differentials on C such that there is an isomorphism as in (2.53), and that (2.54) holds. Define I\underline{ω} as in (2.55). To prove the proposition, it suffices to show that

ε′ ; \underline{ω}ε\underline{ω} + ε\underline{ω}ε′ ; \underline{ω} = [n]\underline{ω}  (2.59)

for some integer n such that −m − m′ ≤ n ≤ m + m′.

We will show that (2.59) is true for some integer n by considering the various possibilities for the endomorphism algebra of ℂ^{2}/Λ_{τ}, and then that −m − m′ ≤ n ≤ m + m′ by applying Theorem 2.7.1.

For considering the various possibilities for the endomorphism algebra of ℂ^{2}/Λ_{τ}, there are two cases to consider: the case where ℂ^{2}/Λ_{τ} is a simple abelian surface, and the case where ℂ^{2}/Λ_{τ} is isogenous to the square of an elliptic curve X.

Suppose that ℂ^{2}/Λ_{τ} is a simple abelian surface. Then, by Theorem 2.7.8 and since ε^{2} = [m], ε^{′2} = [m′] for some positive integers m, m′ such that m, m′, mm′ are not squares (by hypothesis), the endomorphism algebra of ℂ^{2}/Λ_{τ} must be a totally indefinite quaternion algebra of degree 4 over ℚ. So there exist elements α, β in it and integers m_{α}, m_{β} such that

α^{2} = [m_{α}]\underline{ω}, β^{2} = [m_{β}]\underline{ω}, αβ + βα = [0]\underline{ω}

and [1]\underline{ω}, α, β generate it.

If γ is an endomorphism of ℂ^{2}/Λ_{τ} which satisfies γ^{2} = [m_{γ}]\underline{ω} for some integer m_{γ}, then, after a bit of computation, we see that there exist rational numbers a, b, c such that

γ = aα + bβ + cαβ, m_{α}a^{2} + m_{β}b^{2} − m_{α}m_{β}c^{2} = m_{γ}.

In particular, ε\underline{ω} and ε′ ; \underline{ω} are of this form; after a bit of computation, we see that (2.59) holds for some integer n, as required.

Suppose that ℂ^{2}/Λ_{τ} is isogenous to the square of an elliptic curve X. Now the endomorphism algebra of X is isomorphic to ℚ(\sqrt{D}) for some non-positive rational number D (possibly zero). By Corollary 2.7.5, there is a ring isomorphism Ψ from this algebra to the algebra of 2 by 2 matrices over ℚ(\sqrt{D}). Define

( \begin{matrix}a & b \\ c & d \end{matrix} ): = Ψ(ε\underline{ω}), ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ): = Ψ(ε′ ; \underline{ω});

a, b, c, d, a′, b′, c′, d′ ∈ ℚ(\sqrt{D}). Since ε^{2} = [m], ε^{′2} = [m′] for some positive integers m, m′ such that m, m′, mm′ are not squares (by hypothesis), Ψ(ε\underline{ω})^{2}, Ψ(ε′ ; \underline{ω})^{2} are scalar matrices; in particular, d = −a and d′ = −a′, since \sqrt{m}, \sqrt{m′} ∉ ℚ(\sqrt{D}), D being non-positive. After a bit of computation, we see that

Ψ(ε\underline{ω})Ψ(ε′ ; \underline{ω}) + Ψ(ε′ ; \underline{ω})Ψ(ε\underline{ω}) = ( \begin{matrix}2aa′ + bc′ + cb′ & 0 \\ 0 & 2aa′ + bc′ + cb′ \end{matrix} ) ,

a scalar matrix, so, defining

δ\underline{ω}: = Ψ^{−1} ( \begin{matrix}\sqrt{D} & 0 \\ 0 & \sqrt{D} \end{matrix} ) ,

we see that

Ψ(ε\underline{ω})Ψ(ε′ ; \underline{ω}) + Ψ(ε′ ; \underline{ω})Ψ(ε\underline{ω}) = [n_{0}]\underline{ω} + [n_{1}]\underline{ω}δ\underline{ω}

for some rational numbers n_{0}, n_{1}; we will show that n_{1} = 0, so that (2.59) holds for some integer n, as required. Certainly we can take n_{1} = 0 if D = 0, so suppose that D is negative. ε†_{Hτ} ; \underline{ω} , ζ†_{Hτ} ; \underline{ω} are H_{τ}-symmetric, since ε, ε′ are Θ-symmetric (by hypothesis), which implies that either m_{1} = 0 or δ\underline{ω} is H_{τ}- symmetric. But δ2 ; \underline{ω} = [D]\underline{ω} and D is negative, so δ\underline{ω} is not H_{τ}-symmetric by Theorem 2.7.1, and so n_{1} = 0.

Finally, we show that −m−m′ ≤ n ≤ m+m′ by applying Theorem 2.7.1. Now

(ε\underline{ω} + ε′ ; \underline{ω})^{2} = [m + m′ + n],  (2.60)

(ε\underline{ω} − ε′ ; \underline{ω})^{2} = [m + m′ − n],

As we have seen earlier in the proof, ε\underline{ω}, ε′ ; \underline{ω} are H_{τ}-symmetric. In particular,

(ε\underline{ω} + ε′ ; \underline{ω})^{†_{Hτ}} = ε\underline{ω} + ε′ ; \underline{ω},

(ε\underline{ω} − ε′ ; \underline{ω})^{†_{Hτ}} = ε\underline{ω} − ε′ ; \underline{ω},

so, by Theorem 2.7.1 and (2.60), we see that −m − m′ ≤ n ≤ m + m′, as required.□

Finally in this section, we state a theorem about the \overline{K}-endomorphism ring of an abelian variety defined over a subfield K of ℂ, where \overline{K} is an algebraic closure of K, which is well-known.

Theorem 2.8.5 Let A be an abelian variety defined over K. Then, the \overline{K}- endomorphism ring of A is the ℂ-endomorphism ring of A.

Theorem 2.8.5 will be used in conjunction with Corollary 2.7.5, Theorem 2.7.7 and Theorem 2.7.8 in this thesis to discuss the endomorphism ring.

2.9 The Action of a Θ-Symmetric Endomorphism on 2-Torsion

The following problem is of interest in this thesis: given a curve C of genus 2 defined over ℂ, and letting Θ be a theta divisor on the jacobian of C, describe the action of a primitive Θ-symmetric ℂ-endomorphism ε on the 2-torsion of the jacobian with the following properties:

• ε satisfies

ε^{2} + [a]ε + [b] = 0,

where a ∈ {0, −1}, b is an integer and the discriminant Δ: = a^{2} − 4b is positive and not a square.

• ε + [1] is not divisible by [2] in the ℂ-endomorphism ring.

Note that, if O is a subring of the ℂ-endomorphism ring of the jacobian which is an order in a real quadratic extension of ℚ, and the endomorphisms in O are all Θ-symmetric, then they are generated by [1] and an endomorphism ε as above. We will use both complex-analytic and algebraic techniques to tackle this problem, mainly the former.

Let τ ∈ 𝔥_{2}, and consider (ℂ^{2}/Λ_{τ}, ϕ_{H_{τ}}) (defining Λ_{τ}: = Λ_{τ,1_{2}}), which is a principally polarised (not necessarily simple) abelian surface (recall that all principally polarised abelian surfaces are isomorphic to a principally polarised abelian surface of this form), and write

τ: = ( \begin{matrix}τ_{11} & τ_{12} \\ τ_{12} & τ_{22} \end{matrix} ) .

The following lemma, the starting point for tackling our problem, provides useful restrictions on τ given a H_{τ}-symmetric endomorphism of ℂ^{2}/Λ_{τ}:

Lemma 2.9.1 Let ε be a H_{τ}-symmetric endomorphism of ℂ^{2}/Λ_{τ}. Then, there exists a_{1}, a_{2}, a_{3}, a_{4}, b, c ∈ ℤ such that:

ρ_{a}(ε) = ( \begin{matrix}a_{1} − bτ_{12} & a_{3} + bτ_{11} \\ a_{2} − bτ_{22} & a_{4} + bτ_{12} \end{matrix} ) ,  (2.61)

ρ_{r}(ε) = ( \begin{matrix}a_{1} & a_{2} & 0 & b \\ a_{3} & a_{4} & −b & 0 \\ 0 & c & a_{1} & a_{3} \\ −c & 0 & a_{2} & a_{4} \end{matrix} ) ,  (2.62)

ε^{2} − [a_{1} + a_{4}]ε + [a_{1}a_{4} − a_{2}a_{3} + bc] = 0,  (2.63)
a_{2}τ_{11} − a_{3}τ_{22} + (a_{4} − a_{1})τ_{12} + b(τ_{12}^{2} − τ_{11}τ_{22}) + c = 0.  (2.64)

the matrices for ρ_{a}(ε), ρ_{r}(ε) being induced by the standard basis for ℂ^{2} and the basis for Λ_{τ} formed by the columns of Π: = (τ, 1_{2}) (so that ρ_{a}(ε)Π = Πρ_{r}(ε)).

Conversely, if ε is an endomorphism of ℂ^{2}/Λ_{τ} and the four conditions above are satisfied, then ε is H_{τ}-symmetric.

Proof: See [Has]. The proof involves simple algebraic manipulation using the relevant definitions.□

A linear relation between τ_{11}, τ_{12}, τ_{22} of the form

Aτ_{11} + Bτ_{12} + Cτ_{22} + D(τ_{12}^{2} − τ_{11}τ_{22}) + E = 0,  (2.65)

where A, B, C, D, E ∈ ℤ, is called a singular relation for τ, and its discriminant is defined to be

B^{2} − 4AC − 4DE.  (2.66)

The use of the word ‘singular’ to describe such a relation is standard but inappropriate, since the surface ℂ^{2}/Λ_{τ} is no more singular than usual if there is a singular relation for τ.

It is a fundamental and important fact that, given M ∈ Sp_{4}(ℤ), the existence of a singular relation for τ implies the existence of a singular relation for M(τ) with the two discriminants being equal, as is quite easy to show (see [Hud], Chapter XVIII Section 112 p. 203; although note that the language in [Hud] involves linear transformations, common in the 19^{th} century, whereas our language involves the action of Sp_{4}(ℤ) on 𝔥_{2}, common today).

Singular relations arise from Lemma 2.9.1; indeed, if ε is a H_{τ}-symmetric endomorphism of ℂ^{2}/Λ_{τ}, then, by Lemma 2.9.1, (2.64) is a singular relation for τ, whose discriminant equals that of (2.63) w.r.t. ε; this simply follows from finding A, B, C, D, E ∈ ℤ to equate (2.64) with (2.65) and computing (2.66).

Applying Lemma 2.9.1 directly to tackle our problem will probably involve lots of case-by-case manipulations, due to its use of six integers a_{1}, a_{2}, a_{3}, a_{4}, b, c. However, there is a result due to G. Humbert, which will reduce the amount of manipulation we have to do ourselves:

Proposition 2.9.2 Suppose that there is a singular relation for τ with discriminant Δ, and that the integers A, B, C, D, E in (2.65) are coprime. Then, there exists M ∈ Sp_{4}(ℤ) such that, writing

M(τ) = τ′: = ( \begin{matrix}τ′_{11} & τ′_{12} \\ τ′_{12} & τ′_{22} \end{matrix} ) ,

we have \frac{1}{4}Δτ′_{11} = τ′_{22} if Δ is even, and \frac{1}{4}(Δ − 1)τ′_{11} = τ′_{12} + τ′_{22} if Δ is odd; we call these singular relations normalised.

Proof: See [Hum], Theorem 12 p. 245. The proof is elementary but fairly involved, and it uses Dirichlet’s theorem about primes in arithmetic progression (which means that M will probably not be unique). The language in [Hum] involves linear transformations, like that in [Hud].□

Before stating our main corollary of Proposition 2.9.2, which will be used to help tackle our problem, we state a simple corollary which will be used in the proof of our main corollary.

Corollary 2.9.3 Let ε be a primitive H_{τ}-symmetric endomorphism of ℂ^{2}/Λ_{τ} with the properties described at the beginning of this section, and let the integers a_{1}, a_{2}, a_{3}, a_{4}, b, c be as in Lemma 2.9.1.

Then, there exists M ∈ Sp_{4}(ℤ) such that M(τ) has the normalised singular relation with discriminant Δ.

Proof: Lemma 2.9.1 and Proposition 2.9.2 (it is not too hard to see that the properties of ε imply that A, B, C, D, E are coprime).□

The Humbert surface H_{Δ} is a surface which classifies principally polarised abelian surfaces having a primitive endomorphism, fixed by the Rosati involution w.r.t. the polarisation, with the properties described at the beginning of this section. By Corollary 2.9.3, the points on H_{Δ} are in one-to-one correspondence with the matrices in 𝔥_{2} having the normalised singular relation with discriminant Δ.

We temporarily divert from our problem, to discuss briefly a similar proposition for polarised abelian surfaces in general:

Proposition 2.9.4 Let (X, ϕ_{H}) be a polarised abelian surface, let (d_{1}, d_{2}) be the type of λ, and let D be the corresponding 2 × 2 diagonal matrix. Suppose that there is a primitive H-symmetric endomorphism of it with the properties described at the beginning of this section.

Then, (X, ϕ_{H}) is isomorphic to (ℂ^{2}/Λ_{τ′,D}, ϕ_{H′_{τ}}) as polarised abelian varieties, where, writing

τ′: = ( \begin{matrix}τ′_{11} & τ′_{12} \\ τ′_{12} & τ′_{22} \end{matrix} ) ,

τ′_{11} + aτ′_{12} + b\frac{d_{2}}{d_{1}} τ′_{22} = 0 for some integers a, b such that 0 ≤ b ≤ 2\frac{d_{2}}{d_{1}} . Moreover, Δ (as at the beginning of this section) is equal to a^{2} − 4b\frac{d_{2}}{d_{1}} .

Proof: See [VdG], Theorem 2.4 p. 212.□

This proposition can be used (with a bit of elementary ideal theory) to determine the possible values of \frac{d_{2}}{d_{1}} for a given Δ. For example, if Δ is twice a square, and \frac{d_{2}}{d_{1}} is squarefree and does not divide both a and b, then this proposition and elementary ideal theory (taking into account the fact that ℤ[\sqrt{2}] is a principal ideal domain) implies that any prime divisor of \frac{d_{2}}{d_{1}} is either 2 or congruent to ±1 modulo 8. This fact will turn out to be important for our investigation of Conjecture 1.1.2 for jacobians of curves of genus 2 defined over ℚ with \sqrt{2} ℚ-multiplication, as we will see in Section 6.8.

The following corollary of Proposition 2.9.2, which uses the nature of the proof of the proposition as well as the proposition itself, will be used to help tackle our problem, because it allows us to consider endomorphisms which are of specialised forms.

Corollary 2.9.5 Let ε, a_{1}, a_{2}, a_{3}, a_{4}, b, c be as in Corollary 2.9.3, let M, τ′ be as in Proposition 2.9.2, and let ε′ be the endomorphism of ℂ^{2}/Λ_{τ′} such that

ρ_{a}(ε′) = ( \begin{matrix}0 & 1 \\ \frac{1}{4}Δ & 0 \end{matrix} ) , ρ_{r}(ε′) = ( \begin{matrix}0 & \frac{1}{4}Δ & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & \frac{1}{4}Δ & 0 \end{matrix} )  (2.67)

(a_{2} = \frac{1}{4}Δ, a_{3} = 1, a_{1} = a_{4} = b = c = 0 in (2.61), (2.62)) if Δ is even, and

ρ_{a}(ε′) = ( \begin{matrix}1 & 1 \\ \frac{1}{4}(Δ − 1) & 0 \end{matrix} ) , ρ_{r}(ε′) = ( \begin{matrix}1 & \frac{1}{4}(Δ − 1) & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & \frac{1}{4}(Δ − 1) & 0 \end{matrix} ) (2.68)

(a_{1} = a_{3} = 1, a_{2} = \frac{1}{4}(Δ−1), a_{4} = b = c = 0 in (2.61),(2.62)) if Δ is odd. Then, under the isomorphism of principally polarised abelian varieties from ℂ^{2}/Λ_{τ} to ℂ^{2}/Λ_{τ′} whose rational representation is M^{t} ((2.52) with D = 1_{2}), which exists by Proposition 2.7.3, ε corresponds to one of ±ε′ if Δ is even, and to one of ε′, [1] − ε′ if Δ is odd.

Proof: This corollary is an obvious one to conjecture, but it is not immediately obvious how to prove it without tediously extending the manipulations of Proposition 2.9.2. The idea of the proof below is to establish that if the corollary is true for one σ ∈ 𝔥_{2} satisfying (2.64) with τ replaced by σ (as τ does), then it is true for all such σ ∈ 𝔥_{2}, in particular for τ, using the nature of the proof of the proposition. To establish the corollary, we will establish it for a σ ∈ 𝔥_{2} with the additional property that the endomorphism ring of ℂ^{2}/Λ_{σ} is the ring of integers of ℚ(\sqrt{Δ}); this property will make establishing it for this σ easier.

Let A, B, C, D, E ∈ ℤ such that (2.65) is (2.64) (for ε). The proof of Proposition 2.9.2 involves constructing M using A, B, C, D, E, but not using τ_{11}, τ_{12}, τ_{22}. This means that, if σ ∈ 𝔥_{2} satisfies (2.65) with τ replaced by σ (as τ does), then M(σ) has the normalised singular relation with discriminant Δ, and if σ′ ∈ 𝔥_{2} has the normalised singular relation with discriminant Δ (as M(τ) does), then M^{−1}(σ′) satisfies (2.65) with τ replaced by σ. Define endomorphisms ε_{σ}, ε′_{σ} of ℂ^{2}/Λ_{σ}, ℂ^{2}/Λ_{σ′} to be ε, ε′ with τ, τ′ replaced by σ, σ′ respectively. Given the nature of the proof of the proposition, as just described, it suffices to prove, for one σ ∈ 𝔥_{2} as just defined, that under the isomorphism of principally polarised abelian varieties from ℂ^{2}/Λ_{σ} to ℂ^{2}/Λ_{σ′} whose rational representation is M^{t} ((2.52) with D = 1_{2}), which exists by Proposition 2.7.3, ε_{σ} corresponds to one of ±ε′_{σ} if Δ is even, and to one of ε′_{σ}, [1] − ε′_{σ} if Δ is odd; indeed, the rational representation of the endomorphism to which ε_{σ} corresponds is M^{t}ρ_{r}(ε_{σ})M^{t}^{−1}, which is independent of σ.

Choose any ς ∈ 𝔥_{2} such that the endomorphism ring of ℂ^{2}/Λ_{ς} is the ring of integers of ℚ(\sqrt{Δ}) (possible by [L-B], Chapter 9 Section 9 Theorem 9.1), not necessarily satisfying (2.65) with τ replaced by ς. By Corollary 2.9.3, there exists M′ ∈ Sp_{4}(ℤ) such that σ′: = M′(ς) has the normalised singular relation with discriminant Δ, and then σ: = M^{−1}M′(ς) satisfies (2.65) with τ replaced by σ. Since the endomorphism ring of ℂ^{2}/Λ_{σ} is the ring of integers of ℚ(\sqrt{Δ}), we deduce that under the isomorphism of principally polarised abelian varieties from ℂ^{2}/Λ_{σ} to ℂ^{2}/Λ_{σ′} whose rational representation is M^{t} ((2.52) with D = 1_{2}), which exists by Proposition 2.7.3, ε_{σ} corresponds to one of ±ε′_{σ} if Δ is even, and to one of ε′_{σ}, [1] − ε′_{σ} if Δ is odd, as required. □

We now present the final bit of theory required to tackle our problem. Let τ ∈ 𝔥_{2} such that ℂ^{2}/Λ_{τ} is complex-analytically isomorphic to the jacobian of C (recall that C is a curve of genus 2 defined over ℂ) via the Abel-Jacobi isomorphism. The theory described so far in this section would allow us to describe the action of a primitive H_{τ}-symmetric endomorphism with the properties described at the beginning of this section on the 2-torsion of ℂ^{2}/Λ_{τ}, but we want to describe the action of a primitive Θ-symmetric ℂ-endomorphism with these properties on the 2-torsion of the jacobian (recall that Θ is a theta divisor on the jacobian). To do this, we need some way of linking the points in the 2-torsion of ℂ^{2}/Λ_{τ} to the points in the 2-torsion of the jacobian. We will do this via theta-functions, referring to [Mum2] and [Mum3]; the theory presented there applies for hyperelliptic curves of arbitrary genus, but we will be concerned just with curves of genus 2.

Define

ϑ(\underline{z}, τ): = ∑_{\underline{n}∈ℤ^{2}} exp(πi\underline{n}^{t}τ\underline{n} + 2πi\underline{n}^{t} · \underline{z})  (2.69)

(here, t denotes transposition and · denotes dot product). Fortunately, ϑ is holomorphic on ℂ^{2} × 𝔥_{2} (see [Mum2], Proposition 1.1, p. 1.118). Define, for \underline{a}, \underline{b} ∈ ℚ^{2},

ϑ_{\underline{a},\underline{b}}(\underline{z}, τ): = exp(πi\underline{a}^{t}τ\underline{a} + 2πi\underline{a}^{t} · (\underline{z} + \underline{b}))ϑ(\underline{z} + τ\underline{a} + \underline{b}, τ)  (2.70)

(we do not use the exponential term in our thesis, but the standard definition of ϑ_{\underline{a},\underline{b}} includes it). ϑ_{\underline{a},\underline{b}} is quasi-periodic with period Λ_{τ} in the sense that if \underline{z} ≡ \underline{w} mod Λ_{τ}, then ϑ_{\underline{a},\underline{b}}(\underline{z}, τ) = 0 ⇔ ϑ_{\underline{a},\underline{b}}(\underline{w}, τ) = 0. See [Mum2], Chapter II Section 1 for further details.

Let P_{0} be a Weierstrass point on C, and let ω_{1}, ω_{2} be a basis for the vector space of holomorphic differentials on C such that (2.54) is satisfied. The theorem which enables theta functions to help tackle our problem is:

Theorem 2.9.6 There exist \underline{a}, \underline{b} ∈ \frac{1}{2}ℤ^{2} with the following property:

ϑ_{\underline{a},\underline{b}}(\underline{z}, τ) = 0

if and only if there exists P ∈ C such that

\underline{z} ≡ ∫_{P_{0}}^{P} \underline{ω} mod Λ_{τ}

(this makes sense due to the quasi-periodicity discussed above).

Proof: See [Mum3], Theorem 5.3, p. 3.80.□

We will now apply Theorem 2.9.6 to give explicit correspondences, via Φ\underline{ω}, between the points of order 2 on the jacobian of C and the points of order 2 on ℂ^{2}/Λ_{τ}. Let Θ_{0}: = {[P − P_{0}] | P ∈ C}, which is a theta divisor on the jacobian of C. Theorem 2.9.6 implies that there exists \underline{a}, \underline{b} ∈ \frac{1}{2}ℤ^{2} such that the set of points in ℂ^{2} which map modulo Λ_{τ} to Φ\underline{ω}(Θ_{0}) is the zero set of ϑ_{\underline{a},\underline{b}}. By (2.70), there is a translate Θ_{ϑ} of Θ_{0} by a point of order dividing 2 on the jacobian of C such that the set of points in ℂ^{2} which map modulo Λ_{τ} to Φ\underline{ω}(Θ_{ϑ}) is the zero set of ϑ.

From (2.69), we see that τ\underline{a} + \underline{b}, where \underline{a}, \underline{b} ∈ \frac{1}{2}ℤ^{2}, is a zero of ϑ if and only if \underline{a}^{t} · \underline{b} ≡ \frac{1}{4} mod \frac{1}{2}. So the points of order 2 on ℂ^{2}/Λ_{τ} which are images under the natural projection ℂ^{2} ↠ ℂ^{2}/Λ_{τ} of zeros of ϑ, i.e. the points of order 2 on Φ\underline{ω}(Θ_{ϑ}), are

\frac{1}{2} ( \begin{matrix}τ_{11} + 1 \\ τ_{12} + 1 \end{matrix} ) + Λ_{τ}, \frac{1}{2} ( \begin{matrix}τ_{12} \\ τ_{22} + 1 \end{matrix} ) + Λ_{τ}, \frac{1}{2} ( \begin{matrix}τ_{11} + τ_{12} + 1 \\ τ_{12} + τ_{22} \end{matrix} ) + Λ_{τ},

\frac{1}{2} ( \begin{matrix}τ_{11} + 1 \\ τ_{12} \end{matrix} ) + Λ_{τ}, \frac{1}{2} ( \begin{matrix}τ_{12} + 1 \\ τ_{22} + 1 \end{matrix} ) + Λ_{τ}, \frac{1}{2} ( \begin{matrix}τ_{11} + τ_{12} \\ τ_{12} + τ_{22} + 1 \end{matrix} ) + Λ_{τ};

we deduce that Θ_{ϑ} does not contain the identity point on the jacobian of C. Recall that Θ_{0} = {[P − P_{0}] | P ∈ C} and that Θ_{ϑ} is a translate of Θ_{0} by a point of order 2 on the jacobian of C. Since Θ_{ϑ} does not contain the identity point on the jacobian of C, there is a labelling P_{ik}, i ∈ {0, 1, 2}, k ∈ {0, 1} of the Weierstrass points on C such that

Φ\underline{ω}([P_{00} − P_{10}]) = \frac{1}{2} ( \begin{matrix}τ_{11} + 1 \\ τ_{12} + 1 \end{matrix} ) + Λ_{τ}, Φ\underline{ω}([P_{10} − P_{20}]) = \frac{1}{2} ( \begin{matrix}τ_{12} \\ τ_{22} + 1 \end{matrix} ) + Λ_{τ},
Φ\underline{ω}([P_{01} − P_{21}]) = \frac{1}{2} ( \begin{matrix}τ_{11} + 1 \\ τ_{12} \end{matrix} ) + Λ_{τ}, Φ\underline{ω}([P_{11} − P_{21}]) = \frac{1}{2} ( \begin{matrix}τ_{12} + 1 \\ τ_{22} + 1 \end{matrix} ) + Λ_{τ};

(2.71) the reason for our choice of labelling will become apparent.

We will finish tackling our problem with the following proposition:

Proposition 2.9.7 Let C be a curve of genus 2 defined over ℂ, let Θ be a theta divisor on the jacobian of C, and let ε be a primitive Θ-symmetric ℂendomorphism of the jacobian with the properties described at the beginning of this section.

Then, there is a labelling P_{ik}, i ∈ {0, 1, 2}, k ∈ {0, 1} of the Weierstrass points on C such that ε acts on the 2-torsion of the jacobian as follows:

1. Suppose that 8 | Δ. Then

ker ε ∩ Jac(C)[2] = {0, [P_{00} − P_{01}], [P_{10} − P_{11}], [P_{20} − P_{21}]} (2.72)

and

ε([P_{00} − P_{10}]) = [P_{20} − P_{21}], ε([P_{10} − P_{20}]) = [P_{00} − P_{01}].

2. Suppose that 4 || Δ. Then

ε([P_{00} − P_{01}]) = [P_{10} − P_{11}], ε([P_{10} − P_{11}]) = [P_{00} − P_{01}],
ε([P_{00} − P_{10}]) = [P_{11} − P_{21}], ε([P_{10} − P_{20}]) = [P_{01} − P_{21}].

3. Suppose that 8 | Δ − 1. Then

ker ε ∩ Jac(C)[2] = {0, [P_{10} − P_{11}], [P_{10} − P_{21}], [P_{11} − P_{21}]}

and

ε([P_{00} − P_{10}]) = [P_{01} − P_{20}], ε([P_{10} − P_{20}]) = [P_{00} − P_{01}].

4. Suppose that 4 || Δ − 1. Then

ε([P_{00} − P_{01}]) = [P_{10} − P_{21}], ε([P_{10} − P_{11}]) = [P_{00} − P_{21}],
ε([P_{00} − P_{10}]) = [P_{01} − P_{11}], ε([P_{10} − P_{20}]) = [P_{01} − P_{21}].

Proof: We will mostly concentrate on the case where 8 | Δ, this case being the most important case as it includes \sqrt{2} multiplication, the other cases being similar.

Let τ ∈ 𝔥_{2} such that ℂ^{2}/Λ_{τ} is complex-analytically isomorphic to the jacobian of C via the Abel-Jacobi isomorphism, and let ω_{1}, ω_{2} be a basis for the vector space of holomorphic differentials on C such that (2.54) is satisfied. By Proposition 2.9.2 and Corollary 2.9.5, we may choose τ such that ε\underline{ω} (as in (2.56)) satisfies

ρ_{a}(ε\underline{ω}) = ± ( \begin{matrix}0 & 1 \\ \frac{1}{4}Δ & 0 \end{matrix} ) .

Let P_{ik}, i ∈ {0, 1, 2}, k ∈ {0, 1} be a labelling of the Weierstrass points on C such that (2.71) holds. Then, we have

ε([P_{00} − P_{10}]) = Φ−1 ; \underline{ω}ε\underline{ω}Φ\underline{ω}([P_{10} − P_{11}])
= Φ−1 ; \underline{ω}ε\underline{ω} (\frac{1}{2} ( \begin{matrix}τ_{11} + 1 \\ τ_{12} + 1 \end{matrix} ) + Λ_{τ})
= Φ−1 ; \underline{ω} (\frac{1}{2} ( \begin{matrix}τ_{12} + 1 \\ \frac{1}{4}Δ(τ_{12} + 1) \end{matrix} ) + Λ_{τ})
= Φ−1 ; \underline{ω} (\frac{1}{2} ( \begin{matrix}τ_{12} + 1 \\ τ_{22} \end{matrix} ) + Λ_{τ})
= [P_{20} − P_{21}],

the penultimate equality coming from the facts that τ has the normalised singular relation with discriminant Δ, and that 8 | Δ. Similar calculations can be used for the points [P_{00} − P_{01}], [P_{10} − P_{11}], [P_{10} − P_{20}], establishing the case where 8 | Δ. The labelling of the Weierstrass points on C was chosen to ensure that (2.72) holds, which is notationally convenient.

For the case where 8 | Δ − 1, we initially let P_{ik}, i ∈ {0, 1, 2}, k ∈ {0, 1} be a labelling of the Weierstrass points on C such that (2.71) holds, but we will need to swap P_{00} and P_{10}, P_{01} and P_{11}, P_{20} and P_{21} if ρ_{a}(ε\underline{ω}) = [1] − ε′ (ε′ being as in Corollary 2.9.5). Likewise for the case where 4 || Δ − 1, except that we may need to swap P_{00} and P_{01}, P_{10} and P_{11}, P_{21} and P_{22}. □

The shapes of ker ε ∩ Jac(C)[2] are appealing both when 8 | Δ and when 8 | Δ − 1; in the latter case, we have

ker ε ∩ Jac(C)[2] = {0, [P_{10} − P_{11}], [P_{10} − P_{21}], [P_{11} − P_{21}]},
ker([1] − ε) ∩ Jac(C)[2] = {0, [P_{00} − P_{01}], [P_{00} − P_{20}], [P_{01} − P_{20}]}.

If C is the curve Y^{2} = F_{6}(X), where F_{6}(X) ∈ K[X], 5 ≤ ∂F_{6} ≤ 6 and K is a subfield of ℂ, and ε is defined over K, then, in the former case, it is possible for F_{6} to be irreducible over K (we will see in Chapter 3 that F_{6} will usually be irreducible if Δ = 8), but, in the latter case, it is impossible for F_{6} to be irreducible over K.

2.10 Elliptic Curves with Complex Multiplication

In this section, we will very briefly discuss elliptic curves with complex multiplication; the theory presented here will be taken from [Shi2] and used in Chapter 5.

Let E be an elliptic curve defined over \overline{ℚ} such that its \overline{ℚ}-endomorphism algebra is isomorphic to an imaginary quadratic subfield K of \overline{ℚ}. Then, its \overline{ℚ}- endomorphism ring is isomorphic to an order 𝔬 in K, not necessarily maximal. By [Shi2], Proposition 4.8, there is a non-trivial ideal 𝔞 of 𝔬 such that E is complex-analytically isomorphic to ℂ/𝔞 (𝔞 is a lattice in ℂ). Let j(Λ) be the j-invariant of the lattice Λ in ℂ, which is the j-invariant of the elliptic curve ℂ/Λ. Then,

[K(j(𝔞)): K] = [ℚ(j(𝔞)): ℚ] = h_{𝔬},  (2.73)

where h_{𝔬} is the class number of 𝔬, by [Shi2], Proposition 4.8, and Theorem 5.7 (ii) and (iii). Since E is complex-analytically isomorphic to ℂ/𝔞, j(E) = j(𝔞), so, from (2.73),

[K(j(E)): K] = [ℚ(j(E)): ℚ] = h_{𝔬}.  (2.74)

If E′ is an elliptic curve defined over \overline{ℚ} such that its \overline{ℚ}-endomorphism ring is isomorphic to 𝔬, then j(E′) is a G_{ℚ}-conjugate of j(E), by Theorem 5.7 (iii). Moreover, it is well-known that j(E′) is a zero of the Hilbert class polynomial for the discriminant of 𝔬, which can be easily computed using Algorithm 7.6.1 in [Coh].

In Chapter 5, orders 𝔬 for which h_{𝔬} divides 2 will be of interest. The table below lists all orders 𝔬 of imaginary quadratic subfields K of \overline{ℚ} such that h_{𝔬} | 2. For each order, it lists the field of definition of j(𝔞) for all non-trivial ideals 𝔞 of 𝔬 (the field of definition is of degree h_{𝔬} over ℚ and is independent of the choice of 𝔞, by [Shi2], Proposition 4.8, and Theorem 5.7 (ii) and (iii)). 𝔬_{K} is defined to be the maximal order of K. The orders with h_{𝔬} = 1 are taken from [Sil2], p. 483, and the orders with h_{𝔬} = 2 are taken from [Cre2], p. 408.

K  [𝔬_{K}: 𝔬] h_{𝔬} Field of  K  [𝔬_{K}: 𝔬] h_{𝔬} Field of
definition  definition
of j(𝔞)  of j(𝔞)
ℚ(\sqrt{−1})  1  1  ℚ  ℚ(\sqrt{−13})  1  2 ℚ(\sqrt{13})
ℚ(\sqrt{−1})  2  1  ℚ  ℚ(\sqrt{−15})  1  2 ℚ(\sqrt{5})
ℚ(\sqrt{−1})  3  2 ℚ(\sqrt{3}) ℚ(\sqrt{−15})  2  2 ℚ(\sqrt{5})
ℚ(\sqrt{−1})  4  2 ℚ(\sqrt{2}) ℚ(\sqrt{−19})  1  1  ℚ
ℚ(\sqrt{−1})  5  2 ℚ(\sqrt{5}) ℚ(\sqrt{−22})  1  2 ℚ(\sqrt{2})
ℚ(\sqrt{−2})  1  1  ℚ  ℚ(\sqrt{−35})  1  2 ℚ(\sqrt{5})
ℚ(\sqrt{−2})  2  2 ℚ(\sqrt{2}) ℚ(\sqrt{−37})  1  2 ℚ(\sqrt{37})
ℚ(\sqrt{−3})  1  1  ℚ  ℚ(\sqrt{−43})  1  1  ℚ
ℚ(\sqrt{−3})  2  1  ℚ  ℚ(\sqrt{−51})  1  2 ℚ(\sqrt{17})
ℚ(\sqrt{−3})  3  1  ℚ  ℚ(\sqrt{−58})  1  2 ℚ(\sqrt{29})
ℚ(\sqrt{−3})  4  2 ℚ(\sqrt{3}) ℚ(\sqrt{−67})  1  1  ℚ
ℚ(\sqrt{−3})  5  2 ℚ(\sqrt{5}) ℚ(\sqrt{−91})  1  2 ℚ(\sqrt{13})
ℚ(\sqrt{−3})  7  2 ℚ(\sqrt{21}) ℚ(\sqrt{−115})  1  2 ℚ(\sqrt{5})
ℚ(\sqrt{−5})  1  2 ℚ(\sqrt{5}) ℚ(\sqrt{−123})  1  2 ℚ(\sqrt{41})
ℚ(\sqrt{−6})  1  2 ℚ(\sqrt{2}) ℚ(\sqrt{−163})  1  1  ℚ
ℚ(\sqrt{−7})  1  1  ℚ  ℚ(\sqrt{−187})  1  2 ℚ(\sqrt{17})
ℚ(\sqrt{−7})  2  1  ℚ  ℚ(\sqrt{−235})  1  2 ℚ(\sqrt{5})
ℚ(\sqrt{−7})  4  2 ℚ(\sqrt{7}) ℚ(\sqrt{−267})  1  2 ℚ(\sqrt{89})
ℚ(\sqrt{−10})  1  2 ℚ(\sqrt{5}) ℚ(\sqrt{−403})  1  2 ℚ(\sqrt{13})
ℚ(\sqrt{−11})  1  1  ℚ  ℚ(\sqrt{−427})  1  2 ℚ(\sqrt{61})
ℚ(\sqrt{−11})  3  2 ℚ(\sqrt{33})

2.11 Modular Forms

In this section, we briefly discuss modular forms, building blocks for Conjecture 1.1.2.

For a positive integer N, we define Γ(N), the principal congruence group of level N, to be the subgroup of SL(2, ℤ) consisting of the matrices which reduce to the identity matrix modulo N. A subgroup Γ of SL(2, ℤ) is called a congruence group of level N if it contains Γ(N) for some N. A congruence group Γ acts on both the upper half plane 𝔥 and the compactified upper half plane 𝔥^{∗}, defined by

𝔥: = {x ∈ ℂ | ℑ(z) > 0}, 𝔥^{∗}: = {x ∈ ℂ | ℑ(z) > 0} ∪ ℚ ∪ i∞,

(ℚ ∪ i∞ is the set of cusps); indeed, the action corresponding to the matrix

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ

is given by the Möbius transformation

x ↦ \frac{ax + b}{cx + d}.

Y_{Γ}, X_{Γ} are defined as follows:

Y_{Γ} = \frac{𝔥}{Γ}, X_{Γ} = \frac{𝔥^{∗}}{Γ} ;

X_{Γ} is the compactification of Y_{Γ}. Y_{Γ}, X_{Γ} are Riemann surfaces which are both complex-analytically isomorphic to curves defined over ℚ; such a curve is called a modular curve. Define J_{Γ} to be the jacobian of X_{Γ}, thought of as a curve defined over ℚ.

The most important congruence groups, Γ_{1}(N), Γ_{0}(N), are defined as follows:

Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) | c ≡ 0 mod N\} ,
Γ_{1}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, ℤ) | c ≡ 0 mod N, a, d ≡ 1 mod N\} ⊂ Γ_{0}(N);

we define

Y_{0}(N) = Y_{Γ_{0}(N)}, X_{0}(N) = X_{Γ_{0}(N)}, J_{0}(N) = J_{Γ_{0}(N)},
Y_{1}(N) = Y_{Γ_{1}(N)}, X_{1}(N) = X_{Γ_{1}(N)}, J_{1}(N) = J_{Γ_{1}(N)}.

Y_{0}(N), Y_{1}(N) classify isomorphism classes of pairs of the form (E, ⟨P ⟩), (E, P) respectively, where E is an elliptic curve (defined over ℂ) and P is a point of order N on E.

A modular form of weight k on Γ, where k is an even positive integer, is a holomorphic function f on 𝔥 satisfying:

• f(αz) = (cz + d)^{k}f(z) for all α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ,

• For all α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ), the function (cz + d)^{−k}f(αz) has a

series expansion ∑_{n=0}^{∞} f_{n}q^{n/h} with q = e^{2πiz}, h a positive integer, called the Fourier series expansion of f at the cusp α^{−1}(i∞).

If, for each cusp, f_{0} = 0, then f is said to be a cusp form of weight k on Γ. We will denote the ℂ-vector space of cusp forms of weight k on Γ by S_{k}(Γ).

From now on, we will take k = 2 and Γ satisfying Γ_{1}(N) ⊆ Γ ⊆ Γ_{0}(N), these cases being important for this thesis. The transformation z ↦ z + 1 belongs to Γ, so a cusp form f on Γ has period 1, and so its Fourier expansion at the cusp i∞ is of the form f(z) = ∑_{n=1}^{∞} f_{n}q^{n}. From now on, when mentioning the Fourier expansion/coefficients of a cusp form f on Γ, we will mean the Fourier expansion/coefficients of f at the cusp i∞. For later use in this section, we define the L-series of f, L(f, s), to be

L(f, s): = ∑_{n=1}^{∞} f_{n}n^{−s}.

A cusp form f on Γ_{0}(N) is said to be of level N. Throughout this section, ϵ: ℤ → ℂ will be an even Dirichlet character modulo N (if p is a prime, then ϵ(p) is zero if and only if p divides N). In this thesis, ϵ will usually be the trivial character; we will discuss general characters here since they will turn out to be relevant to Conjecture 1.1.2. A cusp form f on Γ_{1}(N) is said to be of level N and character ϵ if

f(Mz) = ϵ(d)(cz + d)^{k}f(z) ∀M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N).

Denote the ℂ-vector space of such forms by S_{2}(N, ϵ). If ϵ is trivial, then S_{2}(N, ϵ) is canonically identifiable with S_{2}(N): = S_{2}(Γ_{0}(N)). We have the direct sum decomposition

S_{2}(Γ_{1}(N)) = ⊕_{ϵ′}S_{2}(N, ϵ′),

the sum being taken over all even Dirichlet characters ϵ′: ℤ → ℂ modulo N.

There are certain operators on S_{2}(N, ϵ), called Hecke operators and denoted by T_{n}, where n is a positive integer; their explicit descriptions are not important for this thesis, so are omitted. It is well-known that T_{n} induces a ℚ-endomorphism ξ_{n} of J_{1}(N).

A cusp form f on Γ_{1}(N) is said to be an eigenform if, for all positive integers n, there exists a complex number c_{n} such that T_{n}(f) = c_{n}f. An eigenform f is said to be a normalised eigenform if f_{1} = 1; since f_{n} = c_{n}f_{1} (see [Miy], Lemma 4.5.15), any non-zero eigenform is some normalised eigenform multiplied by some non-zero complex number.

Let f be a normalised eigenform on Γ_{1}(N), and define K_{f} be the subfield of ℂ generated by the Fourier coefficients of f. We have the following theorem, which will turn out to be relevant to Conjecture 1.1.2:

Theorem 2.11.1 There exists an abelian subvariety A_{f} of J_{1}(N), which is called the abelian variety associated to f via the Shimura construction, and an embedding θ_{f} from K_{f} to the ℚ-endomorphism algebra of A_{f} such that:

• The degree of K_{f} over ℚ is the dimension of A_{f}.

• θ_{f}(f_{n}) is the restriction of ξ_{n} to A_{f} for all positive integers n (where ξ_{n} is the ℚ-endomorphism of J_{1}(N) induced by T_{n}).

• A_{f} is defined over ℚ.

Proof: See [Shi2], Theorem 7.14.□

We now define newforms, which will turn out to be important for Conjecture 1.1.2. S_{2}(N, ϵ) is equipped with a Hermitian inner product, called the Petersson inner product and defined by

⟨f, g⟩: = ∫_{𝔥/Γ_{0}(N)} f(x + iy)\overline{g}(x + iy)dxdy.

We define the old subspace of S_{2}(N, ϵ) to be the subspace generated by the set

∪_{M} ∪_{c} {f(cz) | f(z) ∈ S_{2}(M, ϵ)},

where M runs over all positive integers such that the conductor of ϵ divides M (so that ϵ can be defined modulo M) and M properly divides N, and c runs over all positive divisors of \frac{M}{N}, including both 1 and \frac{M}{N}. We define the new subspace of S_{2}(N, ϵ) to be the orthogonal complement of the old subspace w.r.t. the Petersson inner product. A normalised eigenform in the new subspace is called a newform.

Let f be a newform of level N and character ϵ. We have the following proposition, which gives information about K_{f}:

Proposition 2.11.2 If ϵ is trivial, then K_{f} is a totally real field. If ϵ is nontrivial, then K_{f} is a CM field.

Proof: See [Shi3], Proposition 1.3.□

Let f be a normalised eigenform of level N and character ϵ. We now define the L-series of f at a prime p, and give information on the L-series of A_{f} at p when f is a newform; the latter will be of interest in this thesis. First, we have the following theorem, which allows us to define the L-series of f at p:

Theorem 2.11.3 L(f, s) has the Euler product expansion

L(f, s) = ∏_{p}(1 − f_{p}p^{−s} + ϵ(p)p^{1−2s})^{−1}.

Proof: See [Miy], Theorem 4.5.16. □

We define the L-series of f at p, L_{p}(f, s), to be

L_{p}(f, s): = (1 − f_{p}p^{−s} + ϵ(p)p^{1−2s})^{−1};

then, by Theorem 2.11.3, we have

L(f, s) = ∏_{p} L_{p}(f, s),

which is sensible. Before we give the L-series of A_{f} at a prime p of good reduction when f is a newform, we note that Gal(\overline{ℚ}/ℚ) acts on the set of normalised eigenforms on Γ_{1}(N); indeed, if f: = ∑_{n=1}^{∞} f_{n}q^{n} is a normalised eigenform on Γ_{1}(N), and σ ∈ Gal(ℂ/ℚ), then σ(f): = ∑_{n=1}^{∞} σ(f_{n})q^{n} is a normalised eigenform on Γ_{1}(N), since the actions of Gal(\overline{ℚ}/ℚ) and the Hecke operators on S_{2}(Γ_{1}(N)) commute. We have the following theorem, which gives the L-series of A_{f} at a prime p of good reduction:

Theorem 2.11.4 Suppose that f is a newform. For all primes p of good reduction of A_{f}, we have

L_{p}(A_{f}, s) = ∏_{σ∈Gal(K_{f} /ℚ)}L_{p}(σ(f), s)

(this product is finite by Proposition 2.11.2).

Proof: See [Shi3], Theorem 1.4. □

Finally in this section, we present a proposition due to G. Shimura which implies the existence of a \overline{ℚ}-endomorphism η of A_{f} in the case where f is a newform on Γ_{0}(N) and there is a primitive Dirichlet real non-trivial character χ whose conductor r satisfies r^{2} | N such that χ(n)f_{n} = σ(f_{n}) for all positive integers n and some automorphism σ ∈ Gal(K_{f}/ℚ) other than the identity (see [Shi4]). This proposition will be important in the derivation of the family of curves of genus 2 in Chapter 5.

This proposition has been generalised by K. A. Ribet to newforms on Γ_{1}(N) (see [Rib1], Theorem 5.1), but a disadvantage of his theory is that it implies the existence of a \overline{ℚ}-endomorphism η of A_{g}, where

g(z): = ∑_{(n,N)=1} f_{n}q^{n}

is an eigenform on Γ_{1}(M) for some multiple M of N, rather than of A_{f}; A_{g} turns out to be ℚ-isogenous to A_{f}, so Ribet discusses the \overline{ℚ}-endomorphism algebra rather than the \overline{ℚ}-endomorphism ring.

Proposition 2.11.5 Let f be a newform on Γ_{0}(N), and suppose that there is a primitive Dirichlet real non-trivial character χ, the square of its conductor dividing N, such that χ(n)f_{n} = σ(f_{n}) for all positive integers n and some automorphism σ ∈ Gal(K_{f}/ℚ) other than the identity.

Then, there is an endomorphism η of A_{f} such that

1. η^{2} = χ(−1)r[1], where r is the conductor of χ.

2. ηθ_{f}(a) = θ(σ(a))η for all a ∈ K_{f}.

3. η is defined over the quadratic extension of ℚ corresponding to χ, and σ(η) = −η if σ is the non-trivial automorphism of this extension.

Proof: See [Shi4], Proposition 9.□

2.12 Modular Abelian Varieties

In this section, we briefly discuss modular abelian varieties, with Conjecture 1.1.2 in mind.

Let A be an abelian variety defined over ℚ. Then, A is said to be modular if, for some positive integer M, A is ℚ-isogenous to a factor of J_{1}(M), or, equivalently, if there is a non-constant ℚ-morphism J_{1}(M) → A. For example, if f is a newform on Γ_{1}(N) for some positive integer N, then A_{f} is modular. It is well-known that J_{1}(M) is ℚ-isogenous to the product of the varieties A_{f} for which N divides M.

We say that A is of GL_{2}-type if a field K of degree dim(A) over ℚ embeds into the ℚ-endomorphism algebra of A. Now A_{f} is ℚ-simple and of GL_{2}-type; this follows from Theorem 2.11.1 and the fact that the ℚ-endomorphism algebra of A_{f} is isomorphic to K_{f} (see [Rib1], Corollary 4.2).

From the two paragraphs above, we have the following lemma:

Lemma 2.12.1 A is ℚ-simple and modular if and only if A is ℚ-isogenous to A_{f} for some newform f on Γ_{1}(N) for some positive integer N. When this is the case, A is of GL_{2}-type (with embedding field K, say), and K ≅ K_{f}.

An important corollary of Proposition 2.11.2 is

Corollary 2.12.2 Suppose that A is ℚ-simple and modular, and let K be as in Lemma 2.12.1. Then, A is ℚ-isogenous to A_{f} for some newform f on Γ_{0}(N) for some positive integer N if and only if K is totally real.

An important fact, due to deep work by Deligne, Langlands and Carayol, is that the conductor of A_{f} is N^{dim(A_{f})}. So, since the conductors of A_{1}, A_{2} are equal if A_{1}, A_{2} are ℚ-isogenous abelian varieties, we have the following theorem:

Theorem 2.12.3 Suppose that A is ℚ-simple and modular. Then, the conductor of A is N^{dim(A)}.

In this thesis, we will mostly study the case K_{f} = ℚ(\sqrt{2}). Here, we consider briefly the case K_{f} = ℚ(\sqrt{−3}), out of interest. In [C-G-L-R], a curve C of genus 2 in the family

Y^{2} = X(X − 1)(X^{3} − (3 + A)X^{2} + AX + 1), A ∈ ℚ,

has the following properties:

End_{ℚ}(Jac(C)) ⊗ ℚ ≅ ℚ(\sqrt{−3}),

and the \overline{ℚ}-automorphism group of C is isomorphic to C_{2} × D_{6}, so that the jacobian of C splits over \overline{ℚ}. The jacobian of the curve

Y^{2} = X(X − 1)(X^{3} − 4X^{2} + X + 1)

(A = 1 in the family) is shown to be putatively ℚ-isogenous to A_{f} (the empirical evidence being the equality of the L-series at primes of good reduction for both varieties less than 100), where f is a newform on Γ_{1}(52); more precisely,

f ∈ S_{2}(52, ϵ),

where ϵ is the Dirichlet character modulo 52 defined by

ϵ: ℤ → ℂ, ϵ(13) = 0, ϵ(p) = ω^{6/ deg(p)},

where ω = \frac{1−\sqrt{−3}}{2}, p is a prime not equal to 13 and deg(p) is the residual degree of p in ℚ(cos(\frac{2π}{13}))/ℚ.

Conjecture 1.1.2 has been shown by Ribet in [Rib2] to be implied by Serre’s conjecture, namely that any continuous irreducible homomorphism

\overline{ρ}: G_{ℚ} → GL_{2}(\overline{𝔽_{l}})

with det \overline{ρ}(c) = −1 (c being complex conjugation) is modular. It is known that any continuous irreducible homomorphism

\overline{ρ}: G_{ℚ} → GL_{2}(𝔽_{l}),

where l is 2 or 3, is modular, the proofs making use of the facts that GL_{2}(𝔽_{2}) and GL_{2}(𝔽_{3}) are both soluble.

In [S-T], Theorem 4.2, p. 296, N. I. Shepherd-Barron and R. Taylor prove the following theorem:

Theorem 2.12.4 Let K be a totally real field of degree g, and let μ be a prime of O_{K} with norm 5 (resp. 3). Let A be an abelian variety of dimension g defined over ℚ with semistable reduction at 3 and 5 (resp. at 3). Suppose that there is an isomorphism

K →^{∼} End_{ℚ}(A) ⊗ ℚ,

which gives rise to a representation

\overline{ρ}_{A,μ}: G_{ℚ} → GL_{2}(O_{K}/μ),

well-defined up to semisimplification, and suppose further that \overline{ρ}_{A,μ} is absolutely irreducible when restricted to ℚ(\sqrt{5}) (resp. ℚ(\sqrt{−3})). Then, A is modular.

It is well-known that there is an abelian variety A′ ℚ-isogenous to A such that there is an isomorphism

O_{K} →^{∼} End_{ℚ}(A);

\overline{ρ}_{A,μ} is defined to represent the action of G_{ℚ} on A′[μ]. \overline{ρ}_{A,μ} is independent of the choice of A′.

Unfortunately, the theorem cannot be applied if K = ℚ(\sqrt{2}), the case of interest in this thesis, since there is no prime in O_{ℚ(\sqrt{2})} with norm 5 or 3.

Curves with \sqrt{2} Multiplication

3.1 Introduction

Let K be a subfield of ℂ, and let \overline{K} be the subfield of ℂ which is an algebraic closure of K. In this chapter, we present a family of curves of genus 2 defined over K whose jacobians have \sqrt{2} K-multiplication, and show that any curve of genus 2 defined over K such that the K-endomorphism algebra of its jacobian is a field containing a \sqrt{2} multiplication is K-isomorphic to a curve in the family. For the case K = ℚ, we show that the hypothesis that ‘the Kendomorphism algebra of its jacobian is a field containing a \sqrt{2} multiplication’ can be replaced by the weaker hypothesis that ‘its jacobian is K-simple and has a \sqrt{2} K-multiplication’, which means that every curve of genus 2 defined over ℚ whose jacobian has a \sqrt{2} ℚ-multiplication, is ℚ-simple, and is modular, is ℚ-isomorphic to a curve in our \sqrt{2} multiplication family.

We give a sufficient (but not necessary) condition, easy to apply for an infinite family of curves, for the jacobian of a curve of genus 2 defined over K having a \sqrt{2} K-multiplication to be K-simple, and, for the case K = ℚ, we show that the conclusion that ‘the jacobian is K-simple’ can be replaced by the stronger conclusion that ‘the K-endomorphism ring of the jacobian is isomorphic to ℤ[\sqrt{2}]’.

We look at the Galois group of F_{6}(X) ∈ K[X], 5 ≤ ∂F_{6} ≤ 6 with F_{6} having no multiple zeros such that the curve Y^{2} = F_{6}(X) is a curve in our \sqrt{2} multiplication family.

Penultimately, we strengthen the theorem based on our \sqrt{2} multiplication family under the plausible conjecture that the Richelot isogeny is an isogeny of polarised abelian varieties, the polarisations being the canonical polarisation on the jacobian of C multiplied by [2] and the canonical polarisation on either the jacobian of its Richelot dual or the abelian surface A provided by Proposition 2.6.8 according to whether det(g_{uv}) is non-zero or zero.

CHAPTER 3. CURVES WITH \sqrt{2} MULTIPLICATION 66

Finally, we give a family of principally polarised abelian varieties (presented in the complex-analytic sense) which have \sqrt{2} multiplication, such an abelian variety being in general complex-analytically isomorphic to the jacobian of a curve of genus 2 defined over ℂ which, under the assumption above, is not ℂ-isomorphic to any curve in our \sqrt{2} multiplication family (with K = ℂ).

3.2 The \sqrt{2} Multiplication Family of Curves

In this section, we present our \sqrt{2} multiplication family of curves; more precisely, we present the following theorem:

Theorem 3.2.1 Let C be a curve of genus 2 defined over K, and suppose that the K-endomorphism algebra of its jacobian is a field containing a \sqrt{2} multiplication ε.

Then, there is a labelling P_{ij}, i ∈ {0, 1, 2}, j ∈ {0, 1} of the Weierstrass points on C such that

ker ε = {0, [P_{00} − P_{01}], [P_{10} − P_{11}], [P_{20} − P_{21}]},  (3.1)

and that there is a K-isomorphism from C to a curve of the form

Y^{2} = Δ ∏_{i=0}^{2}(X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R),  (3.2)

where

∏_{i=0}^{2}(X − α_{i}) = X^{3} + AX^{2} + BX + C,  (3.3)

R = 4P, B = \frac{Q(P A − Q) + 4P^{2} + 1}{P^{2}}, C = \frac{4(P A − Q)}{P},  (3.4)

and

Δ, P, Q, A ∈ K, Δ ≠ 0 ≠ P,  (3.5)

which sends the set of Weierstrass points {P_{i0}, P_{i1}} to the set of Weierstrass points {(x, 0), (x′, 0)}, where x, x′ satisfy

G_{i}(X): = (X − x)(X − x′) = X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R.  (3.6)

Conversely, the jacobian of any curve C of the form (3.2), where (3.3), (3.4) and (3.5) hold, has a \sqrt{2} K-multiplication ε satisfying (3.1), where P_{i0} = (x, 0), P_{i1} = (x′, 0), with x, x′ satisfying (3.6); more precisely, there is a Kisomorphism from C to its Richelot dual given by

(x, y) ↦ (\frac{2}{x}, \frac{4y}{x^{3}} ) ,  (3.7)

and

ε: = ι^{−1}ρ_{G_{0},G_{1},G_{2}},  (3.8)

(recall that ρ_{G_{0},G_{1},G_{2}} is the Richelot isogeny), where ι is the K-isomorphism from the jacobian of C to that of its Richelot dual given by

ι([(x, y) − (x′, y′)]) = [(\frac{2}{x}, \frac{4y}{x^{3}} ) − (\frac{2}{x′} , \frac{4y′}{x^{′3}} )] .  (3.9)

A. Brumer has proved that the jacobians of the curves in a similar family had a \sqrt{2} multiplication ε; more precisely, he had

R = SP, B = \frac{4Q(AP − Q) + S(4P^{2} + 1)}{4P^{2}}, C = \frac{S(AP − Q)}{P}

instead of

R = 4P, B = \frac{Q(AP − Q) + 4P^{2} + 1}{P^{2}}, C = \frac{4(AP − Q)}{P}

(S ∈ K^{∗}). His proof uses essentially the same method as the proof of the ‘Conversely’ part of Theorem 3.2.1. For a curve in Brumer’s family, ε is defined over K(\sqrt{S}), with σ^{−1}εσ = −ε if σ is the non-trivial automorphism of K(\sqrt{S}) over K. Our \sqrt{2} multiplication family of curves is a slight simplification of the subfamily of Brumer’s family consisting of curves with S ∈ K^{2} (it seems sensible to use curves of the form Y^{2} = F_{6}(X), F_{6}(X) ∈ K[X], ∂F_{6} = 6 such that the X^{5} coefficient of F_{6} is zero, but this leads to more complicated parameterisations). Much more specialised families of curves with \sqrt{2} multiplication was found by J.-F. Mestre in [Mes1] and G. Humbert in [Hum], together with families for other real multiplications; the former using J. Vélu’s explicit formulae for isogenies between elliptic curves (see [Vel]), and the latter using the geometry of the conic.

Finally in this section, we state a special case (K = ℚ) of Theorem 3.2.1, the most important case in this thesis.

Theorem 3.2.2 Theorem 3.2.1 with the hypothesis ‘the K-endomorphism algebra of its jacobian is a field containing a \sqrt{2} multiplication’ replaced by the weaker hypothesis ‘its jacobian is K-simple and has a \sqrt{2} K-multiplication’ is true for K = ℚ.

Proof: Corollary 2.2.11 and Theorem 3.2.1.□

The assumption that the jacobian of C is ℚ-simple is sufficient for our investigation of Conjecture 1.1.2, due to the fact that if A is an abelian variety defined over ℚ which is ℚ-simple and is modular, then the ℚ-endomorphism algebra of A is of degree dim(A) over ℚ, as remarked in Section 2.12. In particular, if C is a curve of genus 2 defined over ℚ and the jacobian of C has a \sqrt{2} ℚmultiplication, is ℚ-simple and is modular, then C is ℚ-isomorphic to a curve in our \sqrt{2} multiplication family.

3.3 A sufficient condition for K-simplicity

In this section, we present the following proposition, which gives a sufficient (but not necessary) condition for the jacobian of a curve of genus 2 defined over K with a \sqrt{2} K-multiplication to be K-simple, easy to apply for an infinite family of curves. Roughly, the condition implies that the Galois group of the extension of K generated by the 2-torsion of the jacobian over K is too big for the jacobian to split over K; the problem of establishing necessary and sufficient conditions for the jacobian to split over K may well be trickier.

Proposition 3.3.1 Let C be a curve of genus 2 defined over K. Suppose that the jacobian of C has a \sqrt{2} K-multiplication ε, and that the kernel of ε is the only subgroup of order 4 of the 2-torsion of the jacobian defined over K. (Warning: that there is only one subgroup of the 2-torsion of the form (3.1) does not always imply that there is only one subgroup of the 2-torsion of order 4).

Then, the jacobian is K-simple.

Proof: Firstly, we will establish the structure of the kernel of ε, which will be used throughout the proof. Indeed, let P_{00}, P_{01}, P_{10}, P_{11}, P_{20}, P_{21} be the Weierstrass points on C, labelled to satisfy (3.1); this is possible, since if the Weierstrass points could be labelled P_{0}, …, P_{5} such that the kernel of ε was {0, [P_{0} − P_{1}], [P_{1} − P_{2}], [P_{2} − P_{0}]}, then it and {0, [P_{3} − P_{4}], [P_{4} − P_{5}], [P_{5} − P_{3}]} would both be subgroups of order 4 of Jac(C)[2] defined over K, contradicting the subgroup hypothesis in the proposition.

Suppose, for a contradiction, that the jacobian is not K-simple. Then, it is Kisogenous to E_{1} ×E_{2}, where E_{1}, E_{2} are elliptic curves defined over K. Since the jacobian has the \sqrt{2} K-multiplication ε, E_{1} is K-isogenous to E_{2}, so E_{1} × E_{2} is K-isogenous to E_{1}^{2}, so the jacobian is K-isogenous to E_{1}^{2}.

Choose a K-isogeny ϕ from the jacobian to E_{1}^{2}. By factoring out by [2] (the multiplication by 2 map on the jacobian) a finite number of times, we may assume, w.l.o.g., that the kernel of ϕ does not contain the 2-torsion, i.e. that the order of the 2-torsion subgroup H of the kernel of ϕ of divides 8. In fact, we may assume, w.l.o.g., that the order of H divides 2; to show this, we have two cases to consider:

1. The order of H is 4.

2. The order of H is 8.

Consider the first case. Since both ϕ and the 2-torsion are defined over K, H is defined over K. So, being of order 4, H is the kernel of ε, since the kernel of ε is the only subgroup of order 4 of the 2-torsion defined over K (by hypothesis).

By factoring out by ε if necessary, we may assume, w.l.o.g., that the kernel of ϕ does not contain the kernel of ε, and that the order of H is 1.

Consider the second case. It suffices to show that H contains a subgroup of order 4 defined over K; then, it follows that the order of H is 2, by a similar argument to that for the first case.

If H contains the kernel of ε, then H contains a subgroup of order 4 defined over K and we are done, so assume that this is not the case. Then, the intersection of the two groups is generated by a rational point [P_{0} − P_{1}] of order 2 (P_{0}, P_{1} being Weierstrass points on C), since their orders are 8 and 4.

Now the six non-trivial points in H not equal to [P_{0} −P_{1}] can be partitioned into three pairs of two points such that the difference between the two points in any pair is [P_{0} −P_{1}]. There is a Weierstrass point P_{2} on C, distinct from P_{0}, P_{1}, such that the two points in one of the pairs are [P_{0} − P_{2}], [P_{1} − P_{2}]; indeed, if this was false, the pairs would be {[P_{2} − P_{3}], [P_{4} − P_{5}]}, {[P_{2} − P_{4}], [P_{3} − P_{5}]}, {[P_{2} − P_{5}], [P_{3} − P_{4}]}, where P_{2}, P_{3}, P_{4}, P_{5} are the Weierstrass points on C not equal to P_{0}, P_{1}, and so H would contain the kernel of ε (see the first paragraph of the proof), which is a contradiction.

If P_{2} is a rational point, then H contains a subgroup of order 4 defined over K and we are done. Otherwise, there is a Weierstrass point P_{3} on C, distinct from P_{0}, P_{1}, P_{2}, such that P_{2} is sent to P_{3} by some element of G_{K}. Since both ϕ and the 2-torsion are defined over K, H is defined over K, and so we have

H = {0, [P_{0} − P_{1}], [P_{0} − P_{2}], [P_{1} − P_{2}], [P_{0} − P_{3}], [P_{1} − P_{3}], [P_{2} − P_{3}], [P_{4} − P_{5}]}.

Since H is defined over K, the set {P_{2}, P_{3}} is fixed by G_{K}. So H contains a subgroup of order 4 defined over K and we are done.

To summarise, we have two cases:

1. H is trivial.

2. The order of H is 2.

Consider the first case. Let ψ be the dual isogeny of ϕ; since the 2-torsion subgroup of the kernel of ϕ is trivial and ϕ is a K-isogeny, ψ induces a bijection defined over K from the 2-torsion of E_{1}^{2} to the 2-torsion of the jacobian. So there are at least two subgroups of order 4 of the 2-torsion of the jacobian defined over K, which is a contradiction; namely the images under ψ of {(x, 0)|x ∈ E_{1}[2]} and {(0, x)|x ∈ E_{1}[2]}.

Consider the second case. Let P, Q be distinct Weierstrass points on C such that H = {0, [P −Q]}. Since [P −Q] is defined over K, the set {P, Q} is defined over K, so that P, Q are either both rational points on C or conjugate quadratic points on C. Let P_{00}, P_{01}, P_{10}, P_{11}, P_{20}, P_{21} be as in the first paragraph of the proof. If [P − Q] is not in the kernel of ε and P, Q are both rational points on C, then the structure of the kernel and the fact that it is defined over K forces the existence of four rational Weierstrass points on C, contradicting the subgroup hypothesis in the proposition. If [P − Q] is not in the kernel of ε and P, Q are conjugate quadratic points on C, then the structure of the kernel and the fact that it is defined over K forces the existence of four Weierstrass points on C defined over the same quadratic extension of K (but not over K) individually and defined over K as a set, contradicting the subgroup hypothesis in the proposition. So [P − Q] is in the kernel of ε; w.l.o.g., let P_{00}, P_{01} be P, Q respectively.

Since the kernel of ε is defined over K and the set {P_{00}, P_{01}} is defined over K, the set {P_{10}, P_{11}, P_{20}, P_{21}} is defined over K. Also,

ϕ([P_{00} − P_{10}]) = ϕ([P_{01} − P_{10}]), ϕ([P_{00} − P_{11}]) = ϕ([P_{01} − P_{11}]),
ϕ([P_{00} − P_{20}]) = ϕ([P_{01} − P_{20}]), ϕ([P_{00} − P_{21}]) = ϕ([P_{01} − P_{21}])

are distinct, since H = {0, [P_{00} − P_{01}]}. So, for all σ ∈ G_{K}, i, i′ ∈ {1, 2} and j, j′ ∈ {0, 1},

σ(P_{ij}) = P_{i′j′} ⇔ σ(ϕ([P_{00} − P_{ij}])) = ϕ([P_{00} − P_{i′j′}]).  (3.10)

Now E_{1} is K-isomorphic to the elliptic curve Y^{2} = F_{3}(X) for some cubic F_{3} defined over K without multiple factors, so K(E_{1}^{2}[2]) is a splitting field of F_{3} over K. We deduce that the Galois group of K(E_{1}^{2}[2]) over K is the Galois group of F_{3} over K, and so is isomorphic to one of C_{1}, C_{2}, C_{3}, S_{3}. However, since the order of the 2-torsion subgroup of the kernel of ϕ is 2, the 2-torsion of E_{1}^{2} contains a subgroup H of order 8 defined over K, namely the image under ϕ of the 2-torsion of the jacobian, so, in fact, the Galois group of K(E_{1}^{2}[2]) over K is isomorphic to one of C_{1}, C_{2}; indeed, {(x, 0)|x ∈ E_{1}[2]} and {(0, x)|x ∈ E_{1}[2]} are subgroups of the 2-torsion of E_{1}^{2} of order 4 defined over K, and the intersection of at least one of these subgroups with H is a subgroup of order 2 defined over K. If the Galois group is isomorphic to C_{1}, then, by (3.10), P_{10}, P_{11}, P_{20}, P_{21} are all rational points on C, contradicting the subgroup hypothesis in the proposition. If the Galois group is isomorphic to C_{2}, then, by (3.10), P_{10}, P_{11}, P_{20}, P_{21} are all defined over the same quadratic extension of K, contradicting the subgroup hypothesis in the proposition.□

Finally in this section, we state a special case (K = ℚ) of Proposition 3.3.1, the most important case in this thesis.

Proposition 3.3.2 Proposition 3.3.1 with the conclusion ‘the jacobian is Ksimple’ replaced by the stronger conclusion ‘the K-endomorphism ring of the jacobian is isomorphic to ℤ[\sqrt{2}]’ is true for K = ℚ.

Proof: Corollary 2.2.11 and Proposition 3.3.1.□

Let C satisfy the hypotheses of Proposition 3.3.2, except that ker ε need not be the only subgroup of order 4 of the 2-torsion of the jacobian defined over ℚ. If it is true that the ℚ-endomorphism ring of the jacobian is isomorphic to ℤ[\sqrt{2}], then Proposition 2.4.6 can usually be used to prove this; indeed, by Corollary 2.2.11 and the fact that the jacobian has a \sqrt{2} ℚ-multiplication, it suffices to prove that the jacobian is not ℚ-isogenous to the square of an elliptic curve defined over ℚ. However, Proposition 3.3.2 can be used for an infinite family of curves, as we shall see in Sections 4.1 and 5.5, whereas Proposition 2.4.6 cannot be used for an infinite family of curves.

3.4 The Galois Group

We now look at the Galois group of a polynomial F_{6} ∈ K[X] such that Y^{2} = F_{6}(X) is a curve of genus 2 in our \sqrt{2} multiplication family.

First, we establish more general notation, which will be useful later on in this thesis as well as in this section. Let k be a perfect field. We define a compatible isomorphism from the Galois group of a polynomial of degree n defined over k to a permutation group on n letters, and from a permutation group on n letters to the Galois group of a polynomial of degree n defined over k, fixing in advance a labelling of the zeros of the polynomial; this notion will be useful for this thesis.

Fix in advance a labelling x_{0}, …, x_{n−1} of the n zeros of a polynomial F_{n} defined over a field k. A group isomorphism ϕ: S →^{∼} G(F_{n}/k) from a permutation group S on n letters to G(F_{n}/k), the Galois group of F_{n} over k, is called a compatible isomorphism if ϕ maps the permutation τ to the automorphism ϕ(τ) defined by ϕ(τ)(x_{i}) = x_{τ(i)}; if a compatible isomorphism exists, then S is said to be compatibly isomorphic to G(F_{n}/k). So a compatible isomorphism respects the correspondence between the letter i in a permutation and the zero x_{i} of F_{n}; the notation for the letters can be regarded as compatible with the notation for the zeros.

For example, consider

F_{3}: = X^{3} − 1, x_{0}: = 1, x_{1}: = \frac{−1 + \sqrt{−3}}{2}, x_{2}: = \frac{−1 − \sqrt{−3}}{2},

S_{1}: = ⟨(12)⟩, S_{2}: = ⟨(02)⟩.

Here, S_{1} is compatibly isomorphic to G(F_{3}/ℚ) and S_{2} is not compatibly isomorphic to G(F_{3}/ℚ).

Let τ ∈ S (so that ϕ(τ) ∈ G(F_{n}/k)), and let y_{i}: = x_{τ(i)} = ϕ(τ)(x_{i}), generally another labelling of the zeros of F_{n}. Then, S is compatibly isomorphic to G(F_{n}/k) w.r.t. the labelling y_{i} if and only if S is compatibly isomorphic to G(F_{n}/k) w.r.t. the labelling x_{i}. Indeed (we only need to consider the ‘if’ direction since the ‘only if’ direction follows by replacing τ by τ^{−1}), if there is a group isomorphism S →^{∼} G(F_{n}/k) mapping the permutation τ′ to the automorphism σ defined by

σ(x_{i}) = x_{τ′(i)},

then there is a group isomorphism S →^{∼} G(F_{n}/k) mapping the permutation τ^{−1}τ′τ to the automorphism σ defined by

σ(y_{i}) = y_{τ^{−1}τ′τ(i)},

with the same automorphism for both isomorphisms, since

σ(x_{i}) = x_{τ′(i)} ⇔ σ(y_{i}) = y_{τ^{−1}τ′τ(i)}.

So the answer to the question as to whether a compatible isomorphism exists or not is invariant under the action of G(F_{n}/k) on labellings, a nice property.

For example, consider

F_{3}: = X^{3} − 1, S = ⟨(12)⟩,

x_{0}: = 1, x_{1}: = \frac{−1 + \sqrt{−3}}{2}, x_{2}: = \frac{−1 − \sqrt{−3}}{2},

y_{0}: = 1, y_{1}: = \frac{−1 − \sqrt{−3}}{2}, y_{2}: = \frac{−1 + \sqrt{−3}}{2},

Here, S is compatibly isomorphic to G(F_{3}/ℚ) w.r.t. both the labelling x_{i} and the labelling y_{i}.

Consider a curve of the form (3.2), and define

F_{6}: = Δ ∏_{i=0}^{2}(X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R),

a sextic defined over K. Let x_{0}, …, x_{5} be a labelling of the zeros of F_{6} such that

(X − x_{i})(X − x_{i+3}) = X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R

and let P_{i} = (x_{i}, 0); then, P_{0}, …, P_{5} is a labelling of the Weierstrass points on C such that

ker ε = {0, [P_{0} − P_{3}], [P_{1} − P_{4}], [P_{2} − P_{5}]}.  (3.11)

Then, there is a compatible isomorphism from G(F_{6}/K) to a subgroup of the permutation group

⟨(0134), (02)(35)⟩ ≅ S_{4} × C_{2},  (3.12)

because ker ε is defined over K and the elements in this permutation group are precisely the elements in S_{6} that permute the sets {P_{0}, P_{3}}, {P_{1}, P_{4}}, {P_{2}, P_{5}}.

More generally, let C be a curve of the form

C: Y^{2} = F_{6}(X), F_{6}(X) ∈ K[X], 5 ≤ ∂F_{6} ≤ 6,

and let Θ be a theta divisor on the jacobian of C. If ε is a primitive Θ-symmetric K-endomorphism of the jacobian of C which satisfies ε^{2} +[b] = 0 for some even integer b such that −b is not a square, then there is a labelling P_{0}, …, P_{5} of the Weierstrass points on C such that (3.11) is satisfied, by Proposition 2.9.7, and there is a compatible isomorphism from G(F_{6}/K) to a subgroup of 3.12, for a similar reason as before.

3.5 The Outline of the Proof of Theorem 3.2.1

In this section, we state several preparatory propositions and then deduce Theorem 3.2.1 from them. The propositions themselves will be proved in Section 3.6.

Proposition 3.5.1 Let C be a curve of the form

C: Y^{2} = F_{6}(X), F_{6}(X) ∈ K[X], 5 ≤ ∂F_{6} ≤ 6.  (3.13)

Suppose that the Weierstrass points on C can be partitioned into three sets of pairs {P_{i0}, P_{i1}} so that the partitioning is preserved by the action of G_{K}.

Then, taking a permutation of the three sets of pairs {P_{i0}, P_{i1}} if necessary, there is a K-isomorphism from C to a curve C̃ of genus 2 of the form

C̃: Y^{2} = Δ̃ ∏_{i=0}^{2}(X^{2} − α̃_{i}X + P̃α̃_{i}^{2} + Q̃α̃_{i} + R̃),  (3.14)

where

∏_{i=0}^{2}(X − α̃_{i}) = X^{3} + ÃX^{2} + B̃X + C̃  (3.15)

and

Δ̃, P̃, Q̃, R̃, Ã, B̃, C̃ ∈ K, Δ̃ ≠ 0, P̃ ≠ \frac{1}{2},  (3.16)

which sends the set of Weierstrass points {P_{i0}, P_{i1}} to the set of Weierstrass points {P̃_{i0}, P̃_{i1}}, where P̃_{i0} = (x, 0), P̃_{i1} = (x′, 0), with x, x′ satisfying

G̃_{i}(X): = (X − x)(X − x′) = X^{2} − α̃_{i}X + P̃α̃_{i}^{2} + Q̃α̃_{i} + R̃.  (3.17)

Proposition 3.5.2 Let C̃ be a curve of the form (3.14), with (3.15) and (3.16) holding. Suppose that the K-endomorphism algebra of the jacobian of C̃ is a field containing a \sqrt{2} multiplication ε, and that

ker ε = {0, [P̃_{00} − P̃_{01}], [P̃_{10} − P̃_{11}], [P̃_{20} − P̃_{21}]},  (3.18)

where the P̃_{ij}’s are defined as in Proposition 3.5.1. Define

H̃_{i}(Z): = G̃′_{i+1}(Z)G̃_{i+2}(Z) − G̃′_{i+2}(Z)G̃_{i+1}(Z),  (3.19)

and let Q̃_{i0}, Q̃_{i1} be the Weierstrass points on the Richelot dual of C̃ such that their z-coordinates are the zeros of H̃_{i}.

Then, P̃ ≠ 0, and there is a K-isomorphism from C̃ to its Richelot dual which, after swapping Q̃_{i0}, Q̃_{i1} if necessary, sends the point P̃_{ij} to the point Q̃_{ij}, and is not of the form (x, y) ↦ (x, Ny) for any N ∈ K^{∗}.

Proposition 3.5.3 Let C̃ be a curve of the form 3.14, with (3.15) and (3.16) holding, and P̃ ≠ 0. Suppose that there is a K-isomorphism from C̃ to its Richelot dual with the properties described in Proposition 3.5.2.

Then, there is a K-isomorphism from C̃ to a curve C of the form (3.2), with (3.3) and (3.5) holding, the Möbius K-transformation corresponding to the isomorphism sending the zeros of G̃_{i} to the zeros of G_{i}, where G_{i} is defined as in (3.6), such that there is a K-isomorphism from C to its Richelot dual of the form (x, y) ↦ (x, y)|_{M,N} for some N ∈ K^{∗} and some Möbius K-transformation M such that

X|_{M} = \frac{M_{11}}{X}  (3.20)

where M_{11} ∈ K^{∗}, which sends the zeros of G_{i} to the zeros of H_{i}, where

H_{i}(Z): = G′_{i+1}(Z)G_{i+2}(Z) − G′_{i+2}(Z)G_{i+1}(Z).  (3.21)

Proposition 3.5.4 Let C be as in Proposition 3.5.3. Then,

R = S^{2}P, B = \frac{4Q(P A − Q) + S^{2}(1 + 4P^{2})}{4P^{2}}, C = \frac{S^{2}(P A − Q)}{P}  (3.22)

for some S ∈ K^{∗}.

Proposition 3.5.5 Let C be a curve of the form

C: Y^{2} = Δ ∏_{i=0}^{2} G_{i}(X), (G_{i}: = ∑_{j=0}^{2} g_{ij}X^{j}, Δ ∈ K^{∗}, g_{ij} ∈ \overline{K}) ,  (3.23)

with det(g_{uv}) ≠ 0, such that the G_{i}’s are defined over K as a set (not necessarily individually), and suppose that there is a K-isomorphism from C to its Richelot dual of the form (x, y) ↦ (x, y)|_{M,N} for some Möbius non-identity K-transformation M which sends the zeros of G_{i} to the zeros of H_{i}, where H_{i} is defined as in (2.21), and N ∈ K^{∗}. Define ι to be the K-isomorphism from the jacobian of C to that of its Richelot dual given by ι([(x, y) − (x′, y′)]) = [(x, y)|_{M,N} − (x′, y′)|_{M,N}].

Then, ε: = ι^{−1}ρ_{G_{0},G_{1},G_{2}} is a \sqrt{2} K-multiplication of the jacobian of C satisfying (3.1), where the x-coordinate of the Weierstrass point P_{ij} on C is a zero of G_{i}.

In particular, any curve of the form described in Theorem 3.2.1 satisfies the above properties, with the G_{i}’s given by (3.6).

Proof: (of Theorem 3.2.1) By the standard Riemann-Roch procedure for remodelling curves of genus 2 (see [C-F], Chapter 1 Section 3), we may remodel over K a curve C of genus 2 defined over K into a model of the form

C: Y^{2} = F_{6}(X), F_{6}(X) ∈ K[X], 5 ≤ ∂F_{6} ≤ 6,

a Weierstrass model, since K is a perfect field whose characteristic is not 2; if the old model is defined over K, the new model is defined over K and the isomorphism from the old model to the new model can be taken to be defined over K. Such remodelling does not affect the K-endomorphism ring of the jacobian of C, so we can restrict ourselves to Weierstrass models defined over K.

Let Θ be a theta divisor on the jacobian. Since the K-endomorphism algebra of the jacobian is a field, it has precisely \sqrt{2} K-multiplications ±ε, so, by Proposition 2.8.1, ε is Θ-symmetric, and so, by Proposition 2.9.7, there is a labelling P_{ij}, i ∈ {0, 1, 2}, j ∈ {0, 1} of the Weierstrass points on C such that (3.1) is satisfied. Proposition 3.5.1 allows a remodelling of C, defined over K, giving a curve C̃ of a certain form (as described in the proposition), the requirement P̃ ≠ \frac{1}{2} being made to avoid complicated remodelling of C̃ at a later stage. Proposition 3.5.2 implies that C̃ is K-isomorphic to its Richelot dual and indicates how the isomorphism acts on the Weierstrass points on C̃. Proposition 3.5.3 allows a remodelling of C̃, defined over K, giving a curve C (not necessarily the same curve as the curve we started with), such that C is Kisomorphic to its Richelot dual with the Möbius K-transformation corresponding to such an isomorphism being of the form x ↦ \frac{M_{11}}{x} for some M_{11} ∈ K^{∗}. Proposition 3.5.4 gives a parameterisation by utilising this type of isomorphism. Finally, we note that the curve given by A = A′, P = P′, Q = Q′, S = S′ is K-isomorphic to the curve given by A = \frac{2A′}{S′} , P = P′, Q = \frac{2Q′}{S′} , S = 2, an isomorphism from the former curve to the latter curve being (x, y) ↦ (\frac{2x}{S′}, \frac{8y}{S^{′3}} ), so we may assume, w.l.o.g., that S = 2, giving the parameterisation used in Theorem 3.2.1.

The final paragraph of Theorem 3.2.1 follows from Proposition 3.5.5 (which is more general than required for the proof of Theorem 3.2.1, this generality being useful for this thesis). □

3.6 The Proofs of the Propositions

Proposition 3.5.1 Let C be a curve of the form (3.13). Suppose that the Weierstrass points on C can be partitioned into three sets of pairs {P_{i0}, P_{i1}} so that the partitioning is preserved by the action of G_{K}.

Then, taking a permutation of the three sets of pairs {P_{i0}, P_{i1}} if necessary, there is a K-isomorphism from C to a curve C̃ of the form (3.14), with (3.15) and (3.16) holding, which sends the set of Weierstrass points {P_{i0}, P_{i1}} to the set of Weierstrass points {P̃_{i0}, P̃_{i1}}, where P̃_{i0} = (x, 0), P̃_{i1} = (x′, 0), with x, x′ satisfying (3.17).

Proof: By applying a Möbius K-transformation if necessary, we may assume w.l.o.g. that ∞ is not a Weierstrass point on C, so that ∂F_{6} = 6, and by applying a K-translation if necessary, we may assume w.l.o.g. that the coefficient of X^{5} in F_{6} is zero, so that ∑_{i=0}^{2} ∑_{j=0}^{1} x_{ij} = 0. Let x_{ij} ∈ \overline{K} such that P_{ij} = (x_{ij}, 0), and define

α_{i}: = x_{i0} + x_{i1}.

Since G_{K} permutes the three sets {P_{i0}, P_{i1}}, G_{K} permutes the α_{i}’s, so that the α_{i}’s are zeros of a cubic defined over K which we may take to be monic; since ∑_{i=0}^{2} ∑_{j=0}^{1} x_{ij} = 0,

∏_{i=0}^{2}(X − α_{i}) = X^{3} + BX + C

for some B, C ∈ K.

There are two cases to consider: the case where the zeros of X^{3}+BX+C are distinct (which is the main case), and the case where the zeros of X^{3} +BX +C are not distinct (which we will reduce to the main case).

Suppose that the zeros of X^{3} + BX + C are distinct. Now σ ∈ G_{K} induces the same permutation on the triple {x_{00}x_{01}, x_{10}x_{11}, x_{20}x_{21}} as on the triple {α_{0}, α_{1}, α_{2}}, so the system of linear equations

( \begin{matrix}1 & α_{0} & α_{0}^{2} \\ 1 & α_{1} & α_{1}^{2} \\ 1 & α_{2} & α_{2}^{2} \end{matrix} ) ( \begin{matrix}P \\ Q \\ R \end{matrix} ) = ( \begin{matrix}x_{00}x_{01} \\ x_{10}x_{11} \\ x_{20}x_{21} \end{matrix} ) ,

which has a unique solution (P, Q, R) since the zeros of X^{3} + BX + C are distinct, is invariant under the action of G_{K}, so P, Q, R ∈ K.

If P ≠ \frac{1}{2}, then, putting

Δ̃: = Δ, P̃: = P, Q̃: = Q, R̃: = R, Ã: = 0, B̃: = B, C̃: = C,

we see that C̃ is of the form (3.14), with (3.15) and (3.16) holding, as required.

If P = \frac{1}{2}, choose S ∈ K such that, for all i ∈ \{0, 1, 2\},

S^{2} − α_{i}S + \frac{1}{2}α_{i}^{2} + Qα_{i} + R ≠ 0,  (3.24)
S^{2} − (α_{i+1} + α_{i+2} + 2Q)S + P α_{i+1}α_{i+2} − R ≠ 0,  (3.25)
∏_{i=0}^{2} (S^{2} − α_{i}S + \frac{1}{2}α_{i}^{2} + Qα_{i} + R)  (3.26)

− ∏_{i=0}^{2} (S^{2} − (α_{i+1} + α_{i+2} + 2Q)S + \frac{1}{2}α_{i+1}α_{i+2} − R) ≠ 0;

indeed, all these polynomials in S are non-zero. The only one which is not obviously non-zero is (3.26); if (3.26) was zero, then we would have Q = 0, B = 4R, C = 0, and so we would have

∏_{i=0}^{2} (X^{2} − α_{i}X + \frac{1}{2}α_{i}^{2} + Qα_{i} + R) = (X^{2} + R)^{3},

which is absurd, since C is the curve

C: Y^{2} = Δ ∏_{i=0}^{2}(X^{2} − α_{i}X + \frac{1}{2}α_{i}^{2} + Qα_{i} + R).

For i ∈ {0, 1, 2}, define

α̃_{i}: = \frac{α_{i} − 2S}{S^{2} − α_{i}S + \frac{1}{2}α_{i}^{2} + Qα_{i} + R},

P̃: = \frac{1}{2} ∏_{i=0}^{2} \frac{S^{2} − α_{i}S + \frac{1}{2}α_{i}^{2} + Qα_{i} + R}{S^{2} − (α_{i+1} + α_{i+2} + 2Q) + \frac{1}{2}α_{i+1}α_{i+2} − R};

from (3.24), it is possible to define the α̃_{i}’s, from (3.25), the α̃_{i}’s are distinct, and from (3.26), P̃ ≠ \frac{1}{2}. After some computation, we see that C is K-isomorphic to a curve C̃ of the form (3.14), with (3.15) and (3.16) holding, as required, a K-isomorphism C̃ →^{∼} C being

(x, y) ↦ (\frac{1}{x} + S, \frac{y}{x^{3}} ) .

Suppose that the zeros of X^{3} + BX + C are not distinct. It suffices to find a Möbius K-transformation M of the form

x ↦ \frac{M_{11}x + M_{12}}{x + M_{22}} ,

not sending any x_{ij} to ∞ (implying that M_{22} ≠ −x_{ij} ∀i ∈ {0, 1, 2}, j ∈ {0, 1}) such that x_{i0}|_{M} + x_{i1}|_{M} are distinct for i ∈ {0, 1, 2}; we may then apply the case where the zeros of X^{3} + BX + C are distinct, establishing the proposition.

Now, w.l.o.g., we may rearrange the subscripts so that α_{0} = α_{1}. Since ∑_{i=0}^{2} ∑_{k=0}^{1} x_{ij} = 0, we have x_{00} + x_{01} = x_{10} + x_{11} = −\frac{1}{2}(x_{20} + x_{21}), so

x_{11} = x_{00} + x_{01} − x_{10},  (3.27)
x_{21} = −2(x_{00} + x_{01}) − x_{20},  (3.28)

Let M_{11}, M_{12}, M_{22} ∈ K satisfy

M_{11}M_{22} − M_{12} ≠ 0  (3.29)

(so that M is non-singular), and

M_{22} ≠ −x_{ij} ∀i ∈ {0, 1, 2}, j ∈ {0, 1}.  (3.30)

After a bit of computation, using (3.27) and (3.28), we see that

x_{00}|_{M} + x_{01}|_{M} − x_{10}|_{M} − x_{11}|_{M}
= \frac{(x_{00} − x_{10})(x_{01} − x_{10})(M_{11}M_{22} − M_{12})(x_{00} + x_{01} + 2M_{22})}{(x_{00} + M_{22})(x_{01} + M_{22})(x_{10} + M_{22})(x_{00} + x_{01} − x_{10} + M_{22})}
= 0
⇒ M_{22} = x_{20} + x_{21}.  (3.31)

Indeed, the x_{ij}’s are distinct, so x_{00} ≠ x_{10} and x_{01} ≠ x_{10}, and M_{11}M_{22}−M_{12} ≠ 0 by (3.29).

After a bit of computation, using (3.27) and (3.28), we see that

x_{00}|_{M} + x_{01}|_{M} − x_{20}|_{M} − x_{21}|_{M}
= (−3M_{22}^{2}(x_{00} + x_{01}) − 2M_{22}(x_{20}^{2} + x_{00}x_{01} + 2x_{20}(x_{00} + x_{01}))  (3.32)
+ (x_{00} + x_{01})(2(x_{00}x_{01} − x_{20}(x_{00} + x_{01})) − x_{20}^{2}))
× (M_{11}M_{22} − M_{12})
÷ ((x_{00} + M_{22})(x_{01} + M_{22})(x_{20} + M_{22})(2(x_{00} + x_{01}) + x_{20} − M_{22})).

If x_{00}+x_{01} ≠ 0, there exist infinitely many M_{11}, M_{12}, M_{22} ∈ K satisfying (3.29), (3.30) and not satisfying (3.31) such that (3.32) is non-zero, since there is a quadratic expression in M_{22} in the numerator of (3.32) which has non-zero M_{22}^{2} coefficient. If x_{00} + x_{01} = 0, then x_{20}^{2} + x_{00}x_{01} + 2x_{20}(x_{00} + x_{01}) ≠ 0, as is establishable using the fact that the x_{ij}’s are distinct, so there exist infinitely many M_{11}, M_{12}, M_{22} ∈ K satisfying (3.29), (3.30) and not satisfying (3.31) such that (3.32) is non-zero, since there is a linear expression in M_{22} in the numerator of (3.32) which has non-zero M_{22} coefficient. So there exist infinitely many M_{11}, M_{12}, M_{22} ∈ K satisfying (3.29), (3.30) and not satisfying (3.31) such that x_{00}|_{M} + x_{01}|_{M} − x_{20}|_{M} − x_{21}|_{M} ≠ 0. Similarly, there exist infinitely many M_{11}, M_{12}, M_{22} ∈ K satisfying (3.29), (3.30) and not satisfying (3.31) such that x_{10}|_{M} +x_{11}|_{M} −x_{20}|_{M} −x_{21}|_{M} ≠ 0. So there exist M_{11}, M_{12}, M_{22} ∈ K satisfying (3.29), (3.30) and such that x_{i0}|_{M} + x_{i1}|_{M} are distinct for i ∈ {0, 1, 2}, as required. □

Proposition 3.5.2 Let C̃ be a curve of the form (3.14), with (3.15) and (3.16) holding. Suppose that the K-endomorphism algebra of the jacobian of C̃ is a field containing a \sqrt{2} multiplication ε, and that (3.18) holds, where the P̃_{ij}’s are defined as in Proposition 3.5.1. Let Q̃_{i0}, Q̃_{i1} be the Weierstrass points on the Richelot dual of C̃ such that their x-coordinates are the zeros of H̃_{i}, where H̃_{i} is defined as in (3.19).

Then, P̃ ≠ 0, and there is a K-isomorphism from C̃ to its Richelot dual which, after swapping Q̃_{i0}, Q̃_{i1} if necessary, sends the point P̃_{ij} to the point Q̃_{ij}, and is not of the form (x, y) ↦ (x, Ny) for any N ∈ K^{∗}.

Proof: First, note that

det(g̃_{uv}) = | \begin{matrix}P̃α̃_{0}^{2} + Q̃α̃_{0} + R̃ & −α̃_{0} & 1 \\ P̃α̃_{1}^{2} + Q̃α̃_{1} + R̃ & −α̃_{1} & 1 \\ P̃α̃_{2}^{2} + Q̃α̃_{2} + R̃ & −α̃_{2} & 1 \end{matrix} |  (3.33)
= P̃(α̃_{0} − α̃_{1})(α̃_{1} − α̃_{2})(α̃_{2} − α̃_{0}),

so, since the α̃_{i}’s are distinct, det(g̃_{uv}) = 0 if and only if P̃ = 0, dividing the proof into two cases; the case P̃ = 0 will turn out to be impossible.

By Corollary 2.7.5, Theorem 2.7.7, Theorem 2.7.8, Theorem 2.8.5 and the fact that the K-endomorphism algebra of the jacobian of C̃ is a field containing ε (by hypothesis), we see that the conditions on the K-endomorphism algebra in Corollary 2.8.2 with F isomorphic to ℚ(\sqrt{2}) are satisfied; this fact will be useful for both cases.

Suppose, for a contradiction, that P̃ = 0. Let A be the abelian surface defined over K given by Proposition 2.6.8; since there is a K-isogeny from the jacobian of C to A whose kernel is ker ε, there is a K-isomorphism from the former to the latter, and so there is a divisor D defined over K on the jacobian which is the union of two abelian subvarieties which are elliptic curves and meet at precisely one point, and ϕ_{D} is a principal polarisation on the jacobian. By Corollary 2.8.2, ϕ_{D} = ϕ_{Θ}([1]+ε)^{2n} for some integer n, which implies that ϕ_{D} = ϕ_{([1]+ε)^{∗}Θ}, which implies that D, (([1]+ε)^{n})^{∗}Θ are algebraically equivalent. But this is impossible, since D is the union of two abelian subvarieties of the jacobian which are elliptic curves and meet at precisely one point, and (([1] + ε)^{n})^{∗}Θ is a curve of genus 2. So P̃ ≠ 0.

Since P̃ ≠ 0, the Richelot dual of C̃ exists. The Richelot isogeny from the jacobian of C̃ to that of its Richelot dual is defined over K (by Corollary 2.6.4), and its kernel is equal to that of ε (by (2.35)), so there is a K-isomorphism ι of (unpolarised) abelian varieties from the former jacobian to the latter jacobian which satisfies

ρ_{G̃_{0},G̃_{1},G̃_{2}} = ιε.  (3.34)

Also, we have

ρ_{G̃_{0},G̃_{1},G̃_{2}}^{∨} = ει^{−1},  (3.35)

by Lemma 2.6.2. By Corollary 2.8.3, there is an integer n such that

ι_{J}: = ι([1] + ε)^{n}  (3.36)

is a K-isomorphism from the canonically polarised former jacobian to the canonically polarised latter jacobian.

We now define the notation given by Theorem 2.3.2. Define f_{1}, f_{2} by

f_{1}(P̃) = [P̃ − P̃_{00}], f_{2}(Q̃) = [Q̃ − Q̃_{00}].  (3.37)

Choosing the positive sign (possible since C̃ is hyperelliptic), let ι_{C} be the Kisomorphism from C̃ to its Richelot dual such that f_{2} ∘ ι_{C}(P̃) = ι_{J} ∘ f_{1}(P̃) + c for some constant c on the jacobian of the Richelot dual of C̃. Equating both sides at P̃_{00}, we see from (3.37) that c = [ι_{C}(P̃_{00}) − Q̃_{00}], which implies that

f_{2} ∘ ι_{C}(P̃) = ι_{J} ∘ f_{1}(P̃) + [ι_{C}(P̃_{00}) − Q̃_{00}].  (3.38)

We will now show, swapping Q̃_{i0}, Q̃_{i1} if necessary, that ι_{C} sends the point P̃_{ij} to the point Q̃_{ij}.

Since the K-endomorphism algebra of the jacobian of C̃ is a field, it has precisely two \sqrt{2} K-multiplications ±ε, so, by Proposition 2.8.1, ε is Θ-symmetric. By Proposition 2.9.7, (2.35), (2.37) and (3.34) (consider the image under (3.34) of [P̃_{10} − P̃_{20}]), we have ι([P̃_{00} − P̃_{01}]) = [Q̃_{00} − Q̃_{01}]. By Proposition 2.9.7 and (3.36), we have ι_{J}([P̃_{00} − P̃_{01}]) = [Q̃_{00} − Q̃_{01}]. Putting P̃ = P̃_{01} in (3.38), we see that [ι_{C}(P̃_{00}) − ι_{C}(P̃_{01}) + Q̃_{00} − Q̃_{01}] = 0, and so that

{ι_{C}(P̃_{00}), ι_{C}(P̃_{01})} = {Q̃_{00}, Q̃_{01}}.  (3.39)

Also, again from (3.38), we see that, if P̃ is a point on C̃, then

ι_{J}([P̃ − P̃_{00}]) = [ι_{C}(P̃) − ι_{C}(P̃_{00})].  (3.40)

By Proposition 2.9.7, (2.36), (2.38) and (3.35) (consider the image under (3.35) of [Q̃_{00} − Q̃_{10}]), ι([P̃_{00} − P̃_{10}]) is one of the four points [Q̃_{0j_{1}} − Q̃_{1j_{2}}] (j_{1}, j_{2} ∈ {0, 1}). By Proposition 2.9.7, (3.36) and (3.40), ι_{J}([P̃_{00} − P̃_{10}]) is one of the two points [ι_{C}(P̃_{00}) − Q̃_{10}], [ι_{C}(P̃_{00}) − Q̃_{11}]. Swapping Q̃_{10}, Q̃_{11} if necessary, we may assume that it is the former. Putting P̃ = P̃_{10} in (3.38), we see that [2ι_{C}(P̃_{00}) − ι_{C}(P̃_{10}) − Q̃_{10}] = 0, and so that

ι_{C}(P̃_{1j}) = Q̃_{1j}.  (3.41)

Similarly, swapping Q̃_{20}, Q̃_{21} if necessary, we see that

ι_{C}(P̃_{2j}) = Q̃_{2j}.  (3.42)

Together, swapping Q̃_{00}, Q̃_{11} if necessary, (3.39), (3.41), (3.42) ensure that ι_{C} sends the point P̃_{ij} to the point Q̃_{ij}, as required.

Finally, we need to show that ι_{C} is not of the form (x, y) ↦ (x, Ny) for any N ∈ K^{∗}. But if it was, then ι_{J}^{−1}ρ_{G_{0},G_{1},G_{2}} would be a \sqrt{−2} K-multiplication of the jacobian of C̃, by Corollary 2.6.6, and so ([1] + ε)^{−n}ε would be a \sqrt{−2} K-multiplication, by (3.34) and (3.36); this is impossible, since ε is a \sqrt{2} Kmultiplication. □

Proposition 3.5.3 Let C̃ be a curve of genus 2 of the form (3.14), with (3.15) and (3.16) holding, and P̃ ≠ 0. Suppose that there is a K-isomorphism from C̃ to its Richelot dual with the properties described in Proposition 3.5.2. Then, there is a K-isomorphism from C̃ to a curve C of the form (3.2), with (3.3) and (3.5) holding, the Möbius K-transformation corresponding to the isomorphism sending the zeros of G̃_{i} to the zeros of G_{i}, where G_{i} is defined as in (3.6), such that there is a K-isomorphism from C to its Richelot dual of the form (x, y) ↦ (x, y)|_{M,N} for some N ∈ K^{∗} and some Möbius K-transformation M defined by (3.20) which sends the zeros of G_{i} to the zeros of H_{i}, where H_{i} is defined as in (3.21).

Proof: Let M̃ be the Möbius K-transformation and Ñ ∈ K^{∗} such that the K-isomorphism from C̃ to its Richelot dual is given by (x, y) ↦ (x, y)|_{M̃,Ñ}, and let M̃_{11}, M̃_{12}, M̃_{21}, M̃_{22} ∈ K such that

X|M̃ = \frac{M̃_{11}X + M̃_{12}}{M̃_{21}X + M̃_{22}} .

By Corollary 2.6.7 and the fact that X|M̃ ≠ X (by Proposition 3.5.2), we have M̃_{22} = −M̃_{11}. There are two cases to consider, the first of which will be proved impossible:

1. M̃_{21} = 0.

2. M̃_{21} ≠ 0.

\underline{Case 1:} Suppose that M̃_{21} = 0; then, without loss of generality, M̃_{11} = 1, and

X|M̃ = −X − M̃_{12}.  (3.43)

We will show that this case is impossible by showing that P̃ = \frac{1}{2}, contradicting (3.16); the requirement that P̃ ≠ \frac{1}{2} in (3.16) was included to deal with this case quickly. Now

H̃_{i}(X) = G̃′_{i+1}(X)G̃_{i+2}(X) − G̃_{i+1}(X)G̃′_{i+2}(X)  (3.44)
= (2X − α̃_{i+1})(X^{2} − α̃_{i+2}X + P̃α̃_{i+2}^{2} + Q̃α̃_{i+2} + R̃)
− (2X − α̃_{i+2})(X^{2} − α̃_{i+1}X + P̃α̃_{i+1}^{2} + Q̃α̃_{i+1} + R̃)
= 2X^{3} − 2α̃_{i+2}X^{2} + 2P̃α̃_{i+2}^{2}X + 2Q̃α̃_{i+2}X + 2R̃X
− α̃_{i+1}X^{2} + α̃_{i+1}α̃_{i+2}X − P̃α̃_{i+1}α̃_{i+2}^{2} − Q̃α̃_{i+1}α̃_{i+2} − R̃α̃_{i+1}
− 2X^{3} + 2α̃_{i+1}X^{2} − 2P̃α̃_{i+1}^{2}X − 2Q̃α̃_{i+1}X − 2R̃X
+ α̃_{i+2}X^{2} − α̃_{i+2}α̃_{i+1}X + P̃α̃_{i+2}α̃_{i+1}^{2} + Q̃α̃_{i+2}α̃_{i+1} + R̃α̃_{i+2}
= (α̃_{i+1} − α̃_{i+2})(X^{2} − 2P̃(α̃_{i+1} + α̃_{i+2})X − 2Q̃X + P̃α̃_{i+1}α̃_{i+2} − R̃)
= (α̃_{i+1} − α̃_{i+2})(X^{2} + 2X(P̃(Ã + α̃_{i}) − Q̃) + P̃(B̃ + Ãα̃_{i} + α̃_{i}^{2}) − R̃).

Since M̃ sends the zeros of G̃_{i} to the zeros of H̃_{i}, we have, by (2.40), (3.43) and (3.44),

X^{2} − α̃_{i}X + P̃α̃_{i}^{2} + Q̃α̃_{i} + R̃  (3.45)
= C_{i}(α̃_{i+1} − α̃_{i+2})  (3.46)
× (−X − M̃_{12})^{2} + 2(−X − M̃_{12})(P̃(Ã + α̃_{i}) − Q̃) + P̃(B̃ + Ãα̃_{i} + α̃_{i}^{2}) − R̃)

for some constant C_{i} ∈ \overline{K}^{∗}, so, in particular, the product of the coefficient of X^{2} in (3.45) and the coefficient of X in (3.46) equals the product of the coefficient of X in (3.45) and the coefficient of X^{2} in (3.46). Using this fact, we see, after some computation, that (2P̃ − 1)α̃_{i} + 2(P̃Ã − Q̃ − M̃_{12}) = 0. This is true for all i ∈ \{0, 1, 2\}, so, separating coefficients of α̃_{i}, we deduce that P̃ = \frac{1}{2}, as required. Note that we cannot use this method to show that X|M̃ ≠ X, since we would require P̃ = −\frac{1}{2} instead of P̃ = \frac{1}{2}, and the proof of Proposition 3.5.1 with P = \frac{1}{2} replaced by P = −\frac{1}{2} breaks down; indeed, (3.26) with \frac{1}{2} replaced by −\frac{1}{2} is zero if A = Q = 0, B = −4R. In fact, by Corollary 2.6.6 and the subsequent remark, the jacobian of a curve of the form (3.2), with (3.3), (3.5) holding and P = −\frac{1}{2}, A = Q = 0, B = −4R is \overline{K}-isogenous to the square of an elliptic curve defined over \overline{K} whose \overline{K}-endomorphism ring is isomorphic to ℤ[\sqrt{−2}].

\underline{Case 2:} Suppose that M̃_{21} ≠ 0; then, without loss of generality, M̃_{21} = 1, and

X|M̃ = \frac{M̃_{11}X + M̃_{12}}{X − M̃_{11}} .  (3.47)

Define

A: = 6M̃_{11} + Ã, B: = 12M̃_{11}^{2} + 4ÃM̃_{11} + B̃, C: = 8M̃_{11}^{3} + 4ÃM̃_{11}^{2} + 2B̃M̃_{11} + C̃,

P: = P̃, Q: = Q̃ + M̃_{11}(4P̃ − 1), R: = R̃ + M̃_{11}(2Q̃ + M̃_{11}(4P̃ − 1)), Δ: = Δ̃, and define C to be the curve (3.2), where (3.3) holds. After a bit of computation, we see that there is a K-isomorphism from C̃ to C given by (x, y) ↦ (x − M̃_{11}, y), and that the Möbius K-transformation x ↦ x − M̃_{11} sends the zeros of G̃_{i} to the zeros of G_{i}. By Lemma 2.6.5, we see that there is a Kisomorphism from the Richelot dual of C̃ to the Richelot dual of C, given by (z, t) ↦ (z −M̃_{11}, t), and that the Möbius K-transformation z ↦ z −M̃_{11} sends the zeros of H̃_{i} to the zeros of H_{i}. By considering these two K-isomorphisms and the K-isomorphism from C̃ to its Richelot dual from the proposition (recall that the Möbius K-transformation corresponding to it is given by (3.47) and sends the zeros of G̃_{i} to the zeros of H̃_{i}), we see that M_{11}: = M̃_{11}^{2}+M̃_{12}, N: = M̃ gives the required K-isomorphism from C to its Richelot dual. □

Proposition 3.5.4 Let C be as in Proposition 3.5.3. Then, (3.22) holds for some S ∈ K^{∗}.

Proof: In the course of the proof, we will apply some of the equations derived in the proof of Lemma 2.6.5. In order to do this, we first define

G̃_{i} = H_{i},  (3.48)
Δ̃ = \frac{Δ}{det(g_{uv})},  (3.49)

so that C̃ (as in Lemma 2.6.5) is the Richelot dual of C.

Let M be the Möbius K-transformation and N ∈ K^{∗} such that the Kisomorphism from C to its Richelot dual from Proposition 3.5.3 is given by (x, y) ↦ (x, y)|_{M,N}, and let M_{11} ∈ K such that (3.20) holds. Now

H_{i}(X) = (α_{i+1} −α_{i+2})(X^{2} +2X(P (A+α_{i})−Q)+P (B +Aα_{i} +α_{i}^{2})−R) (3.50)

(to see this, drop the tildes from (3.44)). Since M sends the zeros of G_{i} to the zeros of H_{i}, we have, by (2.40) (using (3.48)), (3.20) and (3.50),

X^{2} − α_{i}X + P α_{i}^{2} + Qα_{i} + R  (3.51)
= C_{i}X^{2}(α_{i+1} − α_{i+2})  (3.52)
× ((\frac{M_{11}}{X} )^{2} + 2 (\frac{M_{11}}{X} ) (P (A + α_{i}) − Q) + P (B + Aα_{i} + α_{i}^{2}) − R)

for some constant C_{i} ∈ \overline{K}^{∗}. Equating coefficients of X^{2} in (3.51) and (3.52), we see that

C_{i} = \frac{1}{(α_{i+1} − α_{i+2})(P (α_{i}^{2} + Aα_{i} + B) − R)}.  (3.53)

Also,

det(g_{uv}) = P (α_{0} − α_{1})(α_{1} − α_{2})(α_{2} − α_{0})  (3.54)

(to see this, drop the tildes from (3.33)). By (2.41) (using (3.49)), (3.53) and (3.54), we have

N^{2} = \frac{P^{3}C^{2} − P^{2}RAC + P R^{2}B − R^{3}}{P} .  (3.55)

Denote the coefficient of X^{k} in (3.51) by Φ_{1ki} and the coefficient of X^{k} in (3.52) divided by C_{i}(α_{i+1} −α_{i+2}) by Φ_{2ki} (C_{i}(α_{i+1} −α_{i+2}) ≠ 0 as C_{i} ≠ 0 and α_{i+1}, α_{i+2} are distinct), i.e.

Φ_{10i} = P α_{i}^{2} + Qα_{i} + R, Φ_{11i} = −α_{i}, Φ_{12i} = 1,  (3.56)

Φ_{20i} = M_{11}^{2}, Φ_{21i} = 2M_{11}(P (α_{i} + A) − Q), Φ_{22i} = P (α_{i}^{2} + Aα_{i} + B) − R.

Then, the equation linking (3.51) with (3.52) implies that Φ_{1ki}Φ_{2k′i} = Φ_{1k′i}Φ_{2ki} for all k, k′. Each Φ_{1ki}Φ_{2k′i} − Φ_{1k′i}Φ_{2ki} is a quadratic polynomial in α_{i}, with coefficients in K. Since Φ_{1ki}Φ_{2k′i} = Φ_{1k′i}Φ_{2ki} for each i, the coefficients are zero. We will take i = 0 for our computations, and write Φ_{jk} for Φ_{jk0}.

Now, from (3.56), Φ_{11}Φ_{22}−Φ_{12}Φ_{21} = (R−2M_{11}P)α_{0}+P C −2M_{11}(P A−Q), so, separating the coefficients of α_{0},

R = 2M_{11}P,  (3.57)
C = \frac{2M_{11}(P A − Q)}{P}.  (3.58)

Now, from (3.56), Φ_{10}Φ_{22}−Φ_{12}Φ_{10} = M_{11}(2P^{2}B−2Q(P A−Q)−M_{11}(4P^{2}+1)), so, since M_{11} ≠ 0 (otherwise M would be singular),

B = \frac{2Q(P A − Q) + M_{11}(4P^{2} + 1)}{2P^{2}}.  (3.59)

From (3.55), (3.57), (3.58) and (3.59), we see that N^{2} = 2M_{11}^{3}, so, since N ∈ K^{∗}, M_{11} = \frac{S^{2}}{2} for some S ∈ K^{∗}. From (3.57), (3.58) and (3.59), we deduce that (3.22) holds, as required.□

Proposition 3.5.5 Let C be a curve of the form (3.23), with det(g_{uv}) ≠ 0, such that the G_{i}’s are defined over K as a set (not necessarily individually), and suppose that there is a K-isomorphism from C to its Richelot dual of the form (x, y) ↦ (x, y)|_{M,N} for some non-identity Möbius K-transformation M which sends the zeros of G_{i} to the zeros of H_{i}, where H_{i} is defined as in (2.21), and N ∈ K^{∗}. Define ι to be the K-isomorphism from the jacobian of C to that of its Richelot dual given by ι([(x, y) − (x′, y′)]) = [(x, y)|_{M,N} − (x′, y′)|_{M,N}].

Then, ε: = ι^{−1}ρ_{G_{0},G_{1},G_{2}} is a \sqrt{2} K-multiplication of the jacobian of C satisfying (3.1), where the x-coordinate of the Weierstrass point P_{ij} on C is a zero of G_{i}.

In particular, any curve of the form described in Theorem 3.2.1 satisfies the above properties, with the G_{i}’s given by (3.6).

Proof: Let C be a curve with the properties described in the first paragraph of Proposition 3.5.5. Firstly, we will show that we may assume, w.l.o.g., that C is of the form described in Theorem 3.2.1, to ease computations later. By Proposition 3.5.1, Proposition 3.5.3, Proposition 3.5.4 and Lemma 2.6.5, we may assume, w.l.o.g., that C is as in Proposition 3.5.4 for some S ∈ K^{∗}. Also, the curve given by A = A′, P = P′, Q = Q′, S = S′ is K-isomorphic to the curve given by A = \frac{2A′}{S′} , P = P′, Q = \frac{2Q′}{S′} , S = 2 (A, P, Q, S as in Proposition 3.5.4), a K-isomorphism from the former curve to the latter curve being (x, y) ↦ (\frac{2x}{S′}, \frac{8y}{S^{′3}} ), so we may assume, w.l.o.g., that S = 2, i.e. that C is of the form as described in Theorem 3.2.1.

Let C be a curve of the form as described in Theorem 3.2.1, and let the G_{i}’s be given by (3.6). To finish the proof of Proposition 3.5.5, we need to show that C has the properties described in the first paragraph of Proposition 3.5.5, and that the jacobian of C has a \sqrt{2} K-multiplication satisfying (3.1), where P_{i0} = (x, 0), P_{i1} = (x′, 0), with x, x′ satisfying (3.17).

Since the G_{i}’s are given by (3.6), the G_{i}’s are defined over K as a set. After some computation, we see that there is a K-isomorphism from C to its Richelot dual given by (3.7), as expected from Proposition 3.5.4 (recall that M_{11} = \frac{S^{2}}{2} and N^{2} = 2M_{11}^{3}, so M_{11} = 2 and N = ±4; choose N to be 4). So C has the properties described in the first paragraph of Proposition 3.5.5.

Let ι be the K-isomorphism from the jacobian of C to that of its Richelot dual given by (3.9), and let ι be the K-endomorphism of the jacobian given by (3.8). It satisfies (3.1), by Corollary 2.6.4 and (2.35); we just need to show that it is a \sqrt{2} multiplication. Our procedure for doing this is similar to our procedure for the proof of Lemma 2.6.2, which linked the Richelot isogeny and the dual Richelot isogeny.

Firstly, we will derive ε explicitly. Since M sends the zeros of G_{i} to the zeros of H_{i}, we have

G_{i}(X) = C_{i}X^{2}H_{i} (\frac{2}{X} )  (3.60)

for some constant C_{i} ∈ \overline{K}^{∗}. From (2.22), (2.24), (2.25), (3.9) and (3.60), we see that

ε([(x, y) − (x′, y′)]) = [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})],  (3.61)

where z_{1}, z_{2} are the zeros w.r.t. z of

C_{2}G_{1}(x)G_{1}(z) + C_{1}G_{2}(x)G_{2}(z) = 0  (3.62)

and t_{i} satisfies

4C_{1}yt_{i} = ΔG_{1}(x)G_{1}(z_{i})(z_{i}x − 2),  (3.63)

z′_{i}, t′_{i} having analogous definitions. Specifically, the G_{i}’s,C_{i}’s are given by (3.6), (3.53) respectively (recall that (3.4) holds). After some computation, we see, using these facts, that z_{1}, z_{2} are the zeros w.r.t. z of

C_{3}(C_{2}G_{1}(x)G_{1}(z) + C_{1}G_{2}(x)G_{2}(z))  (3.64)
= P^{2}(2α_{0}^{2} + Aα_{0} + 4)x^{2}z^{2}
− 4(2P^{2}α_{0}^{2} + P (P A + Q)α_{0} + 4P^{2} + Q(P A − Q) + 1)xz
+ 4(2P^{2}α_{0}^{2} − P (Q − 2P A)α_{0} + Q(P A − Q) + 1)
− 4P (x^{2} + z^{2} + ((P (α_{0} + A) − Q)xz − α_{0})(x + z))
= 0,

C_{3} ∈ \overline{K}^{∗} being a constant.

Secondly, we define

M_{11}: = Q − P (α_{0} + A), M_{12}: = R − P (α_{0}^{2} + Aα_{0} + B), M_{21}: = 1, M_{22}: = −M_{11},

and M to be the Möbius \overline{K}-transformation as in Section 2.3; the application of this Möbius transformation will soon become apparent. det(M) is non-zero, since, after some computation, we see that det(M) divides the discriminant of G_{1}(x)G_{2}(x), computed as a quartic, which is non-zero.

Let z_{1}, z_{2}, z′_{1}, z′_{2}, t_{1}, t_{2}, t′_{1}, t′_{2} be as defined in (3.62) and (3.63), so that (3.61) holds. After some computation, using (3.64), we see that

ε([(z_{1}, t_{1}) − (z′_{1}, t′_{1})]) = [(u_{1}, v_{1}) + (u_{3}, v_{3}) − (u′_{1}, v′_{1}) − (u′_{3}, v′_{3})], (3.65)
ε([(z_{2}, t_{2}) − (z′_{2}, t′_{2})]) = [(u_{2}, v_{2}) + (u_{4}, v_{4}) − (u′_{2}, v′_{2}) − (u′_{4}, v′_{4})], (3.66)

where u_{1} = u_{2} = x, u′_{1} = u′_{2} = x′ and u_{3} = u_{4} = x|_{M}, u′_{3} = u′_{4} = x′|_{M}, which implies that

ε^{2}([(x, y) − (x′, y′)])
= [(u_{1}, v_{1}) + (u_{2}, v_{2}) + (u_{3}, v_{3}) + (u_{4}, v_{4})
− (u′_{1}, v′_{1}) − (u′_{2}, v′_{2}) − (u′_{3}, v′_{3}) − (u′_{4}, v′_{4})].

To finish the proof, we just need to show that v_{1} = v_{2} = y, v′_{1} = v′_{2} = y′, v_{3} = −v_{4}, v′_{3} = −v′_{4}.

Consider v_{1}, v_{2}, v′_{1}, v′_{2}; we require v_{1} = v_{2} = y, v′_{1} = v′_{2} = y′. We will concentrate on showing that v_{1} = y; the other three equalities will then follow from similar arguments. From (3.61), (3.65) and (3.66), we have

4C_{1}yt_{1} = ΔG_{1}(x)G_{1}(z_{1})(z_{1}x − 2), 4C_{1}t_{1}v_{1} = ΔG_{1}(z_{1})G_{1}(u_{1})(u_{1}z_{1} − 2);

so, since u_{1} = x, we have v_{1} = y, as required.

Consider v_{3}, v_{4}, v′_{3}, v′_{4}; we require v_{3} = −v_{4}, v′_{3} = −v′_{4}. We will concentrate on showing that v_{3} = −v_{4}; the other equality will then follow from a similar argument. From (3.61), (3.65) and (3.66), we have

4C_{1}yt_{1} = ΔG_{1}(x)G_{1}(z_{1})(z_{1}x − 2), 4C_{1}t_{1}v_{3} = ΔG_{1}(u_{3})G_{1}(z_{1})(u_{3}z_{1} − 2),

4C_{1}yt_{2} = ΔG_{1}(x)G_{1}(z_{2})(z_{2}x − 2), 4C_{1}t_{2}v_{4} = ΔG_{1}(u_{4})G_{1}(z_{2})(u_{4}z_{2} − 2),

and, after some computation, we have

(z_{1}x − 2)(x|_{M}z_{2} − 2) + (z_{2}x − 2)(x|_{M}z_{1} − 2) = 0;

so, since u_{3} = u_{4} = x|_{M}, we have v_{3} = −v_{4}, as required. □

3.7 A Likely Strengthening of Theorem 3.2.1

This section revolves around the following plausible conjecture:

Conjecture 3.7.1 Let C be a curve of genus 2 of the form (2.19) defined over K (with k = K). The Richelot isogeny is a K-isogeny of polarised abelian varieties, the polarisations being the canonical polarisation on the jacobian of C multiplied by [2] and the canonical polarisation on either the jacobian of its Richelot dual or the abelian surface A provided by Proposition 2.6.8 according to whether det(g_{uv}) is non-zero or zero.

The plausibility of this conjecture is due to the fact that if C is a general curve of the form (2.19) defined over K (here, ‘general’ means that the K-endomorphism ring of the jacobian of C is isomorphic to ℤ), then the Richelot isogeny is forced to be a K-isogeny of polarised abelian varieties, by Theorem 2.2.2. It is still plausible for the jacobians in Theorem 3.2.1, for which it will be applied, even though ℤ[\sqrt{2}] embeds into the K-endomorphism ring of such a jacobian.

In this section, we will prove that Conjecture 3.7.1 implies that:

1. The hypothesis in Theorem 3.2.1 that ‘the K-endomorphism algebra of its jacobian is a field containing a \sqrt{2} multiplication ε’ can be replaced by the weaker hypothesis that ‘the K-endomorphism algebra of its jacobian contains a Θ-symmetric \sqrt{2} multiplication ε, where Θ is a theta divisor on it’.

2. In the ‘Conversely’ part of Theorem 3.2.1, ‘\sqrt{2} K-multiplication’ can be replaced by ‘Θ-symmetric \sqrt{2} K-multiplication (where Θ is a theta divisor on the jacobian)’.

This likely strengthening of Theorem 3.2.1 implies that our \sqrt{2} multiplication family is probably a universal family of curves defined over K whose jacobians have \sqrt{2} K-multiplication fixed by the Rosati involution w.r.t. the canonical polarisation; this ties in well with the universal complex-analytic family of principally polarised abelian surfaces with \sqrt{2} multiplication fixed by the Rosati involution w.r.t. the principal polarisation, given by Lemma 2.9.1.

Firstly, we consider the first strengthening of Theorem 3.2.1 described above.

In the proof of Theorem 3.2.1 described in Section 3.5, the former hypothesis is only used in two places; firstly, at the start, when it is used to establish that ε is Θ-symmetric, i.e. the latter hypothesis, and in Proposition 3.5.2. Under Conjecture 3.7.1, the following proposition, replacing Proposition 3.5.2, will allow us to replace the former hypothesis by the latter hypothesis:

Proposition 3.7.2 The version of Proposition 3.5.2 with the hypothesis that ‘the K-endomorphism algebra of the jacobian of C̃ is a field containing a \sqrt{2} multiplication ε’ replaced by the weaker hypothesis that ‘the K-endomorphism algebra of the jacobian of C̃ contains a Θ-symmetric \sqrt{2} multiplication ε’, where Θ is a theta divisor on it’ is true under Conjecture 3.7.1.

Proof: The proof of this proposition is, as one might expect, a simplified version of the proof of Proposition 3.5.2, to which we will refer. Let A be either the jacobian of the Richelot dual or the abelian surface provided by Proposition 2.6.8, and let Θ_{ρ} be either a theta divisor on A or the divisor provided by Proposition 2.6.8 on A, according to whether P̃ is non-zero or zero. Conjecture 3.7.1 can be stated symbolically as follows:

[2]ϕ_{Θ} = ρ̂_{G̃_{0},G̃_{1},G̃_{2}}ϕ_{Θ_{ρ}}ρ_{G̃_{0},G̃_{1},G̃_{2}}.  (3.67)

Since ε is Θ-symmetric, we have

[2]ϕ_{Θ} = ε̂ϕ_{Θ}ε.  (3.68)

There is a K-isomorphism ι of (unpolarised) abelian varieties from the jacobian of C to A such that (3.34) holds. In fact, ι is a K-isomorphism of polarised abelian varieties, with

ϕ_{Θ} = ι̂ϕ_{Θ_{ρ}}ι,  (3.69)

by (3.34), (3.67) and (3.68). By (3.69), Θ, ι^{∗}Θ_{ρ} are algebraically equivalent, and so P̃ ≠ 0 (cf. the proof of Proposition 3.5.2). Finally, define n: = 0; the rest of the proof of Proposition 3.7.2 is the part of the proof of Proposition 3.5.2 from (3.36) onwards.□

Finally, we consider the second strengthening of Theorem 3.2.1 described above. Let ι be defined as in (3.9); then, ι is an isomorphism of polarised abelian varieties, with (3.69) holding. From (3.8), (3.67) and (3.69), we see that (3.68) holds, so that ε is Θ-symmetric, as required.

3.8 A Departure from Θ-Symmetricity

In this section, we give a family of principally polarised abelian varieties, presented in the complex-analytic sense, which have \sqrt{2} multiplication, such an abelian variety being in general complex-analytically isomorphic to the jacobian of a curve of genus 2 defined over ℂ which, under Conjecture 3.7.1, is not ℂ-isomorphic to any curve in our \sqrt{2} multiplication family (with K = ℂ). So, under Conjecture 3.7.1, Theorem 3.2.1 cannot be strengthened to cover all \sqrt{2} multiplications.

In [Has], K. Hashimoto gives families of principally polarised abelian varieties, presented in the complex-analytic sense, whose endomorphism rings are generally orders in totally indefinite quaternion algebras of degree 4 over ℚ (both division algebras and non-division algebras). We will present a family, such that, generally, a surface ℂ^{2}/Λ_{τ} in it (Λ_{τ} as in (2.54)) is quaternionic, simple and has at least one \sqrt{2} multiplication but no H_{τ}-symmetric ones; such a surface will be complex-analytically isomorphic to the jacobian of a curve C of genus 2 defined over ℂ which has at least one \sqrt{2} ℂ-multiplication but no Θ-symmetric ones, where Θ is a theta divisor on the jacobian. Under Conjecture 3.7.1, C will not be ℂ-isomorphic to any curve in our \sqrt{2} multiplication family (with K = ℂ). Note that a curve C in the family derived in Chapter 5 will be ℚ-isomorphic to a curve in the family derived in this chapter; we expect and will show that the jacobian of C has a Θ-symmetric \sqrt{2} ℚ-multiplication.

More precisely, we have the following lemma:

Lemma 3.8.1 Define B to be the division quaternion algebra

B: = (\frac{−26, 5}{ℚ}) = ℚ ⊕ ℚi ⊕ ℚj ⊕ ℚij,

where i, j satisfy

i^{2} = −26, j^{2} = 5, ij = −ji,  (3.70)

† to be the positive anti-involution on B given by

†(i) = −i, †(j) = j, †(ij) = ij,  (3.71)

and O to be the maximal order in B given by

O: = ℤ ⊕ ℤ\frac{1 + j}{2} ⊕ ℤ\frac{i(1 + j)}{2} ⊕ ℤ\frac{(52 + i)j}{5} .  (3.72)

If τ is the matrix

\frac{1}{5z} ( \begin{matrix}\frac{1}{2}(\sqrt{5} − 3) + 104z + 13(3 + \sqrt{5})z^{2} & \frac{1}{2}(1 − \sqrt{5}) − 208z − 13(1 + \sqrt{5})z^{2} \\ \frac{1}{2}(1 − \sqrt{5}) − 208z − 13(1 + \sqrt{5})z^{2} & −1 − 104z + 26z^{2} \end{matrix} )

and z ∈ 𝔥_{2}, then there is an embedding from O to the endomorphism ring of ℂ^{2}/Λ_{τ}, under which † corresponds to †_{H_{τ}}.

Proof: See [Has], Theorem 3.5 (with D = 26, a = 2, p = 5). O is considered in Example 5.5.□

Let τ be as above such that the embedding is an isomorphism, so that ℂ^{2}/Λ_{τ} is quaternionic and simple. To show that it has at least one \sqrt{2} multiplication but no H_{τ}-symmetric ones, we need to show that O has at least one element whose square is 2 but none which are fixed by †.

An element of O whose square is 2 is

−22 + 44\frac{1 + j}{2} + \frac{i(1 + j)}{2} − 2\frac{(52 + i)j}{5} = \frac{1}{2}i + \frac{6}{5}j − \frac{1}{10}ij

(see (3.72) to ensure that this is an element of O); indeed,

(\frac{1}{2}i + \frac{6}{5}j − \frac{1}{10}ij)^{2} = \frac{1}{4}(−26) + \frac{36}{25}(5) + \frac{1}{100}(130) = 2,

the first equality following from (3.70). The elements of O which are fixed by † are precisely the elements of the form

λ + μ\frac{1 + j}{2} + ν \frac{(52 + i)j}{5} = (λ + \frac{μ}{2}) + (\frac{μ}{2} + \frac{52ν}{5} ) j + \frac{ν}{5}ij

with λ, μ, ν ∈ ℤ (see (3.72) to ensure that these are elements of O), by (3.71). For the square of such an element to be 2, we need μ = −2λ, and then we have

((−λ + \frac{52ν}{5} ) j + \frac{ν}{5}ij)^{2} = (−λ + \frac{52ν}{5} )^{2} (5) + \frac{ν^{2}}{25}(130)

= 5 (λ − \frac{52}{5}ν)^{2} + \frac{26}{5}ν^{2} > 2,

the first equality following from (3.70), so O has at least one element whose square is 2 but none which are fixed by †.

Isogenies of Degree 2 Between Curves

4.1 Introduction

Let K be a subfield of ℂ, and let \overline{K} be the subfield of ℂ which is an algebraic closure of K. In this chapter, we present K-isogenies of degree 2 between jacobians of curves of genus 2 defined over K with \sqrt{2} K-multiplication. The curves C we consider are most of the curves in our \sqrt{2} multiplication family (as in Theorem 3.2.1) with the property that the kernel of ε has a non-trivial rational point; we will show, in Theorem 4.3.1, that there is a K-isogeny of degree 2 from the jacobian of C to the jacobian of another curve of genus 2 defined over K whose kernel is generated by this point.

The motivation for discovering Theorem 4.3.1 is as follows. In Section A.1, we list a table of curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor. During the compilation of the table, one notices that, for λ ∈ ℚ such that 4λ + 23 ≠ 0 and Δ ∈ ℚ^{∗}, the jacobians of the curves

C_{1}: Y^{2} = F_{1}(X): = Δ(X^{2} − X + 1)(X^{4} + X^{3} + λX^{2} − 3X + 9)  (4.1)
= Δ(X^{2} − X + 1) ∏_{i=1}^{2}(X^{2} − α_{i}X − 3),
C_{2}: T^{2} = F_{2}(Z): = Δ(Z − 1)(Z + 3)(Z^{4} − 2Z^{3} + (λ − 5)Z^{2} + (6 − λ)Z − 3)
= Δ(Z − 1)(Z + 3) ∏_{i=1}^{2}(Z^{2} − Z + β_{i}),

where α_{1}, α_{2} are the zeros of X^{2} +X +6+λ w.r.t. X and β_{1}, β_{2} are the zeros of Z^{2}+(6−λ)Z −3 w.r.t. Z, which have \sqrt{2} ℚ-multiplication by Proposition 3.5.5 (4λ + 23 ≠ 0 ensures that these curves are really curves of genus 2), have the same conductor; looking at their L-series and applying Theorem 2.5.1, they appear to be ℚ-isogenous as well.

Certainly, for λ ∈ ℚ such that 4λ + 23 ≠ 0 generally, the jacobians are not ℚ-isomorphic; more precisely, if the Galois group of F_{1} over ℚ is isomorphic to D_{8} ×C_{2} and the splitting fields of F_{1}, F_{2} over ℚ (regarded as subfields of \overline{ℚ}) are not isomorphic, then the jacobians are not ℚ-isomorphic, by Proposition 3.3.2 and Corollary 2.8.3.

The thought that there is a ℚ-isogeny of degree 2 from the jacobian of the first curve to that of the second curve seems sensible, since the existence of \sqrt{2} ℚ-multiplications (which are ℚ-endomorphisms of degree 4) of them gives rise to the sensible possibility that there are ℚ-isogenies of degree 2 from the jacobian of the first curve to that of the second curve, and from the jacobian of the second curve to that of the first curve (the compositions giving rise to ℚ-endomorphisms of degree 4 of the jacobians).

Similar comments can be made, for λ ∈ ℚ such that 4λ ≠ 1 and Δ ∈ ℚ^{∗}, the jacobians of the curves

Y^{2} = Δ(X^{2} − X − 1)(X^{4} + X^{3} + λX^{2} + (1 − 4λ)X − 1 + 4λ),  (4.2)
T^{2} = −3Δ(Z − 2)(Z + 2)((1 − 2λ)Z^{4} + 9λZ^{3} + 7(1 − 2λ)Z^{2} + 9λZ + 1 − 2λ)

(4λ ≠ 1 ensures that these curves are really curves of genus 2).

More generally, it is reasonable to ask the following question: what curves C of genus 2 defined over K have the property that there is a K-isogeny of degree 2 from the jacobian of C to the jacobian of another curve of genus 2 defined over K?

If C is a curve of genus 2 defined over K and P is a rational point of order 2 on the jacobian of C, then there is a polarised abelian surface A defined over K and a K-isogeny ϕ from the jacobian of C to A of degree 2 such that the kernel of ϕ is generated by P, but, if the K-endomorphism ring of the jacobian is isomorphic to ℤ, then A cannot be the jacobian of another curve C′ of genus 2 defined over K; indeed, letting Θ, Θ′ be theta divisors on the jacobians of C, C′ respectively, there is a K-polarisation of degree 4 on the jacobian of C, namely ϕ̂ϕ_{Θ′}ϕ, but the K-polarisations on the jacobian of C are ϕ_{Θ}[n] for n ∈ ℤ (of degree n^{4}) by Theorem 2.2.2.

If C is a curve with the properties described in the first paragraph of this section, the thought that there is a K-isogeny of degree 2 from the jacobian of C to that of another curve C′ of genus 2 defined over K seems sensible (as it does for C_{1}, C_{2}, the reasoning being given earlier).

So it is sensible to restrict our attention to the curves with the properties described in the first paragraph of this section.

4.2 A restricted form of Theorem 3.2.1

In this section, we state the restriction of Theorem 3.2.1 to the case where the kernel of ε has a non-trivial rational point.

Proposition 4.2.1 Let C be a curve of genus 2 defined over K, and suppose that the K-endomorphism algebra is a field containing a \sqrt{2} multiplication ε whose kernel has a non-trivial rational point.

Then, for any non-trivial rational point P in the kernel, there is a labelling P_{ij}, i ∈ {0, 1, 2}, j ∈ {0, 1} of the Weierstrass points on C such that (3.1) holds, [P_{00} − P_{01}] = P, and that one of the following is true:

1. There is a K-isomorphism from C to a curve of the form

Y^{2} = Δ(X^{2} + UX + V) ∏_{i=1}^{2} (X^{2} − α_{i}X − \frac{1}{V} (Uα_{i} + 4)) ,  (4.3)

where

∏_{i=1}^{2}(X − α_{i}): = X^{2} + \frac{W (U − V)(U + V) + 4(V^{2} + 4)}{4U}X + W, (4.4)

and

Δ, U, V, W ∈ K, Δ ≠ 0, U ≠ 0, V ≠ 0,  (4.5)

which sends the set of Weierstrass points {P_{00}, P_{01}} to the set of Weierstrass points {(x, 0), (x′, 0)}, where x, x′ satisfy

G_{0}(X): = (X − x)(X − x′) = X^{2} + UX + V  (4.6)

and, for i ≠ 0, the set of Weierstrass points {P_{i0}, P_{i1}} to the set of Weierstrass points {(x, 0), (x′, 0)}, where x, x′ satisfy

G_{i}(X): = (X − x)(X − x′) = X^{2} − α_{i}X − \frac{1}{V} (Uα_{i} + 4).  (4.7)

2. Similar statements hold in this case, with (4.3), (4.4), (4.5), (4.6), (4.7) replaced by

Y^{2} = Δ(X^{2} − 2) ∏_{i=1}^{2}(X^{2} − α_{i}X + 2),  (4.8)

∏_{i=1}^{2}(X − α_{i}): = X^{2} + κX + 8,  (4.9)
Δ, κ ∈ K, Δ ≠ 0,  (4.10)
G_{0}(X): = (X − x)(X − x′) = X^{2} − 2,  (4.11)
G_{i}(X): = (X − x)(X − x′) = X^{2} − α_{i}X + 2  (4.12)

respectively.

Conversely:

1. The jacobian of any curve C of the form (4.3), where (4.4) and (4.5) hold, has a \sqrt{2} K-multiplication ε satisfying (3.1), where P_{i0} = (x, 0), P_{i1} = (x′, 0), with x, x′ satisfying (4.6), (4.7) as is appropriate; more precisely, there is a K-isomorphism from C to its Richelot dual given by (3.7), and (3.8) holds, where ι is as in (3.9).

2. Similar statements hold in this case, with (4.3), (4.4), (4.5), (4.6), (4.7) replaced by (4.8), (4.9), (4.10), (4.11), (4.12) respectively.

Proof: The proof involves Theorem 3.2.1, Proposition 3.5.5 and straightforward computation, which we leave to the reader.□

For Theorem 4.3.1, we will use the curves from the first case with V ≠ 2. It is not too hard to show that the curves from the first case with V = 2 and the curves from the second case have the property that the jacobian of such a curve C is \overline{K}-isogenous to the square of an elliptic curve defined over \overline{K}, using Theorem 14.1.1 from [C-F], which gives rise to the possibility that there is a \overline{K}-isogeny of degree 2 from the jacobian of C to an abelian surface A as in Proposition 2.6.8 (with k = K), but not to the jacobian of another curve of genus 2 defined over K, although we will not investigate this possibility. The curves from the second case are very special, in the sense that they satisfy the conditions in Corollary 2.6.6 (with k = \overline{K}), and so the jacobian of such a curve is \overline{K}-isogenous to the square of an elliptic curve defined over \overline{K} whose \overline{K}-endomorphism ring is isomorphic to ℤ[\sqrt{−2}].

Consider a curve from the first case with V ≠ 2. Note that, if the kernel of ε has three non-trivial rational points, there will be three parameterisations Δ, U, V, W giving the same curve C; this will lead to, in general, three different K-isogenies of degree 2 from the jacobian of C.

For some of the computations involved in the proof of 4.3.1, we will need an explicit form of ε, which we obtain from Chapter 3. Indeed, from (3.62), (3.63), (4.6) and (4.7), we see, after some computation, that ε is given explicitly by

ε([(x, y) − (x′, y′)]) = [(r_{1}, s_{1}) + (r_{2}, s_{2}) − (r′_{1}, s′_{1}) − (r′_{2}, s′_{2})],

where r_{1}, r_{2} are the zeros w.r.t. s of

((U − V)(U + V)x^{2} + 4Ux + 4V)r^{2} + 4(Ux^{2} + (V^{2} + 4)x + UV)r
+ 4(V x^{2} + UV x + (U − 2)(U + 2)) = 0  (4.13)

and s_{i} satisfies

s_{i} = y \frac{(U − V)(U + V) − 4}{(x^{2} + Ux + V)((U − V)(U + V)x^{2} + 4Ux + 4V)^{2}}  (4.14)

× ((U(U^{2} − V (V + 4))x^{3} + 2(U^{2}(2 − V) − V (V^{2} + 8))x^{2}
− 4UV (1 + V)x − 8V^{2})r_{i}
− 4V^{2}x^{3} − 2U(U^{2} + V^{2})x^{2} − 4(U^{2}(2 + V) − 4V)x − 8UV),

r′_{1}, s′_{2}, r′_{1}, s′_{2} having analogous definitions (we use r’s and s’s instead of z’s and t’s to avoid clashing notation when proving Theorem 4.3.1).

4.3 Stating Theorem 4.3.1

Before stating Theorem 4.3.1, we need to set up the notation involved in it.

Firstly, in preparation for defining curves of genus 2 which will appear in Theorem 4.3.1, we define polynomials

F ∈ K[U, V, W, Δ, X], F_{U,V,W} ∈ K[U, V, W, Δ, Z].

Define F to be the polynomial in K[U, V, W, Δ, X] from (4.3), i.e.

F (X): = \frac{Δ}{UV} (X^{2} + UX + V)  (4.15)
× [UV X^{4} + V (\frac{1}{4}W (U − V)(U + V) + V^{2} + 4) X^{3}
+ U (\frac{1}{4}W (U − V)(U + V) + V^{2} + V W − 4) X^{2}
+ (W (U^{2} + V^{2}) − 4(V^{2} + 4))X + UV (W − 4)] ,

and F_{U,V,W} to be the polynomial in K[U, V, W, Δ, Z] from (4.3) with U, V, W, Δ replaced by

U′: = \frac{2(V − U + 2)}{U − 2},  (4.16)
V′: = \frac{2(U − V)}{U − 2},
W′: = \frac{W (U − V)(V^{3} + V^{2}(U − 8) + 4(V (U + 1) − U))}{W (V − 1)(U − V)^{2} − 4(V (U − V)^{2} − (V − 2)^{2})}
+ \frac{4(V^{4} − 8(V^{3} − V^{2}(U + 1) + V (U^{2} − 2(U − 2)) − 2))}{W (V − 1)(U − V)^{2} − 4(V (U − V)^{2} − (V − 2)^{2})},
Δ′: = ΔV (V − 2)U(U − 2)(W (V − 1)(U − V)^{2} − 4(V (U − V)^{2} − (V − 2)^{2}))

respectively, i.e.

F_{U,V,W} (Z): = ΔUV (V − 2)((U − 2)Z^{2} + 2(V − U + 2)Z + 2(U − V)) (4.17)

× [(W (V − 1)(U − V)^{2} − 4(V (U − V)^{2} − (V − 2)^{2}))Z^{4}
+ 2(W (U − V)(2(1 − V)U + V (3V − 2))
+ 4(2UV (U − 2(V + 1)) + (3V − 2)(V^{2} + 4)))Z^{3}
+ (W (U((V^{2} + 4(V − 1))U − 6V (3V − 2)) − V^{2}(V^{2} − 2(7V − 2)))
+ 4(4UV (3(V + 2) − 2U) + (V^{2} + 4)(V^{2} − 2(7V − 2))))Z^{2}
− 2V (W (U(UV + 4(1 − 2V)) + V^{2}(6 − V))
− 4(4U(U − V − 4) − (V − 6)(V^{2} + 4)))Z
+ W V^{2}(U − V)(U + V − 4) − 4V (4U(U − 4) − (V − 4)(V^{2} + 4))].

Note that although the polynomials

U − 2, U − 2, W (V − 1)(U − V)^{2} − 4(V (U − V)^{2} − (V − 2)^{2})

appear in the denominators of the rational functions U′, V′, W′ respectively, they do not appear in the denominators of the coefficients of powers of Z in F_{U,V,W} (Z); they appear in Δ′, allowing cancellation of polynomials.

We need to calculate the discriminants of F, F_{U,V,W} w.r.t. X, Z respectively, to discover the restrictions on U, V, W, Δ required for Y^{2} = F (X), T^{2} = F_{U,V,W} (Z) to be curves of genus 2.

The discriminant of F w.r.t. X is

\frac{(4V − U^{2})^{3}(4 − (U − V)(U + V))^{4}Φ_{1}^{2}Φ_{2}^{3}Δ^{10}}{2^{16}U^{10}V^{7}} ,

and the discriminant of F_{U,V,W} w.r.t. Z is

2^{6}U^{10}V^{12}(V − 2)^{22}(4V − U^{2})^{4}(4 − (U − V)(U + V))^{3}Φ_{1}^{3}Φ_{2}^{2}Δ^{10},

where

Φ_{1}: = W^{2}(U − V)^{2}(U + V)^{2} + 8W ((V − 2)(V + 2)U^{2} − V^{2}(V^{2} + 4)) (4.18)
+ 16(V^{2} + 4)^{2},
Φ_{2}: = W^{2}V ((V − 1)U^{2} − V^{3}) − 4W ((V^{2} + 4(V − 1))U^{2} − 2V^{2}(V^{2} + 4))
+ 16(4V U^{2} − (V^{2} + 4)^{2}).

Let U, V, W, Δ ∈ K such that

U ≠ 0, V ≠ 0, Δ ≠ 0, V ≠ 2, U^{2} ≠ 4V, 4 ≠ (U − V)(U + V), Φ_{1} ≠ 0, Φ_{2} ≠ 0 (4.19)

(which implies that the discriminants of F, F_{U,V,W} (regarded as polynomials in K[X], K[Z]) w.r.t. X, Z respectively are non-zero). Define the curves C, C_{U,V,W} of genus 2 as follows:

C: Y^{2} = F (X), C_{U,V,W}: T^{2} = F_{U,V,W} (Z)  (4.20)

By Proposition 3.5.5, we see that the jacobians of C and C_{U,V,W} both have \sqrt{2} K-multiplication.

We are now in a position to state Theorem 4.3.1:

Theorem 4.3.1 Let U, V, W, Δ ∈ K such that (4.19) holds.

Then, there is a K-homomorphism π_{U,V,W} from the jacobian of C to that of C_{U,V,W} given explicitly as follows:

π_{U,V,W} ([(x, y) − (x′, y′)]) = [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})], (4.21)

((x, y), (x′, y′) ∈ C, (z_{1}, t_{1}), (z_{2}, t_{2}), (z′_{1}, t′_{1}), (z′_{2}, t′_{2}) ∈ C_{U,V,W}), where z_{1}, z_{2} are the zeros w.r.t. z of

∑_{i=0}^{2} ∑_{j=0}^{2} Ψ_{ij}x^{i}z^{j}  (4.22)

and t_{i} satisfies

t_{i} = y \frac{2UV (V − 2)^{2}(Ψ_{1}(x)z_{i} + Ψ_{2}(x))}{(x^{2} + Ux + V) (∑_{i=0}^{2} Ψ_{i2}x^{i})^{2}},  (4.23)

where

( \begin{matrix}Ψ_{00} & Ψ_{10} & Ψ_{20} \\ Ψ_{01} & Ψ_{11} & Ψ_{21} \\ Ψ_{02} & Ψ_{12} & Ψ_{22} \end{matrix} )  (4.24)

: = ( \begin{matrix}2V (U − 2) & 2(U(U − 2) + V (V − 2)) & V (U + V − 4) \\ 2V (4 − U) & 2(U(V − U + 2) − (V − 2)^{2}) & 2(2V − U) \\ 2(U − 2(V − 1)) & U(U − V) − 2(V − 2) & U − V \end{matrix} ) ,

Ψ_{1}(X): = 2(U^{2} − V (V + 2))X^{3}
− (UV^{2} + (U + 4)(6V − U(U + 2)))X^{2}
− (V^{2}(4V − U^{2} − 16) + (U + 2)(2V (U + 10) + U^{3} − 4(2U^{2} − U + 2)))X
− 2(4V^{3} − (U + 2)((U + V^{2})(U + 2) − V (U^{2} − U + 8))),
Ψ_{2}(X): = (8V − U(U + V)(V − U + 2))X^{3}
+ ((U + V)(U − V)(U^{2} − 2V) − 4(3V^{2} − 8V + U^{2}))X^{2}
+ 2(4V^{2}(V − 3) − (U + 2)(V^{2}U + V (U − 8) − U(U − 1)(U − 2)))X
+ 2V (4V^{2} − (U + 2)(V (U + 2) − U^{2} + 2(U − 2))),

with z′_{1}, z′_{2}, t′_{1}, t′_{2} having analogous definitions.

Also, there is a K-homomorphism π_{U,V,W}^{∨} from the jacobian of C_{U,V,W} to that of C given explicitly as follows:

π_{U,V,W}^{∨} ([(z, t) − (z′, t′)]) = [(x_{1}, y_{1}) + (x_{2}, y_{2}) − (x′_{1}, y′_{1}) − (x′_{2}, y′_{2})], (4.25)

((z, t), (z′, t′) ∈ C_{U,V,W} , (x_{1}, y_{1}), (x_{2}, y_{2}), (x′_{1}, y′_{1}), (x′_{2}, y′_{2}) ∈ C), where x_{1}, x_{2} are the zeros w.r.t. x of (4.22) and y_{i} satisfies

y_{i} = t\frac{Ψ_{3}(z)x_{i} − 2Ψ_{4}(z)}{2UV ((U − 2)z^{2} + 2(V − U + 2)z + 2(U − V)) (∑_{j=0}^{2} Ψ_{2j}z^{j})^{2}} , (4.26)

where

Ψ_{3}(Z): = (8V − U(U + V)(V − U + 2))Z^{3}
− (2V^{3} + (U − 2)(U + 2)V^{2} + 2(U − 4)^{2}V − U^{2}(U^{2} − 4(U − 2)))Z^{2}
− (3UV^{3} − 2(U − 4)V^{2} − (U^{2}(U + 10) − 32(U − 1))V + 2U^{2}(U − 2)^{2})Z
− 2(V^{3}(V − 2) + U(U − 2)(4V − U^{2})),
Ψ_{4}(Z): = 2V (V − 2)Z^{3}
+ ((3U − 8)V^{2} + 2(U^{2} − U + 4)V − U^{2}(U + 2))Z^{2}
+ V (2V^{2} + (U^{2} − 4(U − 3))V + U^{2}(U − 6))Z
+ V (UV^{2} + 2(U − 4)V − U^{2}(U − 2)),

with x′_{1}, x′_{2}, y′_{1}, y′_{2} having analogous definitions.

Moreover,

π_{U,V,W}^{∨} π_{U,V,W} = [2] + ε,  (4.27)

where ε is the \sqrt{2} K-multiplication of the jacobian of C as in the ‘Conversely’ part of Proposition 4.2.1, and also the kernels of π_{U,V,W} , π_{U,V,W}^{∨} are generated by [P_{00} − P_{01}], [Q_{00} − Q_{01}] respectively, where:

• Letting x, x′ be the zeros of X^{2} +UX +V w.r.t X, P_{00}, P_{01} are the Weierstrass points on C corresponding to x, x′, not necessarily in that order (so that [P_{00} − P_{01}] is a point of order 2 on the jacobian of C).

• Letting z, z′ be the zeros of (U − 2)Z^{2} + 2(V − U + 2)Z + 2(U − V) w.r.t. Z, Q_{00}, Q_{01} are the Weierstrass points on C_{U,V,W} corresponding to z, z′, not necessarily in that order (so that [Q_{00} − Q_{01}] is a point of order 2 on the jacobian of C_{U,V,W}).

(In particular, π_{U,V,W}^{∨} , π_{U,V,W} are K-isogenies of degree 2).

Proving that the explicit maps given for π_{U,V,W} , π_{U,V,W}^{∨} are correct does not prove that they are isogenies, only that they are homomorphisms; we need to establish that their kernels are finite as well. As we shall see in Section 4.6, the proof of this theorem is mostly of a computational nature; we will omit the computations themselves as they are lengthy, but they are nevertheless routine.

For fixed Δ, λ ∈ K, the first, second curves in (4.1) are K-isomorphic to the curves C, C_{U,V,W} with U = 1, V = 1, W = λ + 12 respectively, and the first, second curves in (4.2) are K-isomorphic to the curves C, C_{U,V,W} with U = 1, V = −1, W = λ + 6 respectively. In the next section, we will find Khomomorphisms π and π^{∨} from the jacobian of C_{1} to that of C_{2} and the jacobian of C_{2} to that of C_{1} respectively (C_{1}, C_{2} being the first, second curves in (4.1) respectively), which will correspond under suitable isomorphisms from C_{1} to C and from C_{2} to C_{U,V,W} (when U = 1, V = 1, W = λ + 12) to π_{U,V,W} and π_{U,V,W}^{∨} respectively.

4.4 The curves C_{1}, C_{2}

In this section, we will find K-homomorphisms between the jacobians of the curves in (4.1) (with λ ∈ K) explicitly, in preparation for the discovery of Theorem 4.3.1, bearing in mind that these homomorphisms should be isogenies of degree 2 (although we will not prove this in this section). The simplicity of the curves makes this task relatively easy. To end this section, we will make observations about these homomorphisms; these observations will be crucial in the discovery of Theorem 4.3.1.

In general, a homomorphism defined over a perfect field k from the jacobian of a curve of genus 2 to the jacobian of a curve of genus 2 can be given explicitly as follows:

[(x, y) − (x′, y′)] ↦ [(z_{1}, t_{1}) + (z_{2}, t_{2}) − (z′_{1}, t′_{1}) − (z′_{2}, t′_{2})],  (4.28)

(x, y), (x′, y′) lying on the first curve and (z_{1}, t_{1}), (z_{2}, t_{2}), (z′_{1}, t′_{1}), (z′_{2}, t′_{2}) lying on the second curve, where z_{1}, z_{2} are the zeros w.r.t. z of a polynomial P defined over \overline{k} (where \overline{k} is an algebraic closure of k) in x, z which is quadratic in z, and t_{i} satisfies

t_{i} = yR(x, z_{i})  (4.29)

for some rational function R defined over \overline{k} in x, z, with z′_{1}, z′_{2}, t′_{1}, t′_{2} having analogous definitions. For example, if the characteristic of k is not 2, and C is a curve of genus 2 defined over k, of the form (2.19), with det(g_{uv}) ≠ 0, then, for the Richelot isogeny, we have

P = G_{1}(x)H_{1}(z) + G_{2}(x)H_{2}(z),
R = \frac{H_{1}(z_{i})(x − z_{i})}{G_{0}(x)G_{2}(x)}.

Note that P is quadratic in x as well as in z, and, if G_{0} is defined over k (which implies that G_{1}, G_{2} are defined over k as a set), then there exists α ∈ \overline{k}^{∗} such that αP ∈ k[x, z], allowing us to choose P ∈ k[x, z] by replacing P by αP, since, given x, the zeros of αP w.r.t. z are the zeros of P w.r.t. z.

The K-homomorphisms found in this section will have P quadratic in x as well as in z. This seems sensible since they should be isogenies of degree 2 (the degree of the Richelot isogeny, which also has P quadratic in x as well as in z, is 4).

Let λ ∈ K such that (4λ+23)(λ^{2}−12λ+48) ≠ 0, so that the curves in (4.1) are really curves of genus 2, let x_{0j}, z_{0j} be the zeros of X^{2} − X + 1, (Z − 1)(Z + 3) respectively, let x_{ij}, z_{ij} (i ∈ {1, 2}) be the zeros of X^{2} + α_{i}X − 3, Z^{2} − Z + β_{i} respectively, and let P_{ij} = (x_{ij}, 0), Q_{ij} = (z_{ij}, 0), which are Weierstrass points on C_{1}, C_{2} respectively. In particular, swapping x_{00}, x_{01} if necessary, we have

P_{00} = (\frac{1 + \sqrt{−3}}{2}, 0) , P_{01} = (\frac{1 − \sqrt{−3}}{2}, 0);  (4.30)

the reason for displaying these points will become apparent. We will first find a K-homomorphism π from the jacobian of C_{1} to that of C_{2} explicitly, bearing in mind that it should be an isogeny of degree 2.

Now G_{K} (Gal(\overline{K}/K)) sends the set \{P_{00}, P_{01}\} to itself, the set \{Q_{00}, Q_{01}\} to itself, the sets {P_{10}, P_{11}}, {P_{20}, P_{21}} to the sets {P_{10}, P_{11}}, {P_{20}, P_{21}}, and the sets {Q_{10}, Q_{11}}, {Q_{20}, Q_{21}} to the sets {Q_{10}, Q_{11}}, {Q_{20}, Q_{21}}. Considering this, we can reasonably expect a K-isogeny of degree 2 from the jacobian of C_{1} to that of C_{2} to act on the 2-torsion of the former as follows:

• It sends [P_{00} − P_{01}] to 0.

• For all i ∈ {1, 2}, it sends [P_{i0} − P_{i1}] to [Q_{00} − Q_{01}].

• For all j_{1}, j_{2} ∈ {0, 1}, there exists i ∈ {1, 2} such that it sends [P_{1j_{1}} −P_{2j_{2}}] to [Q_{i0} − Q_{i1}].

• For all j_{1}, j_{2} ∈ {0, 1} and i ∈ {1, 2}, there exist j_{3}, j_{4} ∈ {0, 1} such that it sends [P_{0j_{1}} − P_{ij_{2}}] to [Q_{1j_{3}} − Q_{2j_{4}}].

This action will be useful for finding π.

To start with, we will consider λ = −6, so that α_{i} ∈ ℚ (this fact will hopefully make finding π easier for λ = −6 than for λ in general). Then, C_{1}, C_{2} are

C_{1}: Y^{2} = (X^{2} − X + 1)(X^{2} − 3)(X^{2} + X − 3),
C_{2}: T^{2} = (Z − 1)(Z + 3)(Z^{4} − 2Z^{3} − 11Z^{2} + 12Z − 3).

Given the above discussion of the action of a K-isogeny of degree 2 from the jacobian of C_{1} to that of C_{2} on the 2-torsion of the former, it seems sensible to choose s_{1}, s_{2} ∈ {±1} and to try to find a polynomial P, quadratic in x and z and defined over ℚ, and constants κ_{1}, κ_{2}, κ_{3}, κ_{4} ∈ \overline{ℚ}, such that

x = \sqrt{3} ⇒ P (x, z) = κ_{1}(z^{2} − z + s_{1}\sqrt{3}(2z − 1)),
x = −\sqrt{3} ⇒ P (x, z) = κ_{2}(z^{2} − z − s_{1}\sqrt{3}(2z − 1)),
x = \frac{−1 + \sqrt{13}}{2} ⇒ P (x, z) = κ_{3}(2(z^{2} − z) + 1 + s_{2}\sqrt{13}(2z − 1)),
x = \frac{−1 − \sqrt{13}}{2} ⇒ P (x, z) = κ_{4}(2(z^{2} − z) + 1 − s_{2}\sqrt{13}(2z − 1))

(the four x’s listed are all x-coordinates of Weierstrass points on C_{1}) and the zeros of the four quadratics in z listed are all z-coordinates of Weierstrass points on C_{2}). Moreover, if we are successful, we then hope that there is a K-isogeny of degree 2 from the jacobian of C_{1} to that of C_{2} which is given explicitly by (4.28), where z_{1}, z_{2} are the zeros w.r.t. z of P, and t_{i} satisfies (4.29) for some rational function R defined over ℚ. Since P is quadratic in x and z, there are 9 coefficients in P to be determined, and the four conditions above give two equations in the coefficients each, so P is uniquely determined by the conditions up to a multiplicative constant. Choosing s_{1} = s_{2} = −1 and taking the coefficient of xz^{2} in P to be 1, we see, after some computation, that

P (x, z) = xz^{2} − (x^{2} + x + 3)z + 3.  (4.31)

We haven’t yet considered the fact that a K-isogeny of degree 2 from the jacobian of C_{1} to that of C_{2} should send [P_{00} − P_{01}] to 0. Now the discriminant of P (x, z) w.r.t. z is (x^{2} − x + 1)(x^{2} + 3x + 9), so, if x = \frac{1+\sqrt{−3}}{2} and x′ = \frac{1−\sqrt{−3}}{2}, then z_{1} = z_{2} and z′_{1} = z′_{2}. Hopefully, t_{1} = −t_{2} and t′_{1} = −t′_{2}; then, if a Kisogeny of degree 2 from the jacobian of C_{1} to that of C_{2} is given explicitly by (4.28), where z_{1}, z_{2} are the zeros w.r.t. z of P, and t_{i} satisfies (4.29) for some rational function R defined over ℚ, as hoped, then it will send [P_{00} − P_{01}] to 0 (from (4.30)), as expected.

We will now try to find a rational function R defined over ℚ such that a Kisogeny of degree 2 from the jacobian of C_{1} to that of C_{2} is given explicitly as described above. It suffices to find a rational function R defined over ℚ such that

(z_{i}−1)(z_{i}+3)(z_{i}^{4}−2z_{i}^{3}−11z_{i}^{2}+12z_{i}−3) = (x^{2}−x+1)(x^{2}−3)(x^{2}+x−3)R(x, z_{i})^{2};

(4.32)

indeed, defining t_{i} as in (4.29) ensures that (z_{i}, t_{i}) ∈ C_{2}, by (4.32). Moreover, there exist rational functions R_{0}(x), R_{1}(x) defined over ℚ such that

R(x, z_{i}) = R_{0}(x) + z_{i}R_{1}(x),

since z_{i} is a zero of (4.31) w.r.t. z (a polynomial of degree 2), so that

(z_{i} − 1)(z_{i} + 3)(z_{i}^{4} − 2z_{i}^{3} − 11z_{i}^{2} + 12z_{i} − 3)
− (x^{2} − x + 1)(x^{2} − 3)(x^{2} + x − 3)(R_{0}(x) + z_{i}R_{1}(x))^{2}
= 0
⇒ \frac{−3(x^{2} − 3)(x^{2} + x − 3)(x^{4} + 3x^{3} + 3x^{2} + 6x + 9)}{x^{5}}
+ \frac{(x^{2} − 3)(x^{2} + x − 3)(x^{3} + x^{2} + 3)(x^{3} + 3x^{2} + 6x + 9)}{x^{5}}z_{i}
= \frac{(x^{2} − 3)(x^{2} + x − 3)(x^{2} − x + 1)(R_{0}^{2}(x)x − 3R_{1}^{2}(x))}{x}
+ \frac{(x^{2} − 3)(x^{2} + x − 3)(x^{2} − x + 1)R_{1}(x)(R_{1}(x)(x^{2} + x + 3) + 2R_{0}(x)x)}{x}z_{i},

again since z_{i} is a zero of (4.31). Separating the coefficients of 1 and z_{i}, we see, after a bit of computation, that

R_{0}(x) = \frac{−3}{x^{2}(x^{2} − x + 1)}, R_{1}(x) = \frac{x^{3} + x^{2} + 3}{x^{2}(x^{2} − x + 1)}

satisfy the above equation. So we can define a K-isogeny π from the jacobian of C_{1} to that of C_{2}, given explicitly by (4.21), (x, y), (x′, y′) lying on C_{1} and (z_{1}, t_{1}), (z_{2}, t_{2}), (z′_{1}, t′_{1}), (z′_{2}, t′_{2}) lying on C_{2}, where z_{1}, z_{2} are the zeros w.r.t. z of (4.31) and t_{i} satisfies

t_{i} = y \frac{(x^{3} + x^{2} + 3)z_{i} − 3}{x^{2}(x^{2} − x + 1)},  (4.33)

as required.

Having found π for λ = −6, it turns out that, for λ in general, we can define a K-isogeny π from the jacobian of C_{1} to that of C_{2}, given explicitly by (4.21), where z_{1}, z_{2} are the zeros w.r.t. z of (4.31) and t_{i} satisfies (4.33). In other words, both the polynomial P and the rational function R are independent of λ; an observation which will be useful when discovering Theorem 4.3.1.

It also turns out that, for λ in general, we can define a K-homomorphism π^{∨} from the jacobian of C_{2} to that of C_{1}, given explicitly by (4.25), (z, t), (z′, t′) lying on C_{2} and (x_{1}, y_{1}), (x_{2}, y_{2}), (x′_{1}, y′_{1}), (x′_{2}, y′_{2}) lying on C_{1}, where x_{1}, x_{2} are the zeros w.r.t. x of (4.31) and t_{i} satisfies

y_{i} = t\frac{(z^{2} − 3)x_{j} − 3(z − 1)}{z(z − 1)(z + 3)}.

The important observation here is that the polynomial whose zeros w.r.t. z are z_{1}, z_{2} (for π) is the polynomial whose zeros w.r.t. x are x_{1}, x_{2} (for π^{∨}); an observation which will be useful when discovering Theorem 4.3.1.

For fixed Δ, λ ∈ K, the curves C_{1}, C_{2} are K-isomorphic to the curves C, C_{U,V,W} with U = 1, V = 1, W = λ + 12 respectively. More precisely, there is a Kisomorphism from C_{1} to C given explicitly by (x, y) ↦ (x − 1, y), and there is a K-isomorphism from C_{2} to C_{U,V,W} given explicitly by (z, y) ↦ (1 − z, 2t). Under these isomorphisms, π, π^{∨} correspond to π_{U,V,W} , π_{U,V,W}^{∨} respectively.

4.5 Discovering Theorem 4.3.1

In this section, we show how Theorem 4.3.1 was discovered, using the observations from the previous section.

For notational convenience, we relax the definitions of C, F, C_{U,V,W} , F_{U,V,W} given in Section 4.3, for this section only; first, we establish useful notation, analogous to that in Section 4.3. For rational functions

U′, V′, W′, \frac{Δ′}{Δ} ∈ K(U, V, W), U′ ≠ 0 ≠ V′  (4.34)

(e.g. those in (4.16)), define F to be the polynomial in K[U, V, W, Δ, X] from (4.3), and F_{U,V,W} to be the polynomial in K[U, V, W, Δ, Z] from (4.3) with U, V, W, Δ replaced by U′, V′, W′, Δ′ respectively, as in Section 4.3 (there, the rational functions were those in (4.16)); the condition U′ ≠ 0 ≠ V′ ensures that the latter definition is possible.

For U, V, W, Δ ∈ K satisfying (4.19) and such that F_{U,V,W} is defined and has a non-zero discriminant, regarded as a sextic in K[X], define C, C_{U,V,W} as in (4.20) (recall that, by (4.19), F is defined and has a non-zero discriminant, regarded as a sextic in K[Z]).

Our aim in this section is to find rational functions as in (4.34) such that, letting U, V, W, Δ ∈ K be as above, there is good reason for the possible existence of a K-isogeny from the jacobian of C to that of C_{U,V,W}, leading to the discovery of Theorem 4.3.1; in fact, the rational functions will turn out to be those in (4.16).

Let U, V, W, Δ ∈ K be as above. A good reason for the possible existence of a K-isogeny from the jacobian of C to that of C_{U,V,W} is that the primes dividing the discriminant of F roughly correspond to the primes dividing the discriminant of F_{U,V,W}, both polynomials regarded as sextics (cf. a similar discussion in Section 2.6).

Recall that, considering the curves C_{1}, C_{2} as defined before, the polynomial P and the rational function R giving π is independent of λ. So it seems a good idea for U′, V′ to be independent of W and for W′ to be a Möbius transformation in W whose coefficients are polynomials in U, V.

Bearing in mind that Φ_{1}, Φ_{2} (as in (4.18)) are factors of the discriminant of F and are quadratic in W in mind, our strategy, decided upon after a bit of thought, will be to try to find rational functions as in (4.34), with U′, V′ independent of W and W′ a Möbius transformation in W whose coefficients are polynomials in U, V, such that, defining Φ_{i,U,V,W} to be Φ_{i} with U, V, W replaced by U′, V′, W′ respectively, the rational functions

\frac{Φ_{1}}{Φ_{2,U,V,W}} W′^{2}, \frac{Φ_{2}}{Φ_{1,U,V,W}} W′^{2} ∈ K(U, V, W)  (4.35)

are independent of W.

After a bit of thought, it seems sensible for rational functions R_{1}, R_{2} ∈ K(U, V) to exist such that

U′^{2} − 4V′ = R_{1}^{2}(4 − (U − V)(U + V)),  (4.36)
4 − (U′ + V′)(U′ − V′) = R_{2}^{2}(U^{2} − 4V),  (4.37)

since the discriminants of Φ_{1}, Φ_{2} w.r.t. W are

2^{10}U^{2}V^{2}(4 − (U − V)(U + V)), 2^{4}U^{2}(V − 2)^{4}(U^{2} − 4V)

respectively. We will initially try to find R_{1}, R_{2}.

Solving (4.36) for U′ and substituting into (4.37), we see that

V′^{2}−4V′−R_{1}^{2}V^{2}+4R_{2}^{2}V +(R_{1}+R_{2})(R_{1}−R_{2})U^{2}−4(R_{1}+1)(R_{1}−1) = 0. (4.38)

By (4.38), the discriminant w.r.t. V′ of the quadratic in (4.38) is the square of a rational function R_{3} ∈ K(U, V). So

R_{3}^{2} = 4(R_{1}^{2}V^{2} − 4R_{2}^{2}V − (R_{1} + R_{2})(R_{1} − R_{2})U + 4R_{1}^{2}).  (4.39)

We see that, for any rational function R_{4} ∈ K(U, V),

R_{1} = R_{4}, R_{2} = R_{4}, R_{3} = 2R_{4}(V − 2)  (4.40)

solves (4.39). We will work with these rational functions R_{1}, R_{2}, R_{3}, their simplicity being appealing, hoping that we can find a rational function R_{4} which gives rational functions U′, V′. In fact, after a bit of thought, we see that

R_{4} = \frac{2}{U − 2}  (4.41)

is a suitable rational function. Defining U′, V′ as in (4.16) ensures that U′, V′ satisfy (4.36) and (4.37), taking (4.40), (4.41) into account. Defining W′ as in (4.16) ensures that the rational functions (4.35) are independent of W, as required.

Having established that suitable U′, V′, W′ are as in (4.16), a suitable strategy for trying to find a polynomial P and a rational function R such that there exists a K-isogeny π_{U,V,W} of degree 2 from the jacobian of C to that of C_{U,V,W}, where U, V, W, Δ satisfy (4.19), given explicitly by (4.21), where z_{1}, z_{2} are the zeros w.r.t. z of P, and t_{i} satisfies (4.29), is similar to that for C_{1}, C_{2}, described in the previous section, albeit computationally more involved. P turns out to be (4.22), and R turns out to be the rational function equating (4.23) with (4.29).

To see the suitability of (4.22) for P when considering the curves C, C_{U,V,W}, suppose that W = 0 (once the suitability is established for W = 0, it is established for W in general, bearing in mind that, considering the curves C_{1}, C_{2} as defined before, the polynomial P and the rational function R giving π is independent of λ, as we saw in the previous section); then, from (4.15), (4.17) and (4.20), we see that C, C_{U,V,W} are the curves

C: Y^{2} = \frac{Δ(X^{2} + UX + V)(V X^{2} − 4)(UX^{2} + (V^{2} + 4)X + UV)}{UV},
C_{U,V,W}: T^{2} = 4ΔUV (V − 2)((U − 2)Z^{2} + 2(V − U + 2)Z + 2(U − V))
× ((−V^{3} + (2U + 1)V^{2} − (U^{2} + 4)V + 4)Z^{4}
+ 2(3V^{3} − 2(2U + 1)V^{2} + 2(U^{2} − 2(U − 3))V − 8)Z^{3}
+ (V^{4} − 14V^{3} + 4(3U + 2)V^{2} − 8(U^{2} − 3U + 7)V + 16)Z^{2}
− 2V (V^{3} − 6V^{2} + 4(U + 1)V − 4(U^{2} − 2(2U − 3)))Z
+ V (V (V − 2)^{2} − 4(U − 2)^{2})).

After some computation, we see that, if x is one of

\frac{2}{\sqrt{V}} , −\frac{2}{\sqrt{V}} ,

\frac{−V^{2} − 4 + \sqrt{(V^{2} + 4)^{2} − 4U^{2}V}}{2UV} , \frac{−V^{2} − 4 − \sqrt{(V^{2} + 4)^{2} − 4U^{2}V}}{2UV} ,

which are all x-coordinates of Weierstrass points on C, then the zeros of (4.22) are all z-coordinates of Weierstrass points on C_{U,V,W}. Moreover, the discriminant of (4.22) w.r.t. z is

4(x^{2} + Ux + V)
×((V^{3} + U^{2}(1 − V))x^{2} + U(2V (V − 2) + 8 − U^{2})x + U^{2}(V − 4) + 16);

the zeros of the first factor are all x-coordinates of Weierstrass points on C. These reasons establish the suitability of (4.22) for P when considering the curves C, C_{U,V,W} with W = 0, bearing in mind the reasons for the suitability of (4.31) for P when considering the curves C_{1}, C_{2} with λ = −6 given in the previous section.

In the previous section, we observed that, for the curves C_{1}, C_{2}, the polynomial whose zeros w.r.t. z are z_{1}, z_{2} (for π) is the polynomial whose zeros w.r.t. x are x_{1}, x_{2} (for π^{∨}). Bearing this in mind, it is reasonable to guess that there is a K-isogeny π_{U,V,W}^{∨} of degree 2 from the jacobian of C_{U,V,W} to that of C, given explicitly by (4.25), where x_{1}, x_{2} are the zeros w.r.t. x of (4.22). This indeed turns out to be true, and y_{i} satisfies (4.26).

4.6 Proving Theorem 4.3.1

In this section, we prove Theorem 4.3.1, using mostly lengthy but nevertheless routine computations. We will describe these computations using Maple instructions; in all cases, the reader is recommended to use specialised values of U, V, W initially, to make identifying typing errors easier.

Proof: (of Theorem 4.3.1) Consider showing that the explicit form given for π_{U,V,W} is correct. Note first that (4.22) is a non-zero polynomial in z for all x ∈ \overline{K} ∪∞, so that it makes sense to talk about the zeros of ψ w.r.t. z; indeed, this is a simple consequence of the fact that the determinant of the matrix in (4.24) is non-zero, due to (4.19) holding, if x ≠ ∞, and, for x = ∞, we see that Ψ_{20}, Ψ_{21}, Ψ_{22} are not all zero, due again to (4.19) holding. Since the explicit form is defined over K, π_{U,V,W} is a K-homomorphism.

We now apply Maple. Define three variables as follows: F to be the polynomial in U, V, W, Δ, z1 which is (4.17) with z1 representing Z, P to be the polynomial in U, V, x, z which is (4.22), and R to be the rational function in U, V, x, z1, y which is the RHS of (4.23) with z1 representing z_{i}. Then, proceed as follows:

z1 := RootOf(P, z); expr1 := simplify(F); expr1 := factor(expr1); expr2 := simplify(R^{2}); simplify(expr1/expr2);

The final output should be the rational function in U, V, W, Δ, x, y which is (4.15) divided by y^{2}, so verifying that the explicit form given for π_{U,V,W} is correct. The vast majority of the computation is involved in the second command above, and to a lesser extent in the third command; the other commands are quick.

Consider showing that the explicit form given for π_{U,V,W}^{∨} is correct. Here, the comments and the Maple commands are similar to those for π_{U,V,W}, so we omit them.

Consider working out an explicit description of π_{U,V,W}^{∨} π_{U,V,W}. The explicit descriptions of π_{U,V,W} and π_{U,V,W}^{∨} described in Theorem 4.3.1 mean that this task is easier than it might be; we will not have to worry about divisors of functions on C. Indeed, if (x, y), (x′, y′) ∈ C and (z_{1}, t_{1}), (z_{2}, t_{2}), (z′_{1}, t′_{1}), (z′_{2}, t′_{2}) ∈ C_{U,V,W} such that (4.21) holds, then these descriptions imply that

π_{U,V,W}^{∨} ([(z_{i}, t_{i}) − (z′_{i}, t′_{i})]) = [(x, y_{i1}) + (x_{i2}, y_{i2}) − (x′, y′_{i1}) − (x′_{i2}, y′_{i2})],

where y_{i1}, y′_{i1} ∈ \overline{K} such that (x, y_{i1}), (x′, y′_{i1}) ∈ C, and (x_{i2}, y_{i2}), (x′_{i2}, y′_{i2}) ∈ C. In fact, we will show that y_{i1} = y, that x_{12}, x_{22} are the roots of (4.13) w.r.t. r, and that s_{i} is y_{i2} in (4.14) when r_{i} is x_{i2}, the arguments implying that analogous results hold for y′_{i1}, x′_{i2}, y′_{i2}, so establishing (4.27), as required.

Consider showing that y_{i1} = y. To do this, we apply Maple. Define three variables as follows: P to be the polynomial in U, V, x, z which is (4.22), R1 to be the rational function in U, V, x, z, y which is the RHS of (4.23) with z representing z_{i}, and R2 to be the rational function in U, V, z, x, t which is the RHS of (4.26) with x representing x_{i}. Then, proceed as follows:

z := RootOf(P, z);

num := simplify(numer(R2)); den := simplify(denom(R2)); t := simplify(R1); expr := simplify(num/den); simplify(expr);

The final output should be y, so verifying that y_{i1} = y. The vast majority of the computation is involved in the third command, and to a lesser extent in the fifth and sixth commands; the other commands are quick. We separate the numerator and the denominator of R2 to ease the computations. num/den needs to be simplified twice by Maple in order to get the required output.

Consider showing that s_{i} is y_{i2} in (4.14) when r_{i} is x_{i2}; it suffices to show that s_{1} is y_{12} in (4.14) when r_{1} is x_{12}, as the computations below will be the same for both roots of (4.13) w.r.t. r. Moreover, having shown already that y_{11} = y, it suffices to show that y_{12} is y_{11} multiplied by the RHS of (4.14) divided by y when r_{1} is x_{12}. After some computation, we see that

x_{12} = \frac{2((U − V)x^{2} + (U^{2} − V (U + 2) + 4)x + 2(U − 2(V − 1)))z_{1}}{((V − U)(V + U)x^{2} − 4Ux − 4V)} (4.42)
+ \frac{2(2V x^{2} − (U^{2} − U(V + 2) − 4V)x + 4V)}{((V − U)(V + U)x^{2} − 4Ux − 4V)}.

We now apply Maple. Define four variables as follows: P to be the polynomial in U, V, x, z which is (4.22), R1 to be the rational function in U, V, x, z1 which is the RHS of (4.42) with z1 representing z_{1}, R2 to be the rational function in U, V, x, z1, t which is the RHS of (4.26) with x representing x_{i} and z1 representing z, R3 to be the rational function in U, V, x, x1 which is the RHS of (4.14) divided by y with x1 representing r_{i}. Then, proceed as follows: x := x0; z1 := RootOf(P, z); x1 := simplify(R1); x := x0; expr1 := factor(simplify(R2)); x := x1; expr2 := factor(simplify(R2)); x := x0; expr3 := factor(simplify(R3)); expr4 := simplify(expr1 ∗ expr3); expr4 := simplify(expr4); expr4 := factor(expr4); simplify(expr2 − expr4);

The final output should be 0, so verifying that y_{12} is y_{11} multiplied by the RHS of (4.14) divided by y when r_{1} is x_{12} (note that we are working out what y_{12} is relative to y_{11}, so we do not need to work out what t is here). The vast majority of the computation is involved in the fifth command. expr1 ∗ expr3 needs to be simplified twice by Maple in order to get it into a suitable form.

Finally, having shown that (4.27) holds, it is an easy matter to work out the kernels of π_{U,V,W} and π_{U,V,W}^{∨}. Indeed, letting P_{ij}, i ∈ {0, 1, 2}, j ∈ {0, 1} be a labelling of the Weierstrass points on C such that P_{ij} corresponds to x, where x is a zero of G_{i} (G_{i} being as in Proposition 4.2.1), and letting Q_{00}, Q_{01} be as in Theorem 4.3.1, it is easy to show that

π_{U,V,W} ([P_{10} − P_{11}]) = π_{U,V,W} ([P_{20} − P_{21}]) = [Q_{00} − Q_{01}],

using the explicit description of π_{U,V,W} given in Theorem 4.3.1, which implies that the kernel of π_{U,V,W} contains [P_{00} − P_{01}]; similarly, we can show that the kernel of π_{U,V,W}^{∨} contains [Q_{00} − Q_{01}]. Then, we see that the kernels of π_{U,V,W} , π_{U,V,W}^{∨} are generated by [P_{00} −P_{01}], [Q_{00} −Q_{01}] respectively, since (4.27) holds and [2] + ε is an endomorphism of degree 4. This finishes the proof of Theorem 4.3.1. □

Curves with (2, −3/ℚ) Multiplication

5.1 Introduction

In this chapter, we present a family of curves of genus 2 defined over ℚ, such that, if C is a curve in it, then C is ℚ-isomorphic to a curve in the \sqrt{2} multiplication family presented in Chapter 3 and the \overline{ℚ}-endomorphism ring of the jacobian of C is a maximal order in a division quaternion algebra over ℚ. More precisely, we state and prove the following theorem:

Theorem 5.1.1 Let N, Δ ∈ ℚ^{∗}, N ≠ ±1. Then, the \overline{ℚ}-endomorphism ring of the jacobian of the curve

Y^{2} = Δ((N −1)X^{6}−6NX^{5}+3(N +1)X^{4}−8N^{2}X^{3}+3(N −1)X^{2}+6NX +N +1)

(5.1)

is a maximal order in the division quaternion algebra (\frac{2,−3}{ℚ}); more precisely,

it is generated by [1], ε, \frac{[1]+η}{[2]}, where ε is a \sqrt{2} ℚ-multiplication, η is a \sqrt{−3} ℚ(\sqrt{−3})-multiplication such that σ^{−1}ησ = −η if σ is the non-trivial automorphism of ℚ(\sqrt{−3}), and ηε + εη = [0].

CHAPTER 5. CURVES WITH (2, −3/ℚ) MULTIPLICATION 110

Note that it is impossible for the ℚ-endomorphism ring of the jacobian of a curve of genus 2 defined over ℚ to be an order in a division quaternion algebra, by Corollary 2.2.10. Our theorem provides a family of curves C of genus 2 defined over ℚ with the closest property to this, namely that the \overline{ℚ}-endomorphism ring of the jacobian of C is an order in a division quaternion algebra and generated by [1], a ℚ-endomorphism ε, and a ℚ(\sqrt{D})-endomorphism η such that σ^{−1}ησ = −σ if σ is the non-trivial automorphism of ℚ(\sqrt{D}), where D is a squarefree integer. To the best of our knowledge, no curve of genus 2 defined over ℚ with this property has been discovered before, although two families of curves C of genus 2 defined over \overline{ℚ} such that the \overline{ℚ}-endomorphism ring of the jacobian of C is an order in a division quaternion algebra have been discovered by K. Hashimoto and N. Murabayashi (see [H-M]), one of which will be used in the proof of our theorem.

More precisely, any curve C from their families is represented by a point on the Humbert surface H_{5} and a point on the Humbert surface H_{8} simultaneously. Recall that the Humbert surface H_{D} is a surface which classifies principally polarised abelian surfaces having a primitive endomorphism, fixed by the Rosati involution w.r.t. the polarisation, with the properties described at the beginning of Section 2.9. So the jacobian of C has \frac{1+\sqrt{5}}{2} and \sqrt{2} Θsymmetric ℂ-multiplications (where Θ is a theta divisor on it), which are in fact \overline{ℚ}-multiplications. For their first family, these endomorphisms and the identity endomorphism generate a maximal order in the division quaternion algebra (\frac{2,−3}{ℚ}). For their second family, a similar thing happens with (\frac{2,−3}{ℚ})

replaced by (\frac{2,5}{ℚ}). Generally, their curves are defined over a quadratic exten-

sion of ℚ, because they assume (w.l.o.g. if working over \overline{ℚ}, but not if working over ℚ) that two of the Weierstrass points are at 0 and at ∞. Usefully, it turns out that each curve in our quaternionic family is \overline{ℚ}-isomorphic (but generally not ℚ-isomorphic) to some curve in their (\frac{2,−3}{ℚ}) family, but the curves in our

family were not discovered by trying to remodel the curves in their (\frac{2,−3}{ℚ}) family; after the curves in our quaternionic family were discovered, the existence of isomorphisms was established using Igusa invariants.

We could attempt to prove our theorem (for N, Δ ∈ K^{∗}, where K is a subfield of ℂ, as well as for N, Δ ∈ ℚ^{∗}, N ≠ ±1), by letting ε be a \sqrt{2} K-multiplication of the jacobian of C given by Proposition 3.5.5, ζ be a Θsymmetric \frac{1+\sqrt{5}}{2} \overline{K}-multiplication (Θ being a theta divisor on the jacobian), deriving ε and ζ as explicit two-to-two correspondences on C (the former derivable as described in the proof of Proposition 3.5.5 and the latter derivable as described in [Wil]), and then performing computations on these correspondences, in particular showing that ζε + εζ = [2s] + ε for some s ∈ {±1}. Theoretically straightforward, the computations are potentially very tedious, and they do not prove that [1], ε, η generate the \overline{K}-endomorphism ring of the jacobian (where η: = [1] + [2]ζ([1] − [s]ε)), only that they generate a subring of it.

Instead, we stick to N, Δ ∈ ℚ^{∗}, N ≠ ±1, and, by using together the endomorphism structure of a complex-analytic abelian surface and the actions of G_{ℚ}, ε, ζ on the 2-torsion of the jacobian, it turns out that the jacobian is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication, ensuring, remarkably, that ζε + εζ = [2s] + ε for some s ∈ {±1}, and that [1], ε, η generate the \overline{K}-endomorphism ring of the jacobian (where η: = [1] + [2]ζ([1] − [s]ε)).

Note that sticking to N, Δ ∈ ℚ^{∗}, N ≠ ±1 is important for our proof of our theorem; in particular, if N, Δ ∈ K^{∗}, then [1], ε, ζ may not generate the \overline{K}-endomorphism ring of the jacobian. For example, if N = ±1, then the jacobian is \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} whose \overline{ℚ}-endomorphism ring is isomorphic to ℤ[\sqrt{−6}], as we now explain. C is the curve

Y^{2} = −2Δ(X − 1)(3X^{4} + 4X^{2} + 4X + 1),

which is \overline{ℚ}-isomorphic to the curve

Y^{2} = X^{6} + (39 + 30\sqrt{2})X^{4} + (3 + 12\sqrt{2})X^{2} − 2\sqrt{2} − 3,

so the jacobian of C is \overline{ℚ}-isogenous to the square of the elliptic curve

E: Y^{2} = X^{3} + (39 + 30\sqrt{2})X^{2} + (3 + 12\sqrt{2})X − 2\sqrt{2} − 3,

whose j-invariant is

j(E) = 2417472 + 1707264\sqrt{2}

(see [Sil1], p. 46 for how to compute the j-invariant). We will show that the \overline{ℚ}-endomorphism ring of E is isomorphic to ℤ[\sqrt{−6}], which implies that the jacobian of C is \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} whose \overline{ℚ}-endomorphism ring is isomorphic to ℤ[\sqrt{−6}], as required. This follows from the fact that j(E) is a zero of the Hilbert class polynomial for the discriminant -24, which is

X^{2} − 4834944X + 14670139392,

by Algorithm 7.6.1 in [Coh] (see Section 2.10).

The main motivation for deriving our quaternionic family is its connection with Conjecture 1.1.2, which is as follows. If the jacobian of a curve in it is modular, then, as we will see, a newform f such that the jacobian of a curve in our quaternionic family is ℚ-isogenous to A_{f} will have the following property:

χ(n)f_{n} = σ(f_{n}),  (5.2)

for all positive integers n coprime to 2 and the primes of bad reduction of the curve, where χ is the primitive Dirichlet character modulo 3 defined by

χ: ℤ → ℂ, n ↦ ( \begin{matrix}n \\ 3 \end{matrix} ) ,  (5.3)

and σ is the non-trivial automorphism of ℚ(\sqrt{2}). To start this chapter, we will prove part of Theorem 5.1.1, under plausible assumptions relating to Conjecture 1.1.2.

5.2 The curve Y^{2} = X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3

In this section, we will prove part of Theorem 5.1.1 for the curve C given by N = \frac{1}{2}, Δ = −2, which is

Y^{2} = X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3,  (5.4)

under the following plausible assumptions relating to Conjecture 1.1.2:

1. There is a ℚ(\sqrt{−3})-endomorphism η of the jacobian of C, with σ^{−1}ησ = −η if σ is the non-trivial automorphism of ℚ(\sqrt{−3}).

2. The jacobian of C is ℚ-isomorphic to A_{f} for some newform f on Γ_{0}(675) with K_{f} = ℚ(\sqrt{2}).

More precisely, we will show that η^{2} = [−3] and ηε+εη = [0]. It suffices to show, under these assumptions, that (5.2) holds for all positive integers n, where f is a newform provided by Assumption 2. Then, we can apply Proposition 2.11.5.

Consider the table in Section A.1, which lists curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor. There are 4 such curves whose jacobians have conductor 675^{2}, namely

C_{1}: Y^{2} = X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3,
C_{−3}: Y^{2} = −3(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3),
C_{5}: Y^{2} = 5(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3),
C_{−15}: Y^{2} = −15(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3).

The first stage in forming our assumptions is to show that Conjecture 1.1.2 implies Assumption 1. We will do this by looking at newforms and L-series.

Now Conjecture 1.1.2, Lemma 2.12.1 and Corollary 2.12.2 imply that the jacobian of C_{1} is ℚ-isogenous to A_{f} for some newform f on Γ_{0}(675) with K_{f} = ℚ(\sqrt{2}) (the last assertion holding due to the fact that the jacobian has a \sqrt{2} ℚ-multiplication, by Proposition 3.5.5), and similarly for C_{−3}, C_{5}, C_{−15}.

However, from the first table in Section 6.8, we see that there are precisely 2 G_{ℚ}-conjugacy classes of newforms f on Γ_{0}(675) with K_{f} = ℚ(\sqrt{2}), so Conjecture 1.1.2, Lemma 2.12.1 and Corollary 2.12.2 imply that the jacobians of C_{1}, C_{−3}, C_{5}, C_{−15} can be partitioned into two pairs, the jacobians in a pair being ℚ-isogenous. This is supported by the equality of the L-series at the primes p, 7 ≤ p ≤ 97, of good reduction for the jacobians of C_{1} and C_{−3}, and by a similar occurrence for the jacobians of C_{5} and C_{−15}, as shown by the table below (taking (2.12) and (2.16) into account). Recall that the a_{p}’s and the b_{p}’s are computed using (2.13) and (2.14) respectively.

\begin{matrix}p & a_{p}(C_{1}), b_{p}(C_{1}) & a_{p}(C_{−3}), b_{p}(C_{−3}) & a_{p}(C_{5}), b_{p}(C_{5}) & a_{p}(C_{−15}), b_{p}(C_{−15}) \\ 7 & −6, 23 & −6, 23 & 6, 23 & 6, 23 \\ 11 & 0, 4 & 0, 4 & 0, 4 & 0, 4 \\ 13 & −6, 35 & −6, 35 & 6, 35 & 6, 35 \\ 17 & 0, 26 & 0, 26 & 0, 26 & 0, 26 \\ 19 & 2, 39 & 2, 39 & 2, 39 & 2, 39 \\ 23 & 0, −4 & 0, −4 & 0, −4 & 0, −4 \\ 29 & 0, 40 & 0, 40 & 0, 40 & 0, 40 \\ 31 & 4, 66 & 4, 66 & 4, 66 & 4, 66 \\ 37 & −18, 155 & −18, 155 & 18, 155 & 18, 155 \\ 41 & 0, 64 & 0, 64 & 0, 64 & 0, 64 \\ 43 & −12, 122 & −12, 122 & 12, 122 & 12, 122 \\ 47 & 0, 86 & 0, 86 & 0, 86 & 0, 86 \\ 53 & 0, 8 & 0, 8 & 0, 8 & 0, 8 \\ 59 & 0, 46 & 0, 46 & 0, 46 & 0, 46 \\ 61 & −26, 291 & −26, 291 & −26, 291 & −26, 291 \\ 67 & 6, 143 & 6, 143 & −6, 143 & −6, 143 \\ 71 & 0, −20 & 0, −20 & 0, −20 & 0, −20 \\ 73 & −18, 227 & −18, 227 & 18, 227 & 18, 227 \\ 79 & −10, 183 & −10, 183 & −10, 183 & −10, 183 \\ 83 & 0, 164 & 0, 164 & 0, 164 & 0, 164 \\ 89 & 0, 178 & 0, 178 & 0, 178 & 0, 178 \\ 97 & −6, 203 & −6, 203 & 6, 203 & 6, 203 \end{matrix}

So, in particular, Conjecture 1.1.2 implies that the jacobians of C_{1} and C_{−3} are ℚ-isogenous, which implies that Assumption 1 holds, as required.

The second stage in forming our assumptions is to give evidence for Assumption 2. We will do this by looking at polarisations on abelian varieties associated to newforms via the Shimura construction.

We have seen that Conjecture 1.1.2, Lemma 2.12.1 and Corollary 2.12.2 imply that the jacobian of C_{1} is ℚ-isogenous to A_{f} for some newform f on Γ_{0}(675) with K_{f} = ℚ(\sqrt{2}). From the second table in Section 6.8, we see that any factor A_{f} of J_{0}(675) which corresponds to some newform f on Γ_{0}(675) with K_{f} = ℚ(\sqrt{2}) has a ℂ-polarisation λ_{f} of type (1, 2), so of degree 4, and a \sqrt{2} ℚ-multiplication. If λ_{f}ε is, in fact, twice a principal ℚ-polarisation, then A_{f} has a principal ℚ-polarisation, so, since A_{f} is ℚ-simple, A_{f} is the jacobian of a curve of genus 2 defined over ℚ. So we have evidence for Assumption 2, as required.

We will now prove part of Theorem 5.1.1 (as described at the beginning of the section) for the curve C given by N = \frac{1}{2}, Δ = −2, under Assumptions 1 and 2. Recall that it suffices to show that (5.2) holds for all positive integers n, where f is a newform provided by Assumption 2.

Firstly, we will show that, if p ≥ 7 is a prime, then a_{p} = 0 if p ≡ 2 mod 3 and a_{p}^{2} = 4(b_{p} − 2p) if p ≡ 1 mod 3, in preparation for showing that (5.2) holds for all positive integers n, where f is a newform provided by Assumption 2. The second statement is the harder, involving Proposition 2.4.6.

Applying Lemma 2.4.4 to C: = C_{1} and C̃: = C_{−3}, we see that ã_{p} = −a_{p} if p ≡ 2 mod 3, since then p is not a quadratic residue modulo 3. By Assumption 1 and Theorem 2.5.1, the L-series of the jacobians of C_{1} and C_{3} are the same at all primes p, so a_{p} = 0 if p ≡ 2 mod 3. This is supported by the table earlier in this section.

In order to show that a_{p}^{2} = 4(b_{p} − 2p) if p ≡ 1 mod 3, we need to show that ε, η do not commute, in preparation for applying Proposition 2.4.3 and Proposition 2.4.6. Suppose, for a contradiction, that they do commute. Then, ℚ(\sqrt{2}) embeds into, but is not isomorphic to, the centraliser of ε in the ℚ(\sqrt{−3})- endomorphism algebra of the jacobian of C, since [1], ε are both defined over ℚ and η is defined over ℚ(\sqrt{−3}) but not over ℚ. To obtain our contradiction, we will show that this algebra embeds simultaneously into the algebra of 2 by 2 matrices M_{−19} over ℚ(\sqrt{−19}) and into the algebra of 2 by 2 matrices M_{−43} over ℚ(\sqrt{−43}).

Consider M_{−19}. Let Jac(C)_{7} be the reduction of Jac(C) at the prime 7. The characteristic polynomial of the Frobenius endomorphism of Jac(C)_{7} is, by (2.12), (1 + 3x + 7x^{2})^{2} (recall that a_{7} = −6, b_{7} = 23 from our table earlier in this section), and so, by a remark in [Tat], p. 141, the 𝔽_{p}-endomorphism algebra of Jac(C)_{7} is isomorphic to M_{−19}, −19 being the discriminant of 1 + 3x + 7x^{2}. Moreover, the ℚ(\sqrt{−3})-endomorphism algebra of Jac(C) embeds into the former algebra, by Proposition 2.4.3 (taking K to be ℚ(\sqrt{−3}) and 𝔭 to be a prime ideal of K above 7ℤ), and so it embeds into the latter algebra, as required. Similarly, taking the prime 13 instead of the prime 7, we see that it embeds into M_{−43}, as required.

We will now show that a_{p}^{2} = 4(b_{p} − 2p) if p ≡ 1 mod 3. Since ε, η do not commute and p ≡ 1 mod 3, i.e. p splits in ℚ(\sqrt{−3}), the 𝔽_{p}-endomorphism ring of the reduction of the jacobian of C at p is not commutative (applying Proposition 2.4.3, taking K, 𝔭 as before), and so a_{p}^{2} = 4(b_{p} − 2p) by Proposition 2.4.6, as required. Again, this is supported by the table earlier in this section.

Secondly, we show that a_{2} = 0, b_{2} = 2 (the jacobian of C has good reduction at 2, since its conductor is 675^{2} which is not divisible by 2, and so a_{2}, b_{2} exist for it), again in preparation for showing that (5.2) holds for all positive integers n, where f is a newform provided by Assumption 2. In (2.13), (2.14), we can take C_{2} to be the curve

Z^{2} + (X_{3} + X_{2} + X_{1} + X_{0})Z = X_{2}X_{3} + X_{2}^{2} + X_{0}^{2}
X_{1}X_{2} = X_{0}X_{3}, X_{1}^{2} = X_{0}X_{2}, X_{2}^{2} = X_{1}X_{3},

which is defined over 𝔽_{2} and non-singular, since C is ℚ-isomorphic to the curve

Z^{2} + (X_{3} + X_{2} + X_{1} + X_{0})Z = X_{2}X_{3} − 3X_{2}^{2} − 2X_{0}X_{1} − X_{0}^{2},
X_{1}X_{2} = X_{0}X_{3}, X_{1}^{2} = X_{0}X_{2}, X_{2}^{2} = X_{1}X_{3};

the former curve is simply the latter curve reduced modulo 2. Let ξ ∈ 𝔽_{4} such that ξ^{2} + ξ + 1 = 0. The points (x_{0} : x_{1} : x_{2} : x_{3} : z) on C_{2} defined over 𝔽_{4} are precisely

(1 : 0 : 0 : 0 : ξ), (1 : 0 : 0 : 0 : ξ + 1), (1 : 1 : 1 : 1 : 1),
(1 : ξ : ξ + 1 : 1 : 0), (1 : ξ : ξ + 1 : 1 : 1), (0 : 0 : 0 : 1 : 0),
(1 : ξ + 1 : ξ : 1 : 0), (1 : ξ + 1 : ξ : 1 : 1), (0 : 0 : 0 : 1 : 1),

three of which are defined over 𝔽_{2}, so |C_{1}(𝔽_{2})| = 3, |C_{1}(𝔽_{4})| = 9. We deduce from (2.13), (2.14) that a_{2} = 0, b_{2} = 2, as required.

To summarise, we have, for p ≥ 7, a_{p} = 0 if p ≡ 2 mod 3, a_{p}^{2} = 4(b_{p} − 2p) if p ≡ 1 mod 3, and a_{2} = 0, b_{2} = 2.

Finally, we use these properties for the a_{p}’s and the b_{p}’s to show that (5.2) holds for all positive integers n, where f is a newform provided by Assumption 2.

For any prime p, write f_{p} = α_{p} + β_{p}\sqrt{2}, α_{p}, β_{p} ∈ ℚ. By Assumption 2 and Theorem 2.5.1, the L-series of the jacobian of C and A_{f} are the same at all primes p. This implies that a_{p} = 2α_{p} and b_{p} = 2p + α_{p}^{2} − 2β_{p}^{2} if p is a prime of good reduction of A_{f}, by Theorem 2.11.3 and Theorem 2.11.4. So, for p ≥ 7, α_{p} = 0 if p ≡ 2 mod 3, and β_{p} = 0 if p ≡ 1 mod 3. Also, α_{2} = 0.

From the relations between the Fourier coefficients of a newform which are listed in [Cre1], p. 15, we see that f_{3^{e}} = f_{5^{e}} = 0 for positive integers e (since 3^{2} | 675, 5^{2} | 675),

f_{p^{e+1}} = f_{p}f_{p^{e}} − pf_{p^{e−1}}

for primes p ∉ {3, 5} and positive integers e, and

f_{nm} = f_{n}f_{m}

for coprime positive integers n, m. We deduce that (5.2) holds for all positive integers n, as required.

Given Theorem 5.1.1, we see, by analogous arguments, that a newform f such that the jacobian of a curve in our quaternionic family is ℚ-isogenous to A_{f} will satisfy (5.2) for all positive integers n coprime to 2 and the primes of bad reduction of the curve.

5.3 Discovering Theorem 5.1.1

Having seen that the jacobian of the curve (5.4) has a \sqrt{2} ℚ-multiplication and probably has a ℚ(\sqrt{−3})-endomorphism η satisfying the properties described at the beginning of Section 5.2, it would be nice to find a family of curves, containing it, which probably have these properties, and then to prove that they have these properties (and also to find the \overline{ℚ}-endomorphism ring of the jacobian of such a curve, as described in Theorem 5.1.1). We will look at discovering Theorem 5.1.1 in this section, and proving Theorem 5.1.1 in Sections 5.4 and 5.5.

Consider the curve (5.4) (which is the curve C given by N = \frac{1}{2}, Δ = −2 in Theorem 5.1.1, and studied in Section 5.2). Defining F_{6} ∈ ℚ[X] such that C is the curve

Y^{2} = F_{6}(X),  (5.5)

we have

F_{6} = −4 ∏_{i=0}^{2}(α_{i}(α_{i} − 1)X^{2} − X + α_{i}(α_{i} + 1)),  (5.6)

where the α_{i}’s are the zeros of X^{3}− \frac{1}{2}, whose discriminant is −3 times a rational square; the discriminant of F_{6} is (2^{6} · 3^{3} · 5^{3})^{2}, which is a rational square.

In the light of the curve (5.4), it seems sensible to search for our family of curves amongst the curves of the form (5.5) which are ℚ-isomorphic to a curve in the \sqrt{2} multiplication family presented in Chapter 3, such that the discriminant of F_{6} is a rational square, and that F_{6} is a constant in ℚ multiplied by the product of three ℚ-conjugate quadratics which are permuted by any automorphism of K, where K ⊂ \overline{ℚ} is a splitting field of a cubic defined over ℚ whose discriminant is −3 times a rational square.

Let C be a curve of the form (5.5) which is ℚ-isomorphic to a curve in our \sqrt{2} multiplication family, let x_{0}, …, x_{5} be the zeros of F_{6} such that P_{i}: = (x_{i}, 0) satisfy (3.11), and let S be a permutation group on 6 letters which is compatibly isomorphic to the Galois group of F_{6} over ℚ. Then, the discriminant of F_{6} is a rational square if and only if the permutations in S are all even (this is not true for F_{6} in general). If the \overline{ℚ}-endomorphism ring of the jacobian of C has the properties described in Theorem 5.1.1, then the restriction of η to the 2-torsion of the jacobian is defined over ℚ, and the action of η on the 2-torsion may well result in a smaller Galois group of the 2-torsion subfield of \overline{ℚ} over ℚ, which is isomorphic to the Galois group of F_{6} over ℚ; hopefully, the permutations in S are all even, so that the discriminant of F_{6} will be a rational square. The idea that F_{6} is a constant in ℚ multiplied by the product of three ℚ-conjugate quadratics which are permuted by any automorphism of K, where K ⊂ \overline{ℚ} is the splitting field of a cubic defined over ℚ whose discriminant is −3 times a rational square, is not as obvious as the idea that the discriminant of F_{6} is a rational square, but it seems a sensible idea, taking into consideration that η is defined over ℚ(\sqrt{−3}).

Even with these restrictions, it is not immediately clear that there is an a priori reason why finding another curve C besides the curve (5.4) such that the \overline{ℚ}-endomorphism ring of the jacobian of C has the properties described in Theorem 5.1.1 should be a simple matter, let alone finding a family of such curves. It turns out, fortunately, that it is a simple matter to find a curve besides (5.4) such that the \overline{ℚ}-endomorphism ring of the jacobian probably has these properties, and then to find a family of such curves, as we will explain below (note that we do not prove in this section that the \overline{ℚ}-endomorphism ring of the jacobian does indeed have these properties).

Searching amongst the other curves from Theorem 3.2.1, we discover that the jacobian of the curve

Y^{2} = X^{6} − 9X^{5} + 6X^{4} − 36X^{3} + 3X^{2} + 9X + 2  (5.7)

(which is the curve C given by N = 3, Δ = \frac{1}{6} in Theorem 5.1.1) has \sqrt{2} ℚmultiplication, by Proposition 3.5.5, and, defining F_{6} ∈ ℚ[X] such that C is the curve (5.5), we have

F_{6} = \frac{1}{6} ∏_{i=0}^{2}(α_{i}(α_{i} − 1)X^{2} − 6X + α_{i}(α_{i} + 1)),  (5.8)

where the α_{i}’s are the zeros of X^{3} − 3, whose discriminant is −3 times a rational square; the discriminant of F_{6} is (2^{7} · 3^{5} · 5^{3})^{2}, which is a rational square. This curve was found as a result of a search for a suitable curve from our \sqrt{2} multiplication family (which did not take too long) and then by taking a ℚ-isomorphism to get a model with smaller coefficients.

Looking at the L-series of the jacobians of the curves

Y^{2} = X^{6} − 9X^{5} + 6X^{4} − 36X^{3} + 3X^{2} + 9X + 2,
Y^{2} = −3(X^{6} − 9X^{5} + 6X^{4} − 36X^{3} + 3X^{2} + 9X + 2),

it seems probable that the jacobians are ℚ-isogenous, using Theorem 2.5.1, and so it seems probable that the \overline{ℚ}-endomorphism ring of the jacobian of the curve 5.7 has the properties described in Theorem 5.1.1 (think of the theory in Section 5.2 being applied to the curve 5.7 instead of the curve 5.4).

So we have two curves, (5.4) and (5.7), such that the \overline{ℚ}-endomorphism ring of the jacobian probably has the properties described in Theorem 5.1.1. Can we find a family of such curves? Looking at the sextics (5.6) and (5.8), it seems a good idea to consider curves of the form (5.1), since the factorisation of the sextic in (5.1) is

Δ((N − 1)X^{6} − 6NX^{5} + 3(N + 1)X^{4}  (5.9)
− 8N^{2}X^{3} + 3(N − 1)X^{2} + 6NX + N + 1)
= \frac{Δ}{N} ∏_{i=0}^{2}(X^{2}α_{i}(α_{i} − 1) − 2NX + α_{i}(α_{i} + 1)),

where the α_{i}’s are the zeros of X^{3} − N; indeed, (5.6) is the above sextic with N = \frac{1}{2}, Δ = −2, and (5.8) is the above sextic with N = 3, Δ = \frac{1}{6}, so two curves of the form (5.1) have already been considered.

A curve of the form (5.1) has \sqrt{2} ℚ-multiplication, by Proposition 3.5.5, the discriminant of (5.9) is (2^{9} · 3^{3}N^{2}(N^{2} + 1)^{3}Δ^{5})^{2}, a rational square, and the discriminant of X^{3} − N is −3 times a rational square. Moreover, looking at the L-series of the jacobians of the curves

Y^{2} = Δ((N − 1)X^{6} − 6NX^{5} + 3(N + 1)X^{4}
− 8N^{2}X^{3} + 3(N − 1)X^{2} + 6NX + N + 1),
Y^{2} = −3Δ((N − 1)X^{6} − 6NX^{5} + 3(N + 1)X^{4}
− 8N^{2}X^{3} + 3(N − 1)X^{2} + 6NX + N + 1),

for fixed N, Δ ∈ ℚ^{∗}, N ≠ ±1, it seems probable that the jacobians are ℚisogenous. So it is reasonable to conjecture Theorem 5.1.1, again thinking of the theory in Section 5.2.

5.4 The Outline of the Proof of Theorem 5.1.1

In this section, we state several preparatory propositions and then deduce Theorem 5.1.1 from them. The propositions themselves will be proved in Section 5.5.

Throughout this section, we will fix N, Δ ∈ ℚ^{∗}, N ≠ ±1, and C, Θ, F_{6} will be (5.1), a theta divisor on the jacobian of (5.1), (5.9) (so that C is the curve (5.5)) respectively.

Moreover, it will be useful to consider the Galois group of F_{6} over ℚ, so we now define appropriate notation. Let x_{0}, …, x_{5} be a labelling of the zeros of F_{6} such that

α_{i}(α_{i} − 1)(X − x_{i})(X − x_{i+3}) = α_{i}(α_{i} − 1)X^{2} − 2NX + α_{i}(α_{i} + 1), (5.10)

where the α_{i}’s are the zeros of X^{3}−N (if N is a rational cube, rearrange the α_{i}’s if necessary to ensure that α_{0} is a rational number). Defining P_{i}: = (x_{i}, 0), a Weierstrass point on C from (5.9), we see that 3.11 holds. Let S be the subgroup of S_{6} such that the Galois group of F_{6} over ℚ is compatibly isomorphic to S w.r.t. our labelling of the zeros of F_{6}. A bit of elementary Galois theory, taking into account the facts that N ≠ ±1 and that α_{0} is a rational number if N is a rational cube, can be used to show that

S = ⟨(03)(14), (12)(45)⟩ ≅ D_{8}  (5.11)

if N is a rational cube, and that

S = ⟨(05)(23), (012)(345)⟩ ≅ S_{4}  (5.12)

if N is not a rational cube; in particular,

S ⊇ ⟨(03)(14), (12)(45)⟩ ≅ D_{8}.  (5.13)

Proposition 5.4.1 The jacobian of C has a \sqrt{2} ℚ-multiplication ε which is Θ-symmetric.

Proposition 5.4.2 The jacobian of C has a \frac{1+\sqrt{5}}{2} \overline{ℚ}-multiplication ζ which is Θ-symmetric.

Proposition 5.4.3

ζε + εζ = [n] + ε  (5.14)

for some integer n such that −3 ≤ n ≤ 3.

Proposition 5.4.4 Let K be the subfield of \overline{ℚ} which is a splitting field of F_{6}, and let n be as in Proposition 5.4.3. If n is even, then any σ ∈ G_{ℚ} which restricts to the automorphism of K corresponding to the permutation (12)(45) has the property that the endomorphisms σ^{−1}ζσ − ζ, σ^{−1}ζσ − ([1] − ζ) are not divisible by [2], and, if n is odd, a similar thing happens with (12)(45) replaced by (14)(25) (note that there exist automorphisms of K corresponding to the permutations (12)(45) and (14)(25), by (5.13)).

Suppose that n is even. Then, for all σ ∈ G_{ℚ} which restrict to the trivial automorphism of ℚ(\sqrt{−3}), the endomorphism σ^{−1}ζσ−ζ is divisible by [2], and, for all σ ∈ G_{ℚ}, the endomorphism σ^{−2}ζσ^{2} − ζ is divisible by [2].

Proposition 5.4.5 Suppose that the jacobian of C is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication. If σ ∈ G_{ℚ}, then

σ^{−1}ζσ ∈ \{ζ, [1] + [\frac{n}{2}] ε − ζ\} ,  (5.15)

n being as in Proposition 5.4.3 (note that σ^{−1}ζσ = [1] + [\frac{n}{2}]ε − ζ implies that n is even, since both sides are \overline{ℚ}-endomorphisms of the jacobian of C).

Proposition 5.4.6 Suppose that the jacobian of C is \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication by an order in ℚ(\sqrt{D}), D being a negative integer, and let δ be an element in the centre of the \overline{ℚ}-endomorphism algebra of the jacobian such that δ^{2} = [D]. Then, δ is a ℚ(\sqrt{D})-endomorphism, and σ^{−1}δσ = −δ if σ is the non-trivial automorphism of ℚ(\sqrt{D}).

If σ ∈ G_{ℚ} restricts to the trivial automorphism of ℚ(\sqrt{D}), then (5.15) holds, n being as in Proposition 5.4.3.

Proposition 5.4.7 The jacobian of C is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication.

Proposition 5.4.1 and Proposition 5.4.2 establish the existence of suitable endomorphisms ε, ζ of the jacobian of C, Proposition 5.4.3 will use complexanalytic techniques to link them together, Proposition 5.4.4 will use the action of G_{ℚ}, η, ζ on the 2-torsion of the jacobian of C, applying Proposition 2.9.7 to ζ, Proposition 5.4.5 and Proposition 5.4.6 will also use these actions, but will apply the structure of the \overline{ℚ}-endomorphism algebra of the jacobian of C instead of Proposition 2.9.7, and Proposition 5.4.7 is the most involved proposition in this section, the proof of it using previous propositions in this section, Proposition 3.3.2 and the theory from Section 2.10.

Proof: (of Theorem 5.1.1) Firstly, we will show that n is even (n being as in Proposition 5.4.3). Suppose, for a contradiction, that n is odd. Then, by Proposition 5.4.7 and Proposition 5.4.5, σ^{−1}ζσ = ζ for all σ ∈ G_{ℚ}, which contradicts Proposition 5.4.4. So n is even.

Secondly, we will discuss σ^{−1}ζσ for σ ∈ G_{ℚ}. By Proposition 5.4.7, Proposition 5.4.4 and Proposition 5.4.5, σ^{−1}ζσ = ζ for all σ ∈ G_{ℚ} which restrict to the trivial automorphism of ℚ(\sqrt{−3}) (so ζ is a ℚ(\sqrt{−3})-endomorphism), and

σ^{−1}ζσ = [1] + [\frac{n}{2}] ε − ζ  (5.16)

for all σ ∈ G_{ℚ} which restrict to the non-trivial automorphism of ℚ(\sqrt{−3}).

Thirdly, we will show that n = ±2. Having already shown that n is even, it suffices to show that n is not divisible by 4, by Proposition 5.4.3. Suppose, for a contradiction, that n is divisible by 4. Then, by Proposition 5.4.7 and Proposition 5.4.5, one of σ^{−1}ζσ − ζ, σ^{−1}ζσ − ([1] − ζ) is divisible by [2] for all σ ∈ G_{ℚ} (the one which is depends on σ), which contradicts Proposition 5.4.4. So n is not divisible by 4. We deduce that there exists s ∈ {±1} such that

ζε + εζ = [2s] + ε  (5.17)

and, from (5.16), that

σ^{−1}ζσ = [1] + [s]ε − ζ  (5.18)

for all σ ∈ G_{ℚ} which restrict to the non-trivial automorphism of ℚ(\sqrt{−3}).

Thirdly, define

η: = [1] + [2]ζ([1] − [s]ε).  (5.19)

Using (5.17), (5.18), (5.19) and the fact that s ∈ {±1}, we see, after a bit of algebraic manipulation, that η is a \sqrt{−3} ℚ(\sqrt{−3})-multiplication such that σ^{−1}ησ = −η if σ is the non-trivial automorphism of ℚ(\sqrt{−3}), and that εη+ηε = [0], as required for Theorem 5.1.1.

Finally, we need to show that [1], ε, \frac{[1]+η}{[2]} generate the \overline{ℚ}-endomorphism ring of the jacobian of C. By Proposition 5.4.7, it is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ}. So, by Corollary 2.7.5, Theorem 2.7.7, Theorem 2.7.8 and Theorem 2.8.5, [1], ε, \frac{[1]+η}{[2]} generate its \overline{ℚ}-endomorphism algebra of the jacobian. By Proposition 2.2.5, they also generate its \overline{ℚ}-endomorphism ring, since they generate over ℤ a maximal order in a division quaternion algebra over ℚ.□

5.5 The Proofs of the Propositions

Throughout this section, as in the previous section, we will fix N, Δ ∈ ℚ^{∗}, N ≠ ±1, and C, Θ, F_{6} will be (5.1), a theta divisor on the jacobian of (5.1), (5.9) (so that C is the curve (5.5)) respectively.

Proposition 5.4.1 The jacobian of C has a \sqrt{2} ℚ-multiplication ε which is Θ-symmetric.

Proof: By Proposition 3.5.5, the jacobian of C has a \sqrt{2} ℚ-multiplication ε. To show that ε is Θ-symmetric, it suffices to show that the kernel of ε is the only subgroup of order 4 of the 2-torsion of the jacobian defined over ℚ; then, we can apply Proposition 3.3.2 and Proposition 2.8.1. In fact, this follows from (5.13). □

Proposition 5.4.2 The jacobian of C has a \frac{1+\sqrt{5}}{2} \overline{ℚ}-multiplication ζ which is Θ-symmetric.

Proof: Consider the curves

C^{HM}: Y^{2} = X(X^{4} − P X^{3} + QX^{2} − RX + 1),
C̃: Y^{2} = (X − 1)(X + 1)
× ((2 − P + Q − R)X^{4} + 2(P − R)X^{3} + 2(6 − Q)X^{2}
+ 2(R − P)Y + 2 + P + Q + R),

defined over \overline{ℚ}, where

P = −2(S + T), R = −2(S − T), Q = \frac{(1 + 2T^{2})(11 − 28T^{2} + 8T^{4})}{3(1 − T^{2})(1 − 4T^{2})},

4S^{2}T^{2} − S^{2} + T^{2} + 2 = 0, S, T ∈ \overline{ℚ};

HM stands for Hayashida and Murabayashi, who found the curves C^{HM} (see [H-M], Theorem 1.3). For fixed S, T ∈ \overline{ℚ}, C^{HM}, C̃ are ℚ-isomorphic, a ℚisomorphism C^{HM} → C̃ being (x, y) ↦ (\frac{x+1}{x−1}, \frac{y}{(x−1)^{3}} ). To prove the proposition, it suffices to show that there exists S, T ∈ \overline{ℚ} such that C̃ is \overline{ℚ}-isomorphic to C, so that C^{HM} is \overline{ℚ}-isomorphic to C, as we now explain. The usage of C^{HM} for this specific S, T is motivated by the fact that Hayashida and Murabayashi have proved that the jacobian of C^{HM} has a \frac{1+\sqrt{5}}{2} \overline{ℚ}-multiplication, fixed by the Rosati involution w.r.t. the canonical polarisation (see [H-M], Theorem 1.3), so that the jacobian of C will have such an endomorphism too, and the usage of C̃ for this specific S, T is motivated by the fact that C̃ does not have a Weierstrass point at ∞ (so that integral invariants of C̃ are definable as in Section 2.3) whereas C^{HM} has a Weierstrass point at ∞ (so that integral invariants of C^{HM} are not definable as in Section 2.3), making showing that C, C̃ are \overline{ℚ}-isomorphic easier than showing that C, C^{HM} are \overline{ℚ}-isomorphic.

We will now show that there exists S, T ∈ \overline{ℚ} such that C̃ is \overline{ℚ}-isomorphic to C, so that C^{HM} is \overline{ℚ}-isomorphic to C. After a bit of computation, we see that

\frac{I_{2}^{5}}{I_{10}} = \frac{128(4N^{4} − 19N^{2} + 4)^{5}}{3N^{4}(N^{2} + 1)^{6}} ,
\frac{I_{2}^{3}I_{4}}{I_{10}} = \frac{−288(4N^{4} − 19N^{2} + 4)^{3}}{N^{2}(N^{2} + 1)^{4}} ,
\frac{I_{2}^{2}I_{6}}{I_{10}} = \frac{−96(5N^{4} − 17N^{2} + 5)(4N^{4} − 19N^{2} + 4)^{2}}{N^{2}(N^{2} + 1)^{4}} ,
\frac{Ĩ_{2}^{5}}{Ĩ_{10}} = \frac{2(16T^{6} − 24T^{4} + 36T^{2} − 1)^{5}(16T^{6} + 24T^{4} − 24T^{2} + 11)^{5}}{3(T − 1)^{6}(T + 1)^{6}(2T − 1)^{6}(2T + 1)^{6}(1 + 2T^{2})^{12}} ,
\frac{Ĩ_{2}^{3}Ĩ_{4}}{Ĩ_{10}} = \frac{18(16T^{6} − 24T^{4} + 36T^{2} − 1)^{3}(16T^{6} + 24T^{4} − 24T^{2} + 11)^{3}}{(T − 1)^{4}(T + 1)^{4}(2T − 1)^{3}(2T + 1)^{3}(1 + 2T^{2})^{8}} ,
\frac{Ĩ_{2}^{2}Ĩ_{6}}{Ĩ_{10}} = \frac{6(16T^{6} − 24T^{4} + 36T^{2} − 1)^{2}(16T^{6} + 24T^{4} − 24T^{2} + 11)^{2}}{(T − 1)^{4}(T + 1)^{4}(2T − 1)^{3}(2T + 1)^{3}(1 + 2T^{2})^{8}}
× (320T^{12} − 480T^{8} + 1568T^{6} − 1116T^{4} + 444T^{2} − 7).

We deduce that a common factor of

\frac{I_{2}^{5}}{I_{10}} − \frac{Ĩ_{2}^{5}}{Ĩ_{10}} , \frac{I_{2}^{3}I_{4}}{I_{10}} − \frac{Ĩ_{2}^{3}Ĩ_{4}}{Ĩ_{10}} , \frac{I_{2}^{2}I_{6}}{I_{10}} − \frac{Ĩ_{2}^{2}Ĩ_{6}}{Ĩ_{10}}

is

(2T + 1)^{3}(2T − 1)^{3}(N^{4} + 1) + 2(2T^{4} + 1)(16T^{8} − 32T^{4} + 24T^{2} + 1)N^{2}.

We have fixed N ∈ ℚ^{∗} − \{1\}; I_{2} ≠ 0, and there exists T ∈ \overline{ℚ} such that the common factor is zero and that Ĩ_{10} is non-zero. Taking S to be any zero of 4S^{2}T^{2} −S^{2} +T^{2} +2, C̃ is \overline{ℚ}-isomorphic to C, by Lemma 5.1.1 and the fact that any Möbius ℂ-transformation which sends six elements of \overline{ℚ} to six elements of \overline{ℚ} is a \overline{ℚ}-transformation.□

Proposition 5.4.3 (5.14) holds for some integer n such that −3 ≤ n ≤ 3.

Proof: Proposition 2.8.4, with ε′: = [2]ζ − [1]. ε is Θ-symmetric by Proposition 5.4.1, and ε′ is Θ-symmetric by Proposition 5.4.2. □

Proposition 5.4.4 Let K be the subfield of \overline{ℚ} which is a splitting field for F_{6}, and let n be as in Proposition 5.4.3. If n is even, then any σ ∈ G_{ℚ} which restricts to the automorphism of K corresponding to the permutation (12)(45) has the property that the endomorphisms σ^{−1}ζσ − ζ, σ^{−1}ζσ − ([1] − ζ) are not divisible by [2], and, if n is odd, a similar thing happens with (12)(45) replaced by (14)(25) (note that there exist automorphisms of K corresponding to the permutations (12)(45) and (14)(25), by (5.13)).

Suppose that n is even. Then, for all σ ∈ G_{ℚ} which restrict to the trivial automorphism of ℚ(\sqrt{−3}), the endomorphism σ^{−1}ζσ−ζ is divisible by [2], and, for all σ ∈ G_{ℚ}, the endomorphism σ^{−2}ζσ^{2} − ζ is divisible by [2].

Proof: The proof involves comparing the actions of ε, ζ, σ ∈ G_{ℚ} on the 2torsion of the jacobian of C, as might be expected.

A basis for the 2-torsion over 𝔽_{2} is

{[P_{0} − P_{3}], [P_{1} − P_{4}], [P_{0} − P_{1}], [P_{0} − P_{2}]},  (5.20)

the first two points as listed lying in the kernel of ε (due to (3.11)). We will represent endomorphisms of the 2-torsion by matrices in M_{4}(𝔽_{2}) corresponding to the basis (5.20). If τ is a permutation in S_{6}, define M_{τ} to be the matrix which corresponds to the endomorphism sending [P_{i} − P_{j}] to [P_{τ(i)} − P_{τ(j)}].

Define M_{ε}, M_{ζ} to be the matrices representing actions of ε, ζ respectively on the 2-torsion. Now ε sends [P_{0} − P_{3}], [P_{1} − P_{4}] to 0, and [P_{0} − P_{1}] to [P_{2} − P_{5}], [P_{0} − P_{2}] to [P_{1} − P_{4}], so

M_{ε}: = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) .

We will now divide the proof of the proposition into the case where n is even and the case where n is odd.

Suppose that n is even. Since ζ^{2} − ζ − [1] = 0, M_{ζ}^{2} = M_{ζ} + 1_{4}. Since ζε + εζ = [n] + ε and n is even, M_{ζ}M_{ε} + M_{ε}M_{ζ} = M_{ε}. After a bit of computation, we see that

M_{ζ} = ( \begin{matrix}a & 1 & 0 & 0 \\ 1 & a + 1 & 0 & 0 \\ b & c & a & 1 \\ b + c & b & 1 & a + 1 \end{matrix} )

for some a, b, c ∈ 𝔽_{2}. If (a, b, c) = (0, 0, 0), then

ζ([P_{0} − P_{1}]) = [P_{0} − P_{2}],

which is impossible by Proposition 2.9.7, since ζ is Θ-symmetric. The cases (a, b, c) = (1, 0, 0), (a, b, c) = (0, 1, 1), (a, b, c) = (1, 1, 1) are similarly impossible. So b = c + 1, and

M_{ζ} = ( \begin{matrix}a & 1 & 0 & 0 \\ 1 & a + 1 & 0 & 0 \\ b & b + 1 & a & 1 \\ 1 & b & 1 & a + 1 \end{matrix} ) .

Now, from the definition of M_{τ} given above where τ is a permutation in S_{6}, we have

M_{(12)(45)} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) ,

so

M_{(12)(45)}^{−1}M_{ζ}M_{(12)(45)} = ( \begin{matrix}a + 1 & 1 & 0 & 0 \\ 1 & a & 0 & 0 \\ b + 1 & b & a + 1 & 1 \\ 1 & b + 1 & 1 & a \end{matrix} ) ∉ \{M_{ζ}, M_{ζ} + 1_{4}\},

so any σ ∈ G_{ℚ} which restricts to the automorphism of K corresponding to the permutation (12)(45) has the property that the endomorphisms σ^{−1}ζσ − ζ, σ^{−1}ζσ − ([1] − ζ) are not divisible by [2], as required.

An automorphism σ ∈ G_{ℚ} restricts to the trivial automorphism of ℚ(\sqrt{−3}) over ℚ if and only if it restricts to an automorphism in K corresponding to a permutation in

S ∩ ⟨(012)(345), (015)(342)⟩.  (5.21)

Now, from the definition of M_{τ} given above where τ is a permutation in S_{6}, we have

M_{(012)(345)} = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , M_{(015)(342)} = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \end{matrix} ) ,

so

M_{(012)(345)}^{−1}M_{ζ}M_{(012)(345)} = M_{ζ}, M_{(015)(342)}^{−1}M_{ζ}M_{(015)(343)} = M_{ζ},

so, if τ is in (5.21), then

M_{τ}^{−1}M_{ζ}M_{τ} = M_{ζ}.

We deduce that, for all σ ∈ G_{ℚ} which restrict to the trivial automorphism of ℚ(\sqrt{−3}) over ℚ, the endomorphism σ^{−1}ζσ − ζ is divisible by [2], as required. Also, for all σ ∈ G_{ℚ}, σ^{2} restricts to the trivial automorphism of ℚ(\sqrt{−3}) over ℚ, so the endomorphism σ^{−2}ζσ^{2} − ζ is divisible by [2], as required. We have now proved the proposition in the case where n is even.

Suppose that n is odd. Since ζ^{2} − ζ − [1] = 0, M_{ζ}^{2} = M_{ζ} + 1_{4}. Since ζε + εζ = [n] + ε and n is odd, M_{ζ}M_{ε} + M_{ε}M_{ζ} = 1_{4} + M_{ε}. After a bit of computation, we see that

M_{ζ}

= ( \begin{matrix}a & b & 1 & 1 \\ c & d & 0 & 1 \\ c(a + b + d + 1) + 1 & b(a + c + d + 1) + 1 & a + c + 1 & a + b + c + d \\ c(a + d + 1) & bc + 1 & c & c + d + 1 \end{matrix} )

for some a, b, c, d ∈ 𝔽_{2}.

Now, from the definition of M_{τ} given above where τ is a permutation in S_{6}, we have

M_{(14)(25)} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \end{matrix} ) ,

so

M_{(14)(25)}^{−1}M_{ζ}M_{(14)(25)}

= ( \begin{matrix}e & f & 1 & 1 \\ g & h & 0 & 1 \\ g(e + f + h + 1) + 1 & f(e + g + h + 1) + 1 & e + g + 1 & e + f + g + h \\ g(e + h + 1) & fg + 1 & g & g + h + 1 \end{matrix} )

∉ {M_{ζ}, M_{ζ} + 1_{4}},

where e = a + 1, f = b, g = c + 1, h = d + 1, so any σ ∈ G_{ℚ} which restricts to the automorphism of K corresponding to the permutation (14)(25) has the property that the endomorphisms σ^{−1}ζσ − ζ, σ^{−1}ζσ − ([1] − ζ) are not divisible by [2], as required. We have now proved the proposition in the case where n is odd.□

Proposition 5.4.5 Suppose that the jacobian of C is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication. If σ ∈ G_{ℚ}, then (5.15) holds, n being as in Proposition 5.4.3 (note that σ^{−1}ζσ = [1] + [\frac{n}{2}]ε − ζ implies that n is even, since both sides are \overline{ℚ}-endomorphisms of the jacobian of C).

Proof: First, we work out the \overline{ℚ}-endomorphism algebra of the jacobian of C. Since it is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication, its \overline{ℚ}-endomorphism algebra is generated by [1], ε, ζ, by Corollary 2.7.5, Theorem 2.7.7, Theorem 2.7.8 and Theorem 2.8.5.

We have established that ε and ζ are Θ-symmetric (Propositions 5.4.1 and 5.4.2 respectively). So the Θ-symmetric \overline{ℚ}-endomorphisms of the jacobian of C lie in the ℚ-vector space

ℚ[1] ⊕ ℚε ⊕ ℚζ,  (5.22)

a subspace of the \overline{ℚ}-endomorphism algebra. We will use this space to establish the proposition, bearing in mind that the action of G_{ℚ} on the \overline{ℚ}-endomorphisms of the jacobian of C preserves the Θ-symmetric ones.

Let σ ∈ G_{ℚ}. Since ζ is Θ-symmetric, σ^{−1}ζσ is Θ-symmetric as well. So, from (5.22),

σ^{−1}ζσ = [a] + [b]ε + [c]ζ  (5.23)

for some a, b, c ∈ ℚ. Now

σ^{−1}ζσε + εσ^{−1}ζσ = σ^{−1}(ζε + εζ)σ = σ^{−1}([n] + ε)σ = [n] + ε  (5.24)

since ε is defined over ℚ, so, after a bit of algebraic manipulation using the fact that ε^{2} = [2] and (5.14), (5.23), (5.24), we see that

[n] + ε = [4b + nc] + [2a + c]ε;  (5.25)

separating the coefficients of [1] and ε, which are linearly independent over ℚ, we deduce that c = 1 − 2a and b = \frac{an}{2}. We deduce, after a bit of algebraic manipulation using (5.14), (5.23) and the facts that ε^{2} = [2], ζ^{2} − ζ − [1] = [0], that

(σ^{−1}ζσ)^{2} = [a^{2} (5 − \frac{n^{2}}{2} ) − a (4 − \frac{n^{2}}{2} ) + 1] + [\frac{an}{2}] ε + [1 − 2a]ζ. (5.26)

Since ζ^{2} −ζ −[1] = [0], (σ^{−1}ζσ)^{2} −σ^{−1}ζσ −[1] = [0]. Separating the coefficients of [1], ε, ζ in (σ^{−1}ζσ)^{2} − σ^{−1}ζσ − [1], which are linearly independent over ℚ, we deduce, using (5.23) and (5.26), that

(n^{2} − 10)a(a − 1) = 0;

since n ∈ ℚ and 10 ∉ ℚ^{2},

a ∈ {0, 1}.

If a = 0, then σ^{−1}ζσ = ζ. If a = 1, then σ^{−1}ζσ = [1] + [\frac{n}{2}]ε − ζ. So (5.15) holds, as required. □

Proposition 5.4.6 Suppose that the jacobian of C is \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication by an order in ℚ(\sqrt{D}), D being a negative integer, and let δ be an element in the centre of the \overline{ℚ}-endomorphism algebra of the jacobian such that δ^{2} = [D]. Then, δ is a ℚ(\sqrt{D})-endomorphism, and σ^{−1}δσ = −δ if σ is the non-trivial automorphism of ℚ(\sqrt{D}).

If σ ∈ G_{ℚ} restricts to the trivial automorphism of ℚ(\sqrt{D}), then (5.15) holds, n being as in Proposition 5.4.3.

Proof: First, we discuss δ. The centre of the \overline{ℚ}-endomorphism algebra of the jacobian of C is generated by 1 and δ, so the only square roots of [D] in the centre of the algebra are ±δ. Let σ ∈ G_{ℚ}; then, σ^{−1}δσ is a square root of D and lies in the centre of the algebra, so σ^{−1}δσ = ±δ. Considering all σ ∈ G_{ℚ}, we deduce that δ is defined over a quadratic extension of ℚ. Now there is a ring embedding

End_{ℚ}(Jac(C)) ⊗ ℚ ↪ M_{2}(ℚ)

by Corollary 2.2.10, and, if K is a subfield of \overline{ℚ} which is a quadratic extension of ℚ, then there is a ring embedding

End_{K}(Jac(C)) ⊗ ℚ ↪ M_{2}(K)

by Corollary 2.2.10. The centre of M_{2}(ℚ) is isomorphic to ℚ and the centre of M_{2}(K) is isomorphic to K. Since σ^{−1}δσ is a square root of [D] and lies in the centre of the algebra, δ is defined over ℚ(\sqrt{D}), and σ^{−1}δσ = −δ if σ is the non-trivial automorphism of ℚ(\sqrt{D}), as required.

We will now find a ℚ-vector space of dimension 4 in which the Θ-symmetric endomorphisms of the jacobian of C lie. As mentioned above, the centre of the \overline{ℚ}-endomorphism algebra of the jacobian of C is generated over ℚ by 1 and δ, so the only square roots of [D] in the centre of the algebra are ±δ. Since δ lies in the centre and is a square root of [D], δ^{†_{Θ}} lies in the centre and is a square root of [D]. By Proposition 2.8.1, δ is Θ-antisymmetric. We have established that ε and ζ are Θ-symmetric (Propositions 5.4.1 and 5.4.2 respectively). Using (5.14), we see, after a bit of algebraic manipulation, that ([n] + ([1] − [2]ζ)ε)δ is also Θ-symmetric. So the Θ-symmetric \overline{ℚ}-endomorphisms of the jacobian of C lie in the ℚ-vector space

ℚ[1] ⊕ ℚε ⊕ ℚζ ⊕ ℚ([n] + ([1] − [2]ζ)ε)δ,  (5.27)

a subspace of the \overline{ℚ}-endomorphism algebra. We will use this space to establish the proposition, bearing in mind that the action of G_{ℚ} on the \overline{ℚ}-endomorphisms of the jacobian of C preserves the Θ-symmetric ones.

Let σ ∈ G_{ℚ}. Since ζ is Θ-symmetric, σ^{−1}ζσ is Θ-symmetric as well. So, from (5.27),

σ^{−1}ζσ = [a] + [b]ε + [c]ζ + [d]([n] + ([1] − [2]ζ)ε)δ  (5.28)

for some a, b, c, d ∈ ℚ. Now

σ^{−1}ζσε + εσ^{−1}ζσ = σ^{−1}(ζε + εζ)σ = σ^{−1}([n] + ε)σ = [n] + ε  (5.29)

since ε is defined over ℚ, so, after a bit of algebraic manipulation using the fact that ε^{2} = [2] and (5.14), (5.28), (5.29), we see that (5.25) holds; separating the coefficients of [1] and ε, which are linearly independent over ℚ, we deduce that c = 1 − 2a and b = \frac{an}{2}, as we did in the proof of Proposition 5.4.6. We deduce, after a bit of algebraic manipulation using (5.14), (5.28) and the facts that ε^{2} = [2], ζ^{2} − ζ − [1] = 0, δ^{2} = [D], that

(σ^{−1}ζσ)^{2}  (5.30)
= [a^{2} (5 − \frac{n^{2}}{2} ) − a (4 − \frac{n^{2}}{2} ) + d^{2}D(n^{2} − 10) + 1]
+ [\frac{an}{2}] ε + [1 − 2a]ζ + [d]([n] + ([1] − [2]ζ)ε)δ.

Since ζ^{2} − ζ − [1] = 0, (σ^{−1}ζσ)^{2} − σ^{−1}ζσ − [1] = 0. Separating the coefficients of [1], ε, ζ, ([n] + ([1] − [2]ζ)ε)δ in (σ^{−1}ζσ)^{2} − σ^{−1}ζσ − [1], which are linearly independent over ℚ, we deduce, using (5.28) and (5.30), that

(n^{2} − 10)(a^{2} − a − 2d^{2}D) = 0;

since n ∈ ℚ and 10 ∉ ℚ^{2}, we have

a^{2} − a − 2d^{2}D = 0,  (5.31)

a relationship which will be useful for the rest of the proof of the proposition.

From now on in the proof of the proposition, we will suppose that σ restricts to the trivial automorphism of ℚ(\sqrt{D}), the case of interest.

Suppose that d = 0. Then, the proposition for this case follows from Proposition 5.4.5.

Suppose that d ≠ 0. We will derive a contradiction, and so finish the proof of the proposition. Since ζ is defined over \overline{ℚ}, ζ is defined over a subfield of \overline{ℚ} which is a number field, so

σ^{−k_{σ}}ζσ^{k_{σ}} = ζ  (5.32)

for some positive integer k_{σ} not equal to 1 (if k_{σ} was equal to 1, then a would be zero by (5.28) and so D would be zero by (5.33), which is absurd).

From (5.31), we have

D = \frac{a(a − 1)}{2d^{2}}.  (5.33)

Define

M: = ( \begin{matrix}1 − 2a & \frac{2a(1−a)}{d} \\ −2d & 1 − 2a \end{matrix} ) ,

and define, for a positive integer k, rational numbers a_{k}, d_{k} as follows:

( \begin{matrix}a_{k} \\ d_{k} \end{matrix} ) = (∑_{j=0}^{k−1} M^{j}) ( \begin{matrix}a \\ d \end{matrix} );  (5.34)

in particular, a_{1} = a, d_{1} = d. After a bit of algebraic manipulation involving induction on k, taking (5.33), (5.34) and the fact that σ^{−1}δσ = δ (since σ restricts to the trivial automorphism of ℚ(\sqrt{D})) into account, we see that

σ^{−k}ζσ^{k} = [a_{k}] + [\frac{a_{k}n}{2}] ε + [1 − 2a_{k}]ζ + [d_{k}]([n] + ([1] − [2]ζ)ε)δ. (5.35)

Now, defining I_{2} to be the identity 2 by 2 matrix, we see, since (5.32) holds, that

( \begin{matrix}a \\ d \end{matrix} ) = (∑_{j=0}^{k_{σ}−1} M^{j}) ( \begin{matrix}a \\ d \end{matrix} ) ,

( \begin{matrix}0 \\ 0 \end{matrix} ) = (∑_{j=1}^{k_{σ}−1} M^{j}) ( \begin{matrix}a \\ d \end{matrix} ) = M (∑_{j=0}^{k_{σ}−2} M^{j}) ( \begin{matrix}a \\ d \end{matrix} )

= (∑_{j=0}^{k_{σ}−2} M^{j}) ( \begin{matrix}a \\ d \end{matrix} ) = (M^{k_{σ}−1} − I_{2})(M − I_{2})^{−1} ( \begin{matrix}a \\ d \end{matrix} ) ,

where the fourth equality comes from the fact that M is invertible, since det(M) = 1, and the fifth equality is allowed due to the fact that M − I_{2} is invertible, since a is non-zero (a is zero implies that D is zero by (5.33), which is absurd).

Now M^{k_{σ}−1} − I_{2} is not invertible, since a is non-zero, which implies that 1 is an eigenvalue of M^{k_{σ}−1}, which implies that a root of unity is an eigenvalue of M. The characteristic equation of M is

λ^{2} + 2λ(2a − 1) + 1 = 0;

λ = 1 ⇒ a = 0 ⇒ D = 0, a contradiction; λ = −1 ⇒ a = 1 ⇒ D = 0, a contradiction; λ = \frac{−1±\sqrt{−3}}{2} ⇒ a = \frac{3}{4}; λ = ±\sqrt{−1} ⇒ a = \frac{1}{2}; λ = \frac{1±\sqrt{−3}}{2} ⇒ a = \frac{1}{4}. All roots of unity which solve a quadratic equation over ℚ have been considered, so

a ∈ \{\frac{3}{4}, \frac{1}{2}, \frac{1}{4}\} .

If a = \frac{3}{4}, then (a_{j}, d_{j}), 1 ≤ j ≤ 3 are

(\frac{3}{4}, d) , (\frac{3}{4}, −d) , (0, 0)  (5.36)

respectively (so σ^{−3}ζσ^{3} = ζ, as might be expected since a = \frac{3}{4} corresponds to λ = \frac{−1±\sqrt{−3}}{2}, the primitive 3^{rd} roots of unity). Now, by (5.35) and (5.36), we have

σ^{−1}ζσ + σ^{−2}ζσ^{2} = [\frac{3}{2}] + [\frac{3n}{4}] ε − ζ,

which is a \overline{ℚ}-endomorphism of the jacobian of C since ζ is. So [3] + [\frac{3n}{2}]ε is a \overline{ℚ}-endomorphism which is divisible by [2], a contradiction.

If a = \frac{1}{2}, then (a_{j}, d_{j}), 1 ≤ j ≤ 4 are

(\frac{1}{2}, d) , (1, 0), (\frac{1}{2}, −d) , (0, 0)  (5.37)

respectively (so σ^{−4}ζσ^{4} = ζ, as might be expected since a = \frac{1}{2} corresponds to λ = ±\sqrt{−1}, the primitive 4^{th} roots of unity). Now, by (5.35) and (5.37), we have

σ^{−1}ζσ + σ^{−3}ζσ^{3} = [1] + [\frac{n}{2}] ε,

which is a \overline{ℚ}-endomorphism of the jacobian of C since ζ is. So n is even. But

σ^{−2}ζσ^{2} = [1] + [\frac{n}{2}] ε − ζ,

since (a_{2}, d_{2}) = (1, 0), which is absurd, since σ^{−2}ζσ^{2} − ζ is divisible by [2] by Proposition 5.4.4.

If a = \frac{1}{4}, then (a_{j}, d_{j}), 1 ≤ j ≤ 6 are

(\frac{1}{4}, d) , (\frac{3}{4}, d) , (1, 0), (\frac{3}{4}, −d) , (\frac{1}{4}, −d) , (0, 0)  (5.38)

respectively (so σ^{−6}ζσ^{6} = ζ, as might be expected since a = \frac{1}{4} corresponds to λ = \frac{1±\sqrt{−3}}{2}, the primitive 6^{th} roots of unity). Now, by (5.35) and (5.38), we have

σ^{−1}ζσ + σ^{−5}ζσ^{5} = [\frac{1}{2}] + [\frac{n}{4}] ε + ζ,

which is a \overline{ℚ}-endomorphism of the jacobian of C since ζ is. So [1] + [\frac{n}{2}] ε is a \overline{ℚ}-endomorphism which is divisible by [2], a contradiction.□

Proposition 5.4.7 The jacobian of C is not \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication.

Proof: Suppose, for a contradiction, that the jacobian of C is \overline{ℚ}-isogenous to the square of an elliptic curve defined over \overline{ℚ} with complex multiplication, and let δ be an endomorphism in the centre of the \overline{ℚ}-endomorphism ring of the jacobian which satisfies δ^{2} = [D] for some negative integer D. Our strategy for obtaining a contradiction will be to show that the jacobian splits over ℚ(\sqrt{−3}, \sqrt{D}), and then to show that this is in fact impossible by Proposition 3.3.2 and the theory from Section 2.10.

Now the \overline{ℚ}-endomorphism algebra of the jacobian is generated by [1], ε, ζ, δ, so, since ε is defined over ℚ and δ is defined over ℚ(\sqrt{D}) (by Proposition 5.4.6), the jacobian splits over ℚ(\sqrt{−3}, \sqrt{D}) if ζ is defined over this field. This is true if n is even; indeed, if σ ∈ G_{ℚ} restricts to the trivial automorphism of this field, then (5.15) holds by Proposition 5.4.6, and then σ^{−1}ζσ = ζ by Proposition 5.4.4.

To show that n is even, it suffices to show that there exists an automorphism in G_{ℚ} which restricts to the trivial automorphism of ℚ(\sqrt{D}) and to the automorphism of K (K being as in Proposition 5.4.4) corresponding to the permutation (14)(25); indeed, if σ is such an automorphism and n is odd, then σ^{−1}ζσ = ζ by Proposition 5.4.6, which contradicts Proposition 5.4.4. Such an automorphism exists if ℚ(\sqrt{D}) ∩ K = ℚ, so suppose that ℚ(\sqrt{D}) ∩ K ⊋ ℚ. If N is a rational cube, then all three subgroups of index 2 of S (recall that S is (5.11)) contain (14)(25). If N is not a rational cube, then the only subgroup of index 2 of S (recall that S is (5.12)) contains (14)(25). So any automorphism in G_{ℚ} which restricts to the automorphism of K corresponding to the permutation (14)(25) also restricts to the trivial automorphism of ℚ(\sqrt{D}), in both cases.

Having shown that the jacobian splits over ℚ(\sqrt{−3}, \sqrt{D}), we will now show that this is in fact impossible by Proposition 3.3.2 and the theory from Section 2.10. As before, let x_{0}, …, x_{5} be a labelling of the zeros of F_{6} such that (5.10) holds. Let S′ be the subgroup of S such that the Galois group of F_{6} over ℚ(\sqrt{−3}, \sqrt{D}) is compatibly isomorphic to S′ w.r.t. our labelling of the zeros of F_{6}.

Suppose that N is not a rational cube. Then, S′ is

⟨(012)(345), (015)(342)⟩ ≅ A_{4},

taking (5.12) into account, so, by Proposition 3.3.2, the jacobian does not split over ℚ(\sqrt{−3}, \sqrt{D}), as required.

Suppose that N is a rational cube. This case is harder to deal with than the previous case, since, as we will see, it is possible for the kernel of ε not to be the only subgroup of order 4 of the 2-torsion of the jacobian defined over ℚ(\sqrt{−3}, \sqrt{D}) in this case. Recall that α_{0} is a rational number, and let r be the discriminant of (5.10) w.r.t. X with i = 0, also a rational number. More precisely,

r = 4α_{0}^{2}(1 − α_{0}^{2} + α_{0}^{4}) = 4α_{0}^{2}((1 − α_{0}^{2})^{2} + α_{0}^{2}),  (5.39)

which is positive since α_{0} is non-zero. Note also that r is not a rational square, by (5.11).

We will now show that \sqrt{D} ∈ ℚ(\sqrt{−3r}), using Proposition 3.3.2 (we will not use this proposition afterwards). If \sqrt{D} ∉ ℚ(\sqrt{−3}, \sqrt{r}), or \sqrt{D} ∈ ℚ(\sqrt{−3}), then S′ is

⟨(03)(14), (03)(25)⟩ ≅ V_{4},

taking ( 5.11) into account, so, by Proposition 3.3.2, the jacobian does not split over ℚ(\sqrt{−3}, \sqrt{D}), a contradiction. Also, \sqrt{D} ∉ ℚ(\sqrt{r}), since D is negative and r is positive. So \sqrt{D} ∈ ℚ(\sqrt{−3r}). Note that S′ is then

⟨(14)(25)⟩ ≅ C_{2},

so Proposition 3.3.2 cannot be used to obtain a contradiction.

We will now apply the theory from Section 2.10. Since the jacobian splits over ℚ(\sqrt{−3}, \sqrt{D}) and \sqrt{D} ∈ ℚ(\sqrt{−3r}), there is an abelian subvariety of the jacobian which is an elliptic curve E defined over ℚ(\sqrt{−3}, \sqrt{r}), which implies that

j(E) ∈ ℚ(\sqrt{−3}, \sqrt{r}),  (5.40)

and that the \overline{ℚ}-endomorphism ring of E is an order 𝔬 in ℚ(\sqrt{−3r}), which implies, by (2.74), that

[ℚ(\sqrt{−3r})(j(E)): ℚ(\sqrt{−3r})] = [ℚ(j(E)): ℚ] = h_{𝔬}.  (5.41)

Putting (5.40) and (5.41) together, we see that h_{𝔬} divides 2. From the table in Section 2.10, we see that j(E) lies in a real quadratic field, so, from (5.40), either j(E) ∈ ℚ or j(E) ∈ ℚ(\sqrt{r}) − ℚ. Recall that the \overline{ℚ}-endomorphism ring of E is an order 𝔬 in ℚ(\sqrt{−3r}). Using the table, we see that, if j(E) ∈ ℚ, then

rℚ^{2} ∈ {ℚ^{2}, 3ℚ^{2}, 6ℚ^{2}, 21ℚ^{2}, 33ℚ^{2}, 57ℚ^{2}, 129ℚ^{2}, 201ℚ^{2}, 489ℚ^{2}},  (5.42)

(since in this case ℚ(\sqrt{−3r}) must have an order 𝔬 such that h_{𝔬} = 1), and, if j(E) ∈ ℚ(\sqrt{r}) − ℚ, then

rℚ^{2} ∈ {2ℚ^{2}, 3ℚ^{2}, 5ℚ^{2}, 17ℚ^{2}, 33ℚ^{2}, 41ℚ^{2}, 89ℚ^{2}},  (5.43)

(since in this case ℚ(\sqrt{−3r}) must have an order 𝔬 such that h_{𝔬} = 2).

The proof of the proposition is now completed by showing that explicitly given rational p, q solve a simple equation, and then by showing that the same p, q do not, which will result in a contradiction.

We have established that r is not a rational square, so, from (5.42) and (5.43), r ∈ (3m + 2)ℚ^{2} for some integer m or r ∈ 3mℚ^{2} for some integer m not divisible by 3. From (5.39) and the fact that α_{0} is non-zero, we see that explicitly given rational p, q solve the equation

q^{2} = t(1 − p^{2} + p^{4}),  (5.44)

where t = 3m + 2 or t = 3m according to whether r ∈ (3m + 2)ℚ^{2} or r ∈ 3mℚ^{2} respectively; more precisely,

p = α_{0}, q = \frac{\sqrt{rt}}{2α_{0}}.

But q^{2}, 1 − p^{2} + p^{4} are non-zero squares in ℚ_{3} (since p, q are rational), and t is not a square in ℚ_{3} (since m is an integer and, if t = 3m, is not divisible by 3), so (5.44) cannot hold. This completes the proof of the proposition.□

Conductors of Curves

6.1 Introduction

In this chapter, we list some curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor, in order to study Conjecture 1.1.2 for curves of genus 2 in the case where K_{f} = ℚ(\sqrt{2}). We will obtain our list by first substituting into our \sqrt{2} multiplication family, and then by taking Richelot isogenies and isogenies of degree 2.

For more general K_{f}, in the case where A_{f} has a principal ℂ-polarisation, X. Wang has a method for finding a curve C of genus 2 such that the jacobian of C is ℂ-isomorphic to A_{f}, which involves calculating the invariants of A_{f} and trying to find a curve with those invariants (see [Wan1]). While his method has the advantage that it is independent of K_{f}, it has the disadvantages that it only works when A_{f} has a principal ℂ-polarisation, and that the curve obtained may not be ℚ-isogenous to A_{f}, although it will be ℂ-isomorphic to A_{f}, by Lemma 2.3.3. For example, he gives the curve

C: Y^{2} = X^{5} − 4X^{4} + 814X^{3} + 964X^{2} + 62969X − 680,

which is ℂ-isomorphic to A_{f} where f is a newform of level 189 such that K_{f} = ℚ(\sqrt{7}), but the odd part of the conductor of the jacobian of C, computed using Q. Liu’s algorithm, is 3^{5} · 7^{3}, which is not a square, implying that the jacobian is not ℚ-isogenous to A_{f}.

6.2 Curves with Low Conductor and \sqrt{2} Multiplication

The table in Section A.1 lists curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductor. The odd parts of the conductors were computed using Q. Liu’s algorithm; discussion about the conductors will appear later in this section.

For most of the putative ℚ-isogeny classes, the isogenies are known, as we will see in Section 6.4, Section 6.5 and Section 6.6, being mostly ℚ-isogenies of degree 2 of the form described in Theorem 4.3.1 and Richelot isogenies, and occasionally coming from Theorem 5.1.1 (if Y^{2} = F_{6}(X) is a curve from this theorem, then the jacobians of the curves Y^{2} = F_{6}(X), Y^{2} = −3F_{6}(X) are ℚ-isogenous).

If C is a curve in the table, then the ℚ-endomorphism ring of its jacobian is isomorphic to ℤ[\sqrt{2}], as we will show in Section 6.7, so that Conjecture 1.1.2 is applicable. One corollary of this is that if C_{1}, C_{2} are distinct curves in the table, then their jacobians are not ℚ-isomorphic, as we will show in Section 6.7.

The table in Section A.2 gives, for each curve in the table in Section A.1, the discriminant of F_{6} w.r.t. X (as a sextic), which can be used to give information about the conductor of the jacobian of C, and the Galois group of F_{6} over ℚ, which can be used to decide whether Conjecture 1.1.2 is applicable or not.

If 2^{12} exactly divides the discriminant of F_{6}, then, from the discussion about curves of genus 2 defined over 𝔽_{2} in Section 2.3, we see that C has good reduction at 2, and so that the conductor of its jacobian is odd. For example, consider the curve 29a_{1}, for which F_{6} is

X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7.  (6.1)

We see that the discriminant of F_{6} w.r.t. X is 2^{12} · 29^{5}, and F_{6} is congruent to X^{6} + 2X^{3} + 1 modulo 4, so, applying the discussion in Section 2.3, we see that the conductor of the jacobian of 29a_{1} is odd.

Whether 2^{12} exactly divides the discriminant of F_{6} or not, the conductor of the jacobian of C will not be divisible by an odd prime p if p does not divide the discriminant of F_{6}, the conductor of the jacobian of C should be a square, thinking of Conjecture 1.1.2 and Theorem 2.12.3, and, bearing in mind that the conductor of an elliptic curve defined over ℚ which has semistable reduction at all primes is squarefree, the exponent of the conductor of the jacobian at C at a prime p should be 2 if it has semistable reduction at p, i.e. (by Theorem 2.4.1 and Lemma 2.4.2) if the zeros of F_{6} modulo p (regarded as a sextic) all have multiplicity ≤ 2. For example, consider the curve 29a_{1}. From (6.1), we see that the discriminant of F_{6} w.r.t. X is 2^{12}·29^{5}, and F_{6} is (X +22)^{2}(X +18)^{2}(X +16)^{2} modulo 29, so the conductor of the jacobian of 29a_{1} should be 29^{2}, as it indeed is.

We now discuss briefly a method of intelligently guessing the conductor of an abelian surface defined over ℚ and likely to be ℚ-isogenous to A_{f} for some newform f on Γ_{0}(N), where N is a positive integer. We can apply this method to intelligently guess the power of 2 exactly dividing the conductor of the jacobian of a curve in the table in Section A.1, having already considered the odd part.

Let A be an abelian surface defined over ℚ, and let f be a newform on Γ_{0}(N), where N is a positive integer, such that A_{f} is an abelian surface. Let D be a positive integer, squarefree and not 1, such that K_{f} = ℚ(\sqrt{D}); for any prime p, write f_{p} = α_{p} + β_{p}\sqrt{D}, α_{p}, β_{p} ∈ ℚ. If A is ℚ-isogenous to A_{f}, which implies that the conductor of A is N^{2} by Theorem 2.12.3, then, by Conjecture 1.1.2 and Theorem 2.5.1, the L-series of A and A_{f} are the same at all primes p. This implies that

a_{p} = 2α_{p}, b_{p} = 2p + α_{p}^{2} − Dβ_{p}^{2}  (6.2)

if p is a prime of good reduction of A_{f}, by Theorem 2.11.3 and Theorem 2.11.4. Up to a chosen bound for our p’s, we can compute the a_{p}’s and the b_{p}’s, and, if we have the α_{p}’s and the β_{p}’s to hand, e.g. from [Tin], we can see whether (6.2) holds for all of our p’s or not; if so, then we have evidence that A is ℚ-isogenous to A_{f}, which implies that the conductor of A is N^{2}, the strength of the evidence increasing as the size of the bound increases.

To illustrate this method, consider the curve 256a_{1}, which is the curve

C: Y^{2} = X^{5} − 4X,

(the odd part of the jacobian of C is 1), and let f be a newform on Γ_{0}(256) such that K_{f} = ℚ(\sqrt{2}). Consider the L-series of the jacobian of C at the primes 3 to 19. After a bit of computation, we see that

|C_{3}(𝔽_{3})| = 4, |C_{5}(𝔽_{5})| = 6, |C_{7}(𝔽_{7})| = 8,
|C_{11}(𝔽_{11})| = 12, |C_{13}(𝔽_{13})| = 14, |C_{17}(𝔽_{17})| = 6, |C_{19}(𝔽_{19})| = 20,
|C_{3}(𝔽_{3^{2}})| = 6, |C_{5}(𝔽_{5^{2}})| = 46, |C_{7}(𝔽_{7^{2}})| = 78,

|C_{11}(𝔽_{11^{2}})| = 150, |C_{13}(𝔽_{13^{2}})| = 222, |C_{17}(𝔽_{17^{2}})| = 286, |C_{19}(𝔽_{19^{2}})| = 294,

where C_{p} is the reduction of C at p. Applying (2.13) and (2.14), we see that

a_{3} = 0, a_{5} = 0, a_{7} = 0, a_{11} = 0, a_{13} = 0, a_{17} = 12, a_{19} = 0,
b_{3} = −2, b_{5} = 10, b_{7} = 14, b_{11} = 14, b_{13} = 26, b_{17} = 70, b_{19} = −34.

Consider the L-series of A_{f} at the primes 3 to 19. As shown in [Tin],

f_{3} = −2s\sqrt{2}, f_{5} = 0, f_{7} = 0, f_{11} = 2s\sqrt{2}, f_{13} = 0, f_{17} = 6, f_{19} = 6s\sqrt{2}

for some s ∈ {±1}. So (6.2), taking D = 2, holds for the primes 3 to 19 (excluding the prime 2 for simplicity), and so it is likely that the conductor of the jacobian of C is 256^{2}, in particular that the power of 2 exactly dividing the conductor is 2^{16}.

Consider a curve C: Y^{2} = F_{6}(X) in the table in Section A.1 corresponding to N = 98 or N = 196. The odd primes of good reduction of C less than or equal to 19, and so probably of the jacobian of C, are 3, 5, 11, 13, 17, 19. The curves Y^{2} = F_{6}(X), Y^{2} = −2F_{6}(X) have the same L-series at each of these primes, and Tingley’s data for the newforms on Γ_{0}(98) and on Γ_{0}(196) only go up to the prime 19, so, considering the method above, it is not obvious whether the jacobian of the former curve or the jacobian of the latter curve has conductor 98^{2} or 196^{2}, although we expect that this is so for one of the jacobians. To resolve the ambiguity, we note that, if N = 98, the correct jacobian should be ℚ(\sqrt{−7})-isogenous to the square of the elliptic curve

Y^{2} − XY − αY = X^{3} + (α − 2)X^{2} + (α − 10)X − α − 8,

and that, if N = 196, the correct jacobian should be ℚ(\sqrt{−7})-isogenous to the square of the elliptic curve

Y^{2} = X^{3} + (1 − α)X^{2} − 4X + 2 + α,

where α = \frac{1+\sqrt{−7}}{2}, from [Cre2], p. 409; by applying Lemma 2.4.5, (2.11) and (2.12) at some odd primes p not equal to 7 for which −7 is a quadratic residue (so that the reduced elliptic curve is defined over 𝔽_{p}), the ambiguity can be resolved.

The conductors of the jacobians of the curves in the table in Section A.1 corresponding to even N ≤ 320 are conjecturably N^{2} by the above considerations. The conductors of the jacobians of the curves corresponding to even N in general are conjecturably N^{2} by Lemma 2.5.2.

6.3 Finding Curves with Low Conductor and \sqrt{2} Multiplication

Recall our \sqrt{2} multiplication family from Theorem 3.2.2. In order to find curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication and low conductors, an obvious procedure is to find suitable curves from this family. A sensible assumption to start off with, in the absence of empirical evidence, is that the smaller the coefficients of a curve C of genus 2 (modelled as Y^{2} = F_{6}(X), F_{6} ∈ ℤ[X]), the smaller the conductor of its jacobian will be, since there is a rough correlation between the size of the coefficients of F_{6} and the size of the discriminant of F_{6} and so the size of the odd prime factors dividing the discriminant, which are the odd primes of bad reduction of C. Looking at the equations defining the family, it therefore seems sensible to start with P = ±1 and with small rational values for A, Q, and use small rational values for P later on. Putting P = ±1 and using small rational values for A, Q usually gives curves with low conductor. Other useful values for P, although not as useful as P = ±1, are small rational values for P such as P = ±1/2, P = ±1/3, P = ±1/5. In general, except for very small rational values for A, P, Q, curves with low conductor tend to be isolated amongst curves with very high conductor, and finding curves with low conductor seems to be a matter of searching for the needles in the haystacks to some extent. In general, in order to find curves C of genus 2 with low odd conductor, one looks for F_{6} such that the power of 2 exactly dividing the discriminant of F_{6} is 12; this often occurs when P has an odd lowest numerator and an odd lowest denominator. Other useful powers of 2 to look for are integers congruent to 12 modulo 30.

As well as our \sqrt{2} multiplication family, other, simpler, families can be usefully used, for example the curves in (4.1) and (4.2) discussed in Chapter 4.

Given a curve

Y^{2} = F_{6}(X)

and λ ∈ ℤ^{∗}, we can consider the curve

Y^{2} = λF_{6}(X)

(twisting by ℚ(\sqrt{λ})). For example, consider the curve

Y^{2} = X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7,

the modular curve X_{0}(29), which has conductor 29^{2}, and which is ℚ-isomorphic to the curve from our \sqrt{2} multiplication family given by A = 2, P = −1, Q = −4, Δ = 1. By fixing A, P, Q and replacing Δ by Δλ, λ ∈ ℤ^{∗} (i.e. twisting X_{0}(29) by ℚ(\sqrt{λ})), one can get curves which have conductors a low square multiple of 29^{2}. For example, putting Δ = −3 gives a curve which has conductor (3^{2} · 29)^{2}. Generally, if λ is congruent to 1 modulo 4, as is the case in the tables above, replacing Δ by Δλ multiplies the conductor by λ^{4} if the original conductor and λ are coprime, and by λ^{2} if the original conductor and λ are not coprime.

If the jacobian of a curve of genus 2 has \sqrt{2} ℚ-multiplication and a rational point of order 2, we can usually find other such curves using ℚ-isogenies of degree 2 of the form described in Theorem 4.3.1 and Richelot isogenies, as we will see in Section 6.4 and Section 6.5.

The table in Section A.3 gives, for each curve in the table in Section A.1, the values of A, P, Q such that it is ℚ-isomorphic to a curve from our \sqrt{2} multiplication family, and a ℚ-isomorphism. The Möbius ℚ-transformations corresponding to the isomorphisms are small integers, indicating that our \sqrt{2} multiplication family generally gives curves with small coefficients, although, for most curves C from it, there is usually a curve ℚ-isomorphic to C with smaller coefficients.

6.4 Taking Isogenies of Degree 2

There are many pairs of curves in the table in Section A.1 with the property that there are ℚ-isogenies of degree 2 coming from Theorem 4.3.1 between their jacobians. The table in Section A.4 shows these pairs.

Note that curves with U = 1, V = 1 and with U = 1, V = −1 appear regularly, indicating that curves from the families (4.1) and (4.2) appear regularly, as expected.

6.5 Taking Richelot Isogenies

In this section, we discuss the Richelot dual curves of a curve of genus 2 defined over ℚ whose jacobian has \sqrt{2} ℚ-multiplication, applying the discussion to the curves in the table in Section A.1.

Let C be a curve of genus 2 whose jacobian has a \sqrt{2} ℚ-multiplication ε such that

ker ε = {0, [P_{00}^{ε} − P_{01}^{ε}], [P_{10}^{ε} − P_{11}^{ε}], [P_{20}^{ε} − P_{21}^{ε}]},

where the P_{ik}^{ε}’s are the Weierstrass points on C. So the jacobian has a subgroup of the form

{0, [P_{00} − P_{01}], [P_{10} − P_{11}], [P_{20} − P_{21}]}  (6.3)

defined over ℚ, where the P_{ik}’s are the Weierstrass points on C, namely ker ε. However, there may be more than one such subgroup. The jacobian of the Richelot dual curve of C w.r.t. such a subgroup Φ satisfying

Φ ≠ ker ε  (6.4)

will be ℚ-isogenous to and have the same conductor as the jacobian of C, but will generally not be ℚ-isomorphic to it. Write

Φ: = {0, [P_{00}^{Φ} − P_{01}^{Φ}], [P_{10}^{Φ} − P_{11}^{Φ}], [P_{20}^{Φ} − P_{21}^{Φ}]},

where the P_{ik}^{Φ}’s are the Weierstrass points on C, and write

G^{Φ}_{i}(X): = (X − X(P_{i0}^{Φ}))(X − X(P_{i1}^{Φ})).

Using the notation of Section 2.6, the jacobian of C_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} has the \sqrt{8} ℚ-multiplication

ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨};

indeed, (ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨})^{2} = [8], since

ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨}ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} = [2], ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} = [2], ε^{2} = [2].

If ε|_{Φ} < Φ, then we see that ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is twice a \sqrt{2} ℚmultiplication. Indeed, the points of order dividing 2 on the jacobian of C_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} are sent by ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} to the points in Φ, which are sent by ε to the points in a subgroup of Φ (by hypothesis), which are sent by ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} to zero. So ker ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} contains all of the points of order dividing 2, and ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is twice a \sqrt{2} ℚ-multiplication.

If ε|_{Φ} ≮ Φ, then we see that ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is a primitive \sqrt{8} ℚ-multiplication. Indeed, the points of order dividing 2 on the jacobian of C_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} are sent by ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} to the points in Φ, at least one of which is sent by ε to a point not in Φ (by hypothesis), which is not sent by ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}} to zero. So ker ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} does not contain all of the points of order dividing 2, and ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is a primitive \sqrt{8} ℚ-multiplication.

If ker ε ∩ Φ is non-trivial, then the order of ker ε ∩ Φ is 2 since Φ ≠ ker ε (by 6.4). Relabelling the P_{ik}’s and the P_{ik}^{Φ}’s if necessary, ensuring that ker ε and Φ are still

{0, [P_{00}^{ε} − P_{01}^{ε}], [P_{10}^{ε} − P_{11}^{ε}], [P_{20}^{ε} − P_{21}^{ε}]},
{0, [P_{00}^{Φ} − P_{01}^{Φ}], [P_{10}^{Φ} − P_{11}^{Φ}], [P_{20}^{Φ} − P_{21}^{Φ}]}

respectively, we may assume without loss of generality that

P_{00}^{Φ} = P_{00}^{ε}, P_{01}^{Φ} = P_{01}^{ε}

and that

ker ε ∩ Φ = {0, [P_{00}^{ε} − P_{01}^{ε}]}.

Suppose that ι is a ℚ-isomorphism from the jacobian of C to that of C_{G_{0},G_{1},G_{2}} such that (3.8) holds, which will exist if the jacobian of C is simple over ℚ, by Theorem 3.2.2. By the definition of ρ_{G_{0},G_{1},G_{2}}, we see that ε|_{Φ} = {0, [P_{00}^{ε} − P_{01}^{ε}]\} < Φ. So ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is twice a \sqrt{2} ℚ-multiplication.

If ker ε ∩ Φ is trivial, then any two different points in Φ are sent by ε to two different points of order dividing 2 on the jacobian of C. Since the points of order dividing 2 on the jacobian of C are sent by ε to ker ε, the points in Φ are sent to ker ε. So ε|_{Φ} = ker ε ≮ Φ. So ρ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}ερ_{G^{Φ}_{0},G^{Φ}_{1},G^{Φ}_{2}}^{∨} is a primitive \sqrt{8} ℚ-multiplication.

For example, consider the curve

C: Y^{2} = X^{5} + 3X^{3} − 4X
= X(X − 1)(X + 1)(X^{2} + 4),

which is the curve 160a_{1} in the table in Section A.1. The jacobian of C has a \sqrt{2} ℚ-multiplication ε, by Proposition 3.5.5. There are three subgroups of the jacobian of the form (6.3), namely:

1. \{0, [∞ − (0, 0)], [(1, 0) − (−1, 0)], [(2\sqrt{−1}, 0) − (−2\sqrt{−1}, 0)]\};

2. \{0, [∞ − (1, 0)], [(0, 0) − (−1, 0)], [(2\sqrt{−1}, 0) − (−2\sqrt{−1}, 0)]\};

3. \{0, [∞ − (−1, 0)], [(0, 0) − (1, 0)], [(2\sqrt{−1}, 0) − (−2\sqrt{−1}, 0)]\}.

Consider the first subgroup. Define (using the notation of Section 2.6)

G_{0}(X): = X, G_{1}(X): = (X − 1)(X + 1), G_{2}(X): = X^{2} + 4.

Then,

H_{0}(X) = G′_{1}(X)G_{2}(X) − G′_{2}(X)G_{1}(X) = 10X,
H_{1}(X) = G′_{2}(X)G_{0}(X) − G′_{0}(X)G_{2}(X) = (X − 2)(X + 2),
H_{2}(X) = G′_{0}(X)G_{1}(X) − G′_{1}(X)G_{0}(X) = −X^{2} − 1,

and, putting G_{i}(X): = g_{i2}X^{2} + g_{i1}X + g_{i0},

det(g_{uv}) = 5.

So the Richelot dual curve of C with respect to the first subgroup, C_{G_{0},G_{1},G_{2}}, is, from (2.20) with Δ = 1,

Y^{2} = −2(X^{5} − 3X^{3} − 4X)
= −2X(X − 2)(X + 2)(X^{2} + 1).

C is ℚ-isomorphic to this curve, the isomorphism being

(x, y) ↦ (\frac{2}{x}, \frac{4y}{x^{3}} ) .

In fact, the first subgroup is ker ε. The second and third subgroups intersect non-trivially with ker ε, so, by the working above, the jacobians of the Richelot dual curves of C with respect to these subgroups have \sqrt{2} ℚ-multiplications. The Richelot dual curve of C with respect to the second subgroup is ℚisomorphic to the curve 160a_{4}, and the Richelot dual curve of C with respect to the third subgroup is ℚ-isomorphic to the curve 160a_{5}; the jacobians of these two curves are ℚ-isogenous to, but not ℚ-isomorphic to, the jacobian of C.

In a similar way, the jacobians of the following pairs of curves are ℚ-isogenous:

(320a_{1}, 320a_{4}), (320a_{1}, 320a_{5}), (800a_{1}, 800a_{4}), (800a_{1}, 800a_{5}),
(1440a_{1}, 1440a_{4}), (1440a_{1}, 1440a_{5}).

Let C_{1}, C_{2}, C_{3}, C_{4} be four curves in the table in Section A.1 within one putative ℚ-isogeny class such that C_{i} is listed before C_{j} if i is less than j. Suppose that the putative ℚ-isogeny class has the property that three of its curves are listed in the table in Section A.4. The jacobian of C_{1} has only one subgroup of the form (6.3), (so that the Richelot dual curve of C w.r.t. such a subgroup is ℚ-isomorphic to C), and, for all i ∈ {2, 3, 4}, the jacobians of C_{i} have three subgroups of the form (6.3), as can be easily seen by looking at the Galois groups in Section A.2. In Section 6.7, we will show, for all i ∈ {2, 3, 4}, that the Richelot dual curve of C_{i} w.r.t. any of the three subgroups exists, and that C_{2}, C_{3}, C_{4} are each ℚ-isomorphic to one of the three Richelot dual curves (so that taking Richelot isogenies gives no more curves than taking isogenies of degree 2 here).

Finally in this section, we derive some curves of genus 2 defined over ℚ whose jacobians have primitive \sqrt{8} ℚ-multiplication from the curves 55a_{4}, 273a_{4}.

Consider the curve 55a_{4}, which is

Y^{2} = −3(X^{6} − 18X^{5} + 63X^{4} + 154X^{3} − 27X^{2} − 48X − 4)
= −3(X^{2} + X − 1)(X^{2} − 8X − 4)(X^{2} − 11X − 1).

Let ε be a \sqrt{2} ℚ-multiplication of the jacobian of C, provided by Proposition 3.5.5. Then, there are four subgroups of the jacobian of the form (6.3) which intersect trivially with the kernel of ε. The Richelot duals of C w.r.t. these subgroups, whose jacobians, by the working above, have primitive \sqrt{8} ℚ-multiplication, are

Y^{2} = −9(3X^{6} − 70X^{5} + 501X^{4} − 890X^{3} − 1137X^{2} − 380X − 40)
= −9(3X^{2} − 28X − 8)(X^{4} − 14X^{3} + 39X^{2} + 30X + 5),
Y^{2} = \frac{3}{19}(109X^{6} + 510X^{5} − 1197X^{4} − 110X^{3} − 3591X^{2} − 540X − 2320)
= \frac{3}{19}(X^{2} + 4X − 16)(109X^{4} + 74X^{3} + 251X^{2} + 70X + 145),
Y^{2} = \frac{3}{35}(125X^{6} + 750X^{5} − 5085X^{4} + 434X^{3} + 10089X^{2} + 828X − 3368)
= \frac{3}{35}(5X^{2} − 4X − 8)(25X^{4} + 170X^{3} − 841X^{2} − 314X + 421),
Y^{2} = −\frac{9}{7}(5X^{6} − 10X^{5} + 51X^{4} + 698X^{3} + 105X^{2} + 196X)
= −\frac{9}{7}X(X + 4)(5X^{4} − 30X^{3} + 171X^{2} + 14X + 49).

Consider the curve 273a_{4}, which is

Y^{2} = −3X^{6} + 6X^{5} − 25X^{4} − 10X^{3} − 55X^{2} − 24X − 36
= −(X^{2} + X + 1)(X^{2} − 3X + 9)(3X^{2} + 4).

Let ε be a \sqrt{2} ℚ-multiplication of the jacobian of C, provided by Proposition 3.5.5. Then, there are four subgroups of the jacobian of the form (6.3) which intersect trivially with the kernel of ε. The Richelot duals of C w.r.t. these subgroups, whose jacobians, by the working above, have primitive \sqrt{8} ℚ-multiplication, are

Y^{2} = −\frac{1}{11}(49X^{6} − 378X^{5} + 755X^{4} + 110X^{3} + 2285X^{2} + 732X + 1368)
= −\frac{1}{11}(7X^{2} + 4X + 8)(7X^{4} − 58X^{3} + 133X^{2} + 6X + 171),
Y^{2} = −\frac{1}{15}(37X^{6} − 186X^{5} − 729X^{4} + 3118X^{3} + 2985X^{2} + 900X)
= −\frac{1}{15}X(X + 4)(37X^{4} − 334X^{3} + 607X^{2} + 690X + 225),
Y^{2} = −\frac{1}{67}(793X^{6} − 186X^{5} + 4075X^{4} + 2910X^{3} + 5125X^{2} + 7764X + 4656)
= −\frac{1}{67}(13X^{2} − 12X + 48)(61X^{4} + 42X^{3} + 127X^{2} + 186X + 97),
Y^{2} = \frac{1}{25}(125X^{6} + 150X^{5} − 865X^{4} + 1518X^{3} + 1217X^{2} − 2004X − 1464)
= −\frac{1}{25}(5X^{2} + 12X − 24)(25X^{4} − 30X^{3} + 19X^{2} + 114X + 61).

Consider the curve 735b_{3}, which is

Y^{2} = −75X^{6} − 60X^{5} + 736X^{4} + 110X^{3} − 1844X^{2} + 1140X − 75
= −(3X^{2} − 9X + 5)(5X^{2} + 13X − 1)(5X^{2} + 6X − 15).

Let ε be a \sqrt{2} ℚ-multiplication of the jacobian of C, provided by Proposition 3.5.5. Then, there are four subgroups of the jacobian of the form (6.3) which intersect trivially with the kernel of ε. The Richelot duals of C w.r.t. these subgroups, whose jacobians, by the working above, have primitive \sqrt{8} ℚ-multiplication, are

Y^{2} = −7(45X^{6} + 60X^{5} − 328X^{4} − 18X^{3} + 812X^{2} − 840X + 245)
= −7(5X^{2} + 10X − 7)(9X^{4} − 6X^{3} − 41X^{2} + 70X − 35),
Y^{2} = −\frac{1}{43}(3825X^{6} + 14820X^{5} − 34792X^{4} − 23250X^{3}
+ 83852X^{2} − 19920X − 33415)
= −\frac{1}{43}(5X^{2} + 34X + 41)(765X^{4} − 2238X^{3} + 1987X^{2} + 190X − 815),
Y^{2} = −\frac{1}{29}(5015X^{6} + 3900X^{5} − 19784X^{4} − 40974X^{3}
+ 49636X^{2} + 33000X − 7585)
= −\frac{1}{29}(17X^{2} − 30X + 5)(295X^{4} + 750X^{3} + 73X^{2} − 2502X − 1517),
Y^{2} = −\frac{7}{9}(85X^{6} − 180X^{5} − 1272X^{4} − 450X^{3} + 3612X^{2} − 720X − 1075)

= −\frac{7}{9}(X − 1)(X − 5)(85X^{4} + 330X^{3} + 283X^{2} − 402X − 215).

6.6 The Other Putative ℚ-Isogeny Classes

In Section 6.4 and Section 6.5, we established the existence of ℚ-isogenies between the jacobians of any pair of curves in the table in Section A.1 within one putative ℚ-isogeny class, with the exception of the following pairs:

(29a_{1}, 29a_{2}), (43a_{1}, 43a_{2}), (82a_{1}, 82a_{2}), (82a_{3}, 82a_{4}),
(82a_{1}, 82a_{4}), (82a_{2}, 82a_{3}), (98a_{1}, 98a_{2}), (98a_{3}, 98a_{4}),
(98a_{1}, 98a_{4}), (98a_{2}, 98a_{3}), (256a_{1}, 256a_{2}), (261a_{1}, 261a_{2}),
(387a_{1}, 387a_{2}), (529a_{1}, 529a_{2}), (529b_{1}, 529b_{2}), (675a_{1}, 675a_{2}),
(675b_{1}, 675b_{2}), (738a_{1}, 738a_{2}), (738a_{3}, 738a_{4}), (738a_{1}, 738a_{4}),
(738a_{2}, 738a_{3}), (841a_{1}, 841a_{2}), (882a_{1}, 882a_{2}), (882a_{3}, 882a_{4}),
(882a_{1}, 882a_{3}), (882a_{2}, 882a_{4}), (1075a_{1}, 1075a_{2}), (1421a_{1}, 1421a_{2}),
(1849a_{1}), (1849a_{2}).

The curves 675a_{1}, 675a_{2} are ℚ-isogenous, by Theorem 5.1.1; similarly, the curves 675b_{1}, 675b_{2} are ℚ-isogenous. The jacobians of the curves in any of the other pairs above appear to be ℚ-isogenous, looking at their L-series. In fact, if one can show that the jacobians of the curves in any of the boldfaced pairs are ℚ-isogenous, then the jacobians of the curves in any of the pairs are ℚ-isogenous, by the transitive, symmetric properties of ℚ-isogenies and by twisting curves. The curves 256a_{1}, 256a_{2} are listed in [Sma], in his list of curves of genus 2 defined over ℚ with good reduction away from 2.

6.7 The ℚ-Endomorphism Rings of the Curves

In this section, we show that, if C is a curve in the table in Section A.1, then the ℚ-endomorphism ring of the jacobian of C is isomorphic to ℤ[\sqrt{2}]. To do this, we use Proposition 3.3.2 and Proposition 2.4.6. For the usage of Proposition 2.4.6, note that, by Corollary 2.2.11 and the fact that the jacobian of C has a \sqrt{2} ℚ-multiplication, it suffices to prove that the ℚ-endomorphism ring of the jacobian of C is commutative.

Proposition 3.3.2 can be used for the curves such that the Galois group of F_{6} over ℚ is one of the groups

C_{2}^{3A}, D_{8}^{A}, D_{8}^{B}, D_{12}, D_{8} × C_{2}, S_{4}^{+}, S_{4}^{−}, S_{4} × C_{2}.

Consider the curves

39a_{1}, 55a_{1}, 160a_{1}, 165a_{1}, 256a_{1}, 256a_{2},
273a_{1}, 315a_{1}, 735a_{1}, 777a_{1}, 1045a_{1}, 1197a_{1}, 1885b_{1}.

If C: Y^{2} = F_{6}(X) is a curve in the table in Section A.1 such that the Galois group of F_{6} over ℚ is not one of the groups above, then at least one of the following occurs:

1. There exists a ℚ-isomorphism from the jacobian of C to that of one of the curves above.

2. For some N ∈ ℚ^{∗}, there exists a \overline{ℚ}-isomorphism of the form

[(x, y) − (x′, y)] ↦ [(x, \sqrt{N}y) − (x′, \sqrt{N}y′)]

from the jacobian of C to that of one of the curves above.

So, to show that the ℚ-endomorphism ring of the jacobian of C is isomorphic to ℤ[\sqrt{2}], it suffices to show that the ℚ-endomorphism ring of the jacobian of C′ is isomorphic to ℤ[\sqrt{2}], where C′ ranges across the curves above. This can be done using the table below and Corollary 2.2.11.

\begin{matrix}Curve & p & a_{p} & b_{p} & a_{p}^{2} − 4(b_{p} − 2p) \\ 39a_{1} & 5 & 0 & 2 & 32 \\ 55a_{1} & 3 & 0 & −2 & 32 \\ 160a_{1} & 3 & 0 & −2 & 32 \\ 165a_{1} & 7 & −4 & 10 & 32 \\ 256a_{1} & 3 & 0 & −2 & 32 \\ 256a_{2} & 3 & 0 & −2 & 32 \\ 273a_{1} & 17 & 4 & 30 & 32 \\ 315a_{1} & 11 & −4 & 18 & 32 \\ 735a_{1} & 11 & 0 & 14 & 32 \\ 777a_{1} & 5 & 0 & 2 & 32 \\ 1045a_{1} & 7 & 0 & 6 & 32 \\ 1197a_{1} & 11 & 4 & 18 & 32 \\ 1885a_{1} & 3 & 0 & −2 & 32 \end{matrix}

Note that the fact that a_{p}^{2} − 4(b_{p} − 2p) = 32 for each curve C′ is purely coincidental; however, it is reasonable for a_{p}^{2} − 4(b_{p} − 2p) to be a power of 2. Indeed, a_{p}^{2} − 4(b_{p} − 2p) is a factor of the discriminant of (2.12) w.r.t. X, and the splitting field of (2.12) w.r.t. X contains ℚ(\sqrt{2}), since the 𝔽_{p}-endomorphism ring of the reduction of the jacobian of C′ at p is commutative and contains a \sqrt{2} 𝔽_{p}-multiplication.

Let C_{1}, C_{2} be two distinct curves in the table in Section A.1. Then, their jacobians are not ℚ-isomorphic. Indeed, suppose, for a contradiction, that they are ℚ-isomorphic. By Corollary 2.8.3, taking into account the fact that the ℚendomorphism ring of the jacobian of C_{1} is isomorphic to ℤ[\sqrt{2}], C_{1} and C_{2} are ℚ-isomorphic. This is not true, as can be easily verified; recall the discussion on isomorphisms between curves of genus 2 in Section 2.3.

Let C_{1}, C_{2}, C_{3}, C_{4} be as described in Section 6.5. Recall that, for all i ∈ {2, 3, 4}, the jacobians of C_{i} have three subgroups of the form (6.3). We will show, for all i ∈ {2, 3, 4}, that the Richelot dual curve of C_{i} w.r.t. any of the three subgroups exists, and that C_{2}, C_{3}, C_{4} are each ℚ-isomorphic to one of the three Richelot dual curves.

For all i ∈ {2, 3, 4}, there are rational numbers U_{i}, V_{i}, W_{i}, Δ_{i} such that C_{1} is ℚ-isomorphic to C and C_{i} is ℚ-isomorphic to C_{U_{i},V_{i},W_{i}}, using the notation of Section 4.1; note that C is independent of i. Let ϕ_{C} be a ℚ-isomorphism from C_{1} to C, let ϕ_{C,i} be a ℚ-isomorphism from C_{i} to C_{U,V,W}, and let ϕ_{J}, ϕ_{J,i} be the ℚ-isomorphisms of the jacobians induced by the ℚ-isomorphisms of the curves, so that ϕ_{J,i}^{−1}π_{U_{i},V_{i},W_{i}}ϕ_{J} is a ℚ-isogeny of degree 2 from the jacobian of C_{1} to that of C_{i}, and that ϕ_{J}^{−1}π_{U_{i},V_{i},W_{i}}^{∨}ϕ_{J,i} is a ℚ-isogeny of degree 2 from the jacobian of C_{i} to that of C_{1}.

Now

ϕ_{J,2}^{−1}π_{U_{2},V_{2},W_{2}}π_{U_{i},V_{i},W_{i}}^{∨}ϕ_{J,i},
ϕ_{J,3}^{−1}π_{U_{3},V_{3},W_{3}}π_{U_{i},V_{i},W_{i}}^{∨}ϕ_{J,i},
ϕ_{J,4}^{−1}π_{U_{4},V_{4},W_{4}}π_{U_{i},V_{i},W_{i}}^{∨}ϕ_{J,i}

are ℚ-isogenies of degree 4 from the jacobian of C_{i} to those of C_{2}, C_{3}, C_{4} respectively. From the explicit descriptions of the π’s and π_{U_{i},V_{i},W_{i}}^{∨}, we see that the set of the kernels of these isogenies is the set of the three subgroups of the jacobian of C_{i} of the form (6.3), so that, for all j ∈ {2, 3, 4}, the jacobian of C_{j} is ℚ-isomorphic to the jacobian of the Richelot dual curve of C w.r.t. the kernel of ϕ_{J,2}^{−1}π_{U_{2},V_{2},W_{2}}π_{U_{i},V_{i},W_{i}}^{∨}ϕ_{J,i} or to the abelian surface A provided by Proposition 2.6.8 as is appropriate. Since the ℚ-endomorphism ring of the jacobian of C_{j} is isomorphic to ℤ[\sqrt{2}], the Richelot dual curve of C w.r.t. this kernel exists (cf. the start of the proof of Proposition 3.5.2), and C_{j} is ℚ-isomorphic to it, by Corollary 2.8.3.

6.8 Comments on Modularity

In this section, we compare the positive integers N ≤ 2000 for which there are G_{ℚ}-conjugacy classes of newforms f of level N such that K_{f} = ℚ(\sqrt{2}) with the positive integers N ≤ 2000 for which there are putative ℚ-isogeny classes of curves appearing in the table in Section A.1 (given an isogeny class, there is a conjugacy class such that the jacobian of any curve in the isogeny class, being ℚ-simple and of GL_{2}-type, is conjecturally ℚ-isogenous to A_{f} for any newform f in the conjugacy class), to see how successful we have been in generating conjecturally modular ℚ-simple abelian surfaces from our \sqrt{2} mulitplication family of curves.

The first table in this section lists the positive integers N ≤ 2000 for which there are G_{ℚ}-conjugacy classes of newforms f of level N such that K_{f} = ℚ(\sqrt{2}), with the number of N’s in the table being the number of conjugacy classes of newforms of level N. The table is taken from the table in [Mul], which lists the positive integers N ≤ 2000 for which there are G_{ℚ}-conjugacy classes of newforms f of level N such that K_{f} is a quadratic extension of ℚ. The number of N’s in the table in boldface are the number of isogeny classes of curves corresponding to N appearing in the table in Section A.1.

In [Wan1], X. Wang discusses A_{f} complex-analytically, and so in particular has a method for producing a ℂ-polarisation on A_{f}. The second table in this section gives, for newforms f of level N ≤ 750 such that K_{f} = ℚ(\sqrt{2}), \frac{d_{2}}{d_{1}} , where (d_{1}, d_{2}) is the type of the ℂ-polarisation on A_{f}, and the conductor c of an order of ℤ[\sqrt{2}] embedding into the ℚ-endomorphism ring of A_{f}. The table is taken from the table in [Wan2]. When \frac{d_{2}}{d_{1}} and c are coprime, the level N is boldfaced; we shall see why this is so in the second paragraph after the tables.

Note that we don’t know which abelian variety A_{f} the jacobian of a given curve appearing in the table in Section A.1 is putatively ℚ-isogenous to, since neither [Mul] nor [Wan1] contain Fourier coefficients of newforms, so the two tables need to be separate.

CHAPTER 6. CONDUCTORS OF CURVES 148

\begin{matrix}N & N & N & N & N & N & N & N & N & N \\ 29 & 39 & 43 & 55 & 65 & 82 & 85 & 91 & 94 & 98 \\ 117 & 123 & 129 & 145 & 147 & 147 & 160 & 165 & 172 & 178 \\ 183 & 196 & 203 & 207 & 207 & 209 & 218 & 223 & 226 & 227 \\ 232 & 237 & 245 & 245 & 245 & 245 & 249 & 256 & 259 & 261 \\ 262 & 265 & 273 & 275 & 276 & 278 & 285 & 285 & 302 & 303 \\ 315 & 315 & 320 & 325 & 325 & 325 & 332 & 334 & 335 & 339 \\ 339 & 345 & 357 & 362 & 369 & 380 & 385 & 387 & 387 & 389 \\ 390 & 392 & 392 & 393 & 397 & 397 & 424 & 425 & 426 & 429 \\ 430 & 434 & 436 & 437 & 438 & 441 & 441 & 445 & 464 & 464 \\ 465 & 488 & 490 & 490 & 495 & 495 & 497 & 507 & 512 & 512 \\ 512 & 512 & 512 & 512 & 517 & 517 & 517 & 518 & 520 & 526 \\ 529 & 529 & 533 & 534 & 537 & 542 & 544 & 545 & 547 & 549 \\ 561 & 578 & 578 & 578 & 579 & 582 & 583 & 585 & 590 & 591 \\ 597 & 598 & 605 & 606 & 606 & 609 & 618 & 618 & 621 & 621 \\ 621 & 621 & 624 & 637 & 639 & 639 & 645 & 646 & 646 & 646 \\ 652 & 656 & 661 & 669 & 670 & 670 & 673 & 674 & 675 & 675 \\ 678 & 680 & 682 & 688 & 688 & 703 & 705 & 706 & 711 & 715 \\ 725 & 725 & 728 & 733 & 735 & 735 & 735 & 735 & 736 & 736 \\ 737 & 738 & 741 & 742 & 742 & 742 & 747 & 748 & 752 & 753 \\ 755 & 759 & 760 & 765 & 767 & 768 & 768 & 770 & 777 & 777 \\ 779 & 784 & 784 & 784 & 784 & 790 & 792 & 792 & 794 & 794 \\ 797 & 798 & 800 & 814 & 814 & 819 & 819 & 822 & 825 & 825 \\ 825 & 828 & 830 & 834 & 836 & 837 & 837 & 840 & 841 & 844 \\ 845 & 846 & 848 & 850 & 850 & 855 & 855 & 861 & 866 & 867 \\ 867 & 870 & 870 & 872 & 874 & 876 & 880 & 882 & 882 & 882 \\ 885 & 888 & 895 & 897 & 909 & 910 & 921 & 921 & 928 & 928 \\ 931 & 931 & 931 & 931 & 934 & 936 & 936 & 942 & 945 & 945 \\ 950 & 950 & 961 & 961 & 961 & 969 & 972 & 973 & 975 & 976 \\ 980 & 980 & 984 & 985 & 986 & 992 & 992 & 996 & 1002 & 1010 \\ 1017 & 1017 & 1022 & 1024 & 1024 & 1024 & 1024 & 1024 & 1024 & 1032 \\ 1035 & 1035 & 1035 & 1038 & 1040 & 1040 & 1045 & 1055 & 1058 & 1067 \\ 1067 & 1071 & 1074 & 1074 & 1075 & 1078 & 1078 & 1078 & 1078 & 1078 \\ 1078 & 1078 & 1078 & 1088 & 1094 & 1102 & 1104 & 1105 & 1112 & 1113 \\ 1122 & 1122 & 1127 & 1127 & 1130 & 1132 & 1133 & 1142 & 1145 & 1146 \\ 1146 & 1155 & 1156 & 1162 & 1164 & 1165 & 1166 & 1170 & 1176 & 1176 \\ 1176 & 1176 & 1178 & 1178 & 1179 & 1183 & 1185 & 1195 & 1197 & 1197 \\ 1201 & 1203 & 1209 & 1218 & 1221 & 1225 & 1225 & 1225 & 1225 & 1225 \\ 1225 & 1240 & 1245 & 1265 & 1266 & 1266 & 1274 & 1274 & 1275 & 1275 \\ 1278 & 1280 & 1280 & 1280 & 1280 & 1280 & 1280 & 1287 & 1288 & 1290 \end{matrix}

N  N  N  N  N  N  N  N  N  N
1293 1295 1305 1306 1312 1314 1314 1314 1320 1320
1325 1325 1325 1328 1330 1342 1360 1360 1361 1385
1395 1408 1408 1408 1408 1408 1408 1408 1408 1409
1410 1421 1421 1424 1425 1425 1426 1428 1430 1430
1430 1430 1433 1440 1445 1446 1446 1446 1448 1455
1456 1456 1466 1470 1470 1470 1470 1472 1472 1479
1482 1482 1482 1484 1485 1485 1520 1520 1521 1526
1526 1527 1533 1536 1536 1536 1536 1536 1536 1536
1536 1536 1536 1536 1536 1540 1548 1554 1556 1568
1568 1568 1568 1568 1568 1568 1568 1575 1575 1578
1578 1580 1582 1584 1584 1587 1587 1587 1594 1594
1594 1596 1598 1598 1600 1602 1602 1608 1611 1617
1617 1634 1641 1650 1650 1664 1664 1664 1664 1666
1666 1675 1680 1681 1683 1695 1698 1713 1713 1716
1725 1734 1734 1737 1738 1738 1743 1744 1744 1744
1746 1755 1755 1758 1764 1768 1768 1773 1776 1778
1784 1785 1791 1792 1792 1792 1792 1794 1794 1808
1810 1813 1815 1818 1818 1827 1827 1830 1830 1833
1849 1854 1854 1856 1856 1856 1856 1856 1856 1862
1862 1866 1866 1870 1870 1872 1872 1872 1876 1881
1885 1885 1887 1891 1896 1898 1900 1911 1911 1911
1911 1917 1917 1918 1925 1935 1945 1950 1960 1960
1960 1960 1962 1964 1968 1968 1983 1984 1984 1997
1998 1998 1998 1998

\begin{matrix}N & c & \frac{d_{2}}{d_{1}} & N & c & \frac{d_{2}}{d_{1}} & N & c & \frac{d_{2}}{d_{1}} \\ 29 & 1 & 1 & 39 & 1 & 2 & 43 & 1 & 2 \\ 55 & 1 & 2 · 7 & 65 & 1 & 1 & 82 & 1 & 2 \\ 85 & 1 & 2 & 91 & 1 & 2 & 94 & 2 & 2 · 47 \\ 98 & 1 & 2 & 117 & 1 & 1 & 123 & 1 & 2 \\ 129 & 1 & 7 & 145 & 2 & 2 · 7 & 147 & 1 & 1 \\ 147 & 1 & 7 & 160 & 2 & 2 & 165 & 1 & 2 \\ 172 & 1 & 2 & 178 & 1 & 2 & 183 & 1 & 2 \\ 196 & 2 & 2 · 7 & 203 & 2 & 7 & 207 & 1 & 2 \\ 207 & 1 & 2 & 209 & 1 & 1 & 218 & 1 & 7 \\ 223 & 1 & 7 & 226 & 1 & 7 & 227 & 2 & 7 \\ 232 & 1 & 2 & 237 & 1 & 31 & 245 & 1 & 2 \\ 245 & 1 & 2 · 7 & 245 & 2 & 7 & 245 & 2 & 7 \\ 249 & 1 & 31 & 256 & 2 & 1 & 259 & 1 & 7 · 17 \\ 261 & 1 & 23 & 262 & 1 & 2 · 313 & 265 & 1 & 2 · 7 \\ 273 & 1 & 2 · 7 & 275 & 1 & 7 & 276 & 1 & 2 · 7 \\ 278 & 1 & 7 & 285 & 1 & 2 · 47 & 285 & 1 & 2 · 7 \\ 302 & 1 & 7 & 303 & 1 & 2 & 315 & 2 & 2 \\ 315 & 2 & 2 & 320 & 2^{2} & 2 & 325 & 1 & 7 \\ 325 & 1 & 7 & 325 & 2 & 1 & 332 & 1 & 2 \\ 334 & 2 & 7^{2} & 335 & 1 & 1 & 339 & 2 & 7 · 71 \\ 339 & 1 & 7 & 345 & 1 & 1 & 357 & 1 & 1 \\ 362 & 1 & 2 · 17 · 23 & 369 & 1 & 2 & 380 & 1 & 7 \\ 385 & 1 & 2 & 387 & 1 & 2 · 7 & 387 & 1 & 7 \\ 389 & 1 & 1 & 390 & 2^{2} & 2 · 7^{2} & 392 & 2 & 2^{3} · 7 \\ 392 & 2^{2} & 2 · 7 & 393 & 1 & 23^{2} & 397 & 1 & 23 \\ 397 & 1 & 31 & 424 & 1 & 1 & 425 & 1 & 2 \\ 426 & 2 & 7 · 31 & 429 & 1 & 7 · 23 & 430 & 1 & 2 · 7 \\ 434 & 1 & 7 · 17 & 436 & 2 & 31 & 437 & 1 & 2 \\ 438 & 2 & 2 · 7 · 23 & 441 & 1 & 1 & 441 & 3 & 7 \\ 445 & 1 & 2 · 7 & 464 & 2^{2} & 1 & 464 & 2 & 2 \\ 465 & 1 & 23 & 488 & 1 & 1 & 490 & 1 & 2 \\ 490 & 1 & 2 · 7 & 495 & 1 & 7 & 495 & 1 & 2 \\ 497 & 1 & 2 · 7 & 507 & 1 & 1 & 512 & 3 & 1 \\ 512 & 3 & 1 & 512 & 1 & 1 & 512 & 3 & 1 \\ 512 & 3 & 2 & 512 & 3 & 2 & 517 & 1 & 1 \\ 517 & 1 & 7 · 41 & 517 & 1 & 2 · 7 · 17 · 31 & 518 & 1 & 2 · 7 · 17 \\ 520 & 1 & 2 · 7 & 526 & 2 & 23 · 31 & 529 & 1 & 2 \\ 529 & 1 & 2 · 23 & 533 & 1 & 127 & 534 & 1 & 7 · 17 \\ 537 & 2 & 2^{3} · 7 & 542 & 1 & 2 · 7 · 23 & 544 & 1 & 1 \\ 545 & 1 & 23 · 113 & 547 & 1 & 103 & 549 & 1 & 2 \\ 561 & 1 & 2 & 578 & 1 & 17 & 578 & 2^{2} & 17 \\ 578 & 2 & 17 & 579 & 1 & 7^{2} & 582 & 3 & 41 \end{matrix}

N  c  \frac{d_{2}}{d_{1}}  N c  \frac{d_{2}}{d_{1}}  N c  \frac{d_{2}}{d_{1}}
583 1  7  585 1  1  590 1  97
591 1  17 597 1 7 · 17 598 1  7
605 3  7  606 1  71  606 1 7 · 17
609 1  2 · 7 618 1 2 · 97 618 1 2 · 7 · 17
621 2  7  621 2  7  621 1  7
621 1  7  624 2^{2}  7  637 1  2
639 1 2 · 17 639 1 2 · 17 645 1  1
646 1 2 · 47 646 1  2 · 7 646 2  2^{3}
652 1 2 · 17 656 1  2  661 1  607
669 2  2 · 7 670 1 7 · 17^{2} 670 1  7
673 2  7  674 2  2  675 1  2
675 1  2  678 2  2 · 7 680 1  1
682 1 2 · 241 688 1  2 · 7 688 1  2
703 1  2  705 1  7  706 1 23 · 199
711 1  31 715 1  7^{2}  725 2^{2} 2 · 7
725 2 · 3  7  728 3 2 · 7 · 17 733 1  31
735 2^{2}  7  735 2^{2}  7  735 2  1
735 2  7^{2} 736 1  2  736 1  2
737 1  17 738 1  2  741 1 2 · 7^{2}
742 2  1  742 2 2 · 23 742 2^{2} 2 · 7
747 1  31 748 1  2

We have been pretty successful in our task for small N, say N ≤ 500, but generally increasingly less successful as N increases. From the point of view of substitution into the family of curves described in Theorem 3.2.1, a reason for this could well be the reasonable intuition that, very roughly, bigger values for the parameters give bigger values for N, and the values for the parameters used in our search for curves are necessarily bounded. One notices too that integers in the table above tend to be even rather than odd, so that searching mainly for curves whose jacobians have odd conductor is rather limiting.

As part of a more general result (see [Wil]), J. M. Wilson has shown that if an abelian surface A defined over ℚ has a ℚ-polarisation of degree d, and the order of conductor c of ℤ[\sqrt{2}] embeds into the ℚ-endomorphism ring of A, then, provided (d, c) = 1, A is ℚ-isogenous to an abelian surface with d = c = 1, and so is hopefully ℚ-isogenous to the jacobian of a curve of genus 2 in our \sqrt{2} multiplication family; this explains the boldfaced levels in the second table in this section. Most of the levels in this table are boldfaced; however, we know that the polarisations are defined over ℂ, but not that they are defined over ℚ. Also, some levels which are not boldfaced are boldfaced in the first table in this section, for example N = 196.

In [Gal], there are lists of modular curves and the quotients of modular curves by Atkin-Lehner involutions (see [Gal] for the definition of these involutions). Some of these curves also appear in the table above. More precisely,

X_{0}(29): Y^{2} = X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7

is the curve 29a_{1},

X_{0}(39)/W_{13}: Y^{2} = X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9

is the curve 39a_{2},

X_{0}(85)/⟨W_{5}, W_{17}⟩: Y^{2} = X^{6} + 2X^{5} + 7X^{4} + 6X^{3} + 13X^{2} − 8X + 4

is the curve 85a_{1}, and

X_{0}(165)/⟨W_{3}, W_{5}, W_{11}⟩: Y^{2} = X^{6} + 6X^{5} + 11X^{4} + 14X^{3} + 5X^{2} − 12X

is the curve 165a_{1}. The jacobian of a quotient of the modular curve X_{0}(N) is a factor of J_{0}(N), and so is modular in the sense of Section 2.12. We deduce that the jacobians of the curves

29a_{1}, 39a_{1}, 39a_{2}, 39a_{3}, 39a_{4}, 85a_{1}, 85a_{2}, 165a_{1}, 165a_{2}, 165a_{3}, 165a_{4}

are all modular.

Conclusion

An obvious question to ask about the table in Section A.1 is: how complete is it? Also, one can ask whether there is another way of determining curves of genus 2 whose jacobians have \sqrt{2} ℚ-multiplication, besides using our \sqrt{2} multiplication family and taking Richelot isogenies and ℚ-isogenies of degree 2.

In [C-F], Chapter 11, the use of the Richelot isogeny to compute the Mordell-Weil group of the jacobian of a curve of genus 2 defined over ℚ via 2-descent is discussed. We can ask whether the ℚ-isogenies of degree 2 derived in Chapter 4 can be used in a similar manner, albeit for a limited family of curves. Also, can isogenies between the jacobians of any pair of curves listed in Section 6.6 be found explicitly? The explicit maps on curves giving our ℚ-isogenies of degree 2 are conveniently defined over ℚ, but this need not be true for ℚ-isogenies in general.

We may be fairly close to obtaining all genus 2 curves defined over ℚ whose jacobians have \sqrt{2} ℚ-multiplication. Indeed, we know about the curves which are ℚ-simple, so we need to investigate the curves which split over ℚ. We know that there is more than one subgroup of order 4 of the 2-torsion of the jacobian of such a curve defined over ℚ, but it is not clear that this helps us much.

Having very briefly touched upon curves of genus 2 defined over a subfield K of ℂ whose jacobians have primitive \sqrt{8} K-multiplication in Chapter 6 (with K = ℚ), it would be nice to find a family of such curves which is fairly universal, as in the case of \sqrt{2} K-multiplication.

Appendix A

Tables

A.1 Curves With \sqrt{2} Multiplication and Low Conductor

The table below lists curves of genus 2 defined over ℚ whose jacobians have \sqrt{2} multiplication defined over ℚ and low conductor. Associated to each curve C is an integer N; if N is odd, the conductor of its jacobian is N, and if N is even, the conductor of its jacobian is conjectured to be N. The curves are divided into putative ℚ-isogeny classes, different letters being used for different classes; for example, for N = 735, there are four putative ℚ-isogeny classes; one class consists of the curves a_{1}, a_{2}, a_{3}, a_{4}; one class consists of the curves b_{1}, b_{2}, b_{3}, b_{4}; one class consists of the curve c_{1}; one class consists of the curve d_{1}.

N C F_{6}
29 a_{1} X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7
a_{2} X^{6} − 2X^{5} + 7X^{4} − 6X^{3} + 13X^{2} − 4X + 8
39 a_{1} X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9

= (X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9

= (X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)

a_{3} X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9

= (X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)

a_{4} −5(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)

= −5(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)

43 a_{1} −3X^{6} − 2X^{5} + 7X^{4} − 4X^{3} − 13X^{2} + 10X − 7
a_{2} −3(X^{6} + 8X^{3} − 36X^{2} + 12X − 4)
55 a_{1} −3X^{6} + 4X^{5} + 16X^{4} − 2X^{3} − 4X^{2} + 4X − 3

= −(X + 1)(X − 3)(X^{2} + X − 1)(3X^{2} − X + 1)

a_{2} −3(X^{6} − 6X^{5} + 27X^{4} + 2X^{3} − 27X^{2} − 36X − 16)

= −3(X^{2} − X − 1)(X^{4} − 5X^{3} + 23X^{2} + 20X + 16)

a_{3} −3X^{6} − 2X^{5} − 21X^{4} − 6X^{3} − 39X^{2} + 8X − 12

= −(X^{2} + 4)(X^{2} + X + 3)(3X^{2} − X + 1)

a_{4} −3(X^{6} − 18X^{5} + 63X^{4} + 154X^{3} − 27X^{2} − 48X − 4)

= −3(X^{2} + X − 1)(X^{2} − 8X − 4)(X^{2} − 11X − 1)

65 a_{1} −3(5X^{6} − 12X^{4} − 10X^{3} − 24X^{2} + 13)

= −3(X^{2} + X − 1)(5X^{4} − 5X^{3} − 2X^{2} − 13X − 13)

a_{2} X^{6} + 2X^{5} + 3X^{4} − 6X^{3} − 27X^{2} − 8X − 4

= (X − 2)(X + 2)(X^{4} + 2X^{3} + 7X^{2} + 2X + 1)

N C F_{6}
82 a_{1} −12X^{6} − 4X^{5} − 87X^{4} − 100X^{3} − 69X^{2} − 26X − 6

= −(12X^{2} + 4X + 3)(X^{4} + 7X^{2} + 6X + 2)

a_{2} −3(4X^{6} − 36X^{5} + 81X^{4} + 50X^{3} − 207X^{2} − 204X − 52)

= −3(X^{2} − 4X − 4)(4X^{4} − 20X^{3} + 17X^{2} + 38X + 13)

a_{3} −3(4X^{6} − 36X^{5} + 189X^{4} − 500X^{3} + 423X^{2} − 66X + 2)

= −3(2X^{2} − 7X + 1)(2X^{4} − 11X^{3} + 55X^{2} − 52X + 2)

a_{4} 4X^{6} + 4X^{5} − 11X^{4} + 2X^{3} + 29X^{2} + 4X − 12

= (X + 1)(2X + 3)(2X^{4} − 3X^{3} − X^{2} + 8X − 4)

85 a_{1} X^{6} + 4X^{5} + 8X^{4} − 2X^{3} − 12X^{2} + 4X + 1

= (X^{2} + X − 1)(X^{4} + 3X^{3} + 6X^{2} − 5X − 1)

a_{2} X^{6} + 2X^{5} + 7X^{4} + 6X^{3} + 13X^{2} − 8X + 4

= (X^{2} + 4)(X^{4} + 2X^{3} + 3X^{2} − 2X + 1)

91 a_{1} −3X^{6} − 2X^{5} − 13X^{4} − 12X^{3} − 17X^{2} − 14X − 7
98 a_{1} 2X^{6} + 4X^{5} + 7X^{4} + 48X^{3} + 20X^{2} − 20X + 3

= (X + 1)(X + 3)(2X^{4} − 4X^{3} + 17X^{2} − 8X + 1)

a_{2} −7(2X^{6} + 4X^{5} + 7X^{4} + 48X^{3} + 20X^{2} − 20X + 3)

= −7(X + 1)(X + 3)(2X^{4} − 4X^{3} + 17X^{2} − 8X + 1)

a_{3} 3(4X^{6} − 36X^{5} + 63X^{4} + 86X^{3} − 9X^{2} + 12X + 20)

= 3(X^{2} − 4X − 4)(4X^{4} − 20X^{3} − X^{2} + 2X − 5)

a_{4} −21(4X^{6} − 36X^{5} + 63X^{4} + 86X^{3} − 9X^{2} + 12X + 20)

= −21(X^{2} − 4X − 4)(4X^{4} − 20X^{3} − X^{2} + 2X − 5)

N C F_{6}
117 a_{1} −3(X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9)

= −3(X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} −3(X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9)

= −3(X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)

a_{3} −3(X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9)

= −3(X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)

a_{4} 15(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)

= 15(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)

123 a_{1} −3X^{6} + 6X^{5} − 17X^{4} + 20X^{3} − 29X^{2} + 18X − 15
129 a_{1} −3X^{6} − 12X^{5} − 16X^{4} − 10X^{3} − 4X^{2} + 12X − 3
145 a_{1} X^{6} − 8X^{4} + 30X^{3} − 44X^{2} + 25

= (X^{2} − X − 1)(X^{2} − 2X + 5)(X^{2} + 3X − 5)

a_{2} −3(13X^{6} + 24X^{5} − 36X^{4} + 38X^{3} + 24X^{2} − 36X − 27)

= −3(X − 1)(X + 3)(13X^{4} − 2X^{3} + 7X^{2} + 18X + 9)

a_{3} X^{6} + 12X^{5} + 28X^{4} − 6X^{3} − 32X^{2} + 12X + 1

= (X^{2} + X − 1)(X^{4} + 11X^{3} + 18X^{2} − 13X − 1)

a_{4} 3(27X^{6} + 84X^{4} − 50X^{3} + 72X^{2} − 60X + 35)

= 3(3X^{2} + 2X + 7)(9X^{4} − 6X^{3} + 11X^{2} − 10X + 5)

147 a_{1} X^{6} − 4X^{4} + 2X^{3} + 8X^{2} − 12X + 9

= (X^{2} − X + 1)(X^{4} + X^{3} − 4X^{2} − 3X + 9)

a_{2} X^{6} − 16X^{4} − 2X^{3} + 44X^{2} − 36X + 9

= (X − 1)(X + 3)(X^{4} − 2X^{3} − 9X^{2} + 10X − 3)

b_{1} −7(X^{6} − 4X^{4} + 2X^{3} + 8X^{2} − 12X + 9)

= −7(X^{2} − X + 1)(X^{4} + X^{3} − 4X^{2} − 3X + 9)

b_{2} −7(X^{6} − 16X^{4} − 2X^{3} + 44X^{2} − 36X + 9)

= −7(X − 1)(X + 3)(X^{4} − 2X^{3} − 9X^{2} + 10X − 3)

N C F_{6}
160 a_{1} X^{5} + 3X^{3} − 4X

= X(X − 1)(X + 1)(X^{2} + 4)

a_{2} 6(2X^{6} + 24X^{5} + 117X^{4} + 301X^{3} + 441X^{2} + 360X + 130)

= 6(X^{2} + 2X + 2)(X^{2} + 5X + 5)(2X^{2} + 10X + 13)

a_{3} 2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3

= (X − 1)(X − 3)(2X − 1)(X^{2} − 3X + 1)

a_{4} 3(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= 3(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

a_{5} −3(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= −3(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

165 a_{1} X^{6} + 6X^{5} + 11X^{4} + 14X^{3} + 5X^{2} − 12X

= X(X + 4)(X^{2} + X − 1)(X^{2} + X + 3)

a_{2} X^{6} + 8X^{4} − 10X^{3} + 20X^{2} − 12X + 9

= (X^{2} − X + 1)(X^{4} + X^{3} + 8X^{2} − 3X + 9)

a_{3} −3(5X^{6} − 18X^{5} + 15X^{4} + 54X^{3} − 135X^{2} + 72X − 20)

= −3(X − 2)(X + 2)(X^{2} − 3X + 5)(5X^{2} − 3X + 1)

a_{4} X^{6} − 12X^{5} + 32X^{4} + 34X^{3} − 52X^{2} − 60X − 15

= (X^{2} − X − 1)(X^{2} − 5X − 5)(X^{2} − 6X − 3)

178 a_{1} 4X^{6} − 4X^{5} − 3X^{4} − 4X^{3} + 7X^{2} + 2X + 2
183 a_{1} X^{6} + 8X^{4} + 10X^{3} + 20X^{2} + 24X + 9
196 a_{1} 2(3X^{6} + 26X^{5} + 69X^{4} + 108X^{3} + 33X^{2} − 50X − 18)

= 2(X^{2} + 6X + 2)(3X^{4} + 8X^{3} + 15X^{2} + 2X − 9)

N C F_{6}
207 a_{1} X^{6} − 2X^{3} + 12X^{2} − 12X + 9
= (X^{2} − X + 1)(X^{4} + X^{3} − 3X + 9)
a_{2} X^{6} − 12X^{4} + 2X^{3} + 24X^{2} − 24X + 9
= (X − 1)(X + 3)(X^{4} − 2X^{3} − 5X^{2} + 6X − 3)
b_{1} −3(X^{6} − 2X^{3} + 12X^{2} − 12X + 9)
= −3(X^{2} − X + 1)(X^{4} + X^{3} − 3X + 9)
b_{2} −3(X^{6} − 12X^{4} + 2X^{3} + 24X^{2} − 24X + 9)
= −3(X − 1)(X + 3)(X^{4} − 2X^{3} − 5X^{2} + 6X − 3)
209 a_{1} X^{6} + 4X^{5} + 8X^{4} + 8X^{3} + 8X^{2} − 4X + 4
223 a_{1} X^{6} + 4X^{5} − 4X^{4} − 22X^{3} + 8X^{2} + 8X − 71
227 a_{1} X^{6} − 8X^{4} − 16X^{3} + 12X^{2} + 100X + 52
232 a_{1} X^{6} − 4X^{5} + 6X^{4} − 2X^{2} + 2X + 1
237 a_{1} −27X^{6} − 72X^{5} − 100X^{4} − 70X^{3} − 40X^{2} − 12X − 3
245 a_{1} X^{6} − 2X^{5} + 7X^{4} + 4X^{3} + 3X^{2} + 10X + 5
b_{1} −7(X^{6} − 2X^{5} + 7X^{4} + 4X^{3} + 3X^{2} + 10X + 5)
c_{1} 5X^{6} − 10X^{5} − X^{4} + 2X^{3} + 45X^{2} − 76X + 40
d_{1} −7(5X^{6} − 10X^{5} − X^{4} + 2X^{3} + 45X^{2} − 76X + 40)
249 a_{1} 9X^{6} + 30X^{5} + 19X^{4} − 38X^{3} − 47X^{2} + 12X + 24
256 a_{1} X^{5} − 4X
= X(X^{2} + 2)(X^{2} − 2)
a_{2} 3X^{6} + 2X^{5} − 5X^{4} + 20X^{3} + 5X^{2} + 2X − 3
= (X^{2} + 2X − 1)(X^{2} − 2X + 3)(3X^{2} + 2X + 1)

N C F_{6}
261 a_{1} −3(X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7)
a_{2} −3(X^{6} − 2X^{5} + 7X^{4} − 6X^{3} + 13X^{2} − 4X + 8)
265 a_{1} X^{6} + 10X^{5} + 23X^{4} − 10X^{3} + 5X^{2} + 8X + 4
273 a_{1} 5X^{6} + 6X^{5} − 25X^{4} − 30X^{3} + 5X^{2} − 24X − 36

= (X + 2)(5X + 6)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} 5(9X^{6} + 66X^{5} + 127X^{4} − 58X^{3} − 251X^{2} + 204X − 48)

= 5(X^{2} + 3X − 3)(9X^{4} + 39X^{3} + 37X^{2} − 52X + 16)

a_{3} 5(9X^{6} + 102X^{5} + 211X^{4} − 298X^{3} − 563X^{2} + 528X − 108)

= 5(X^{2} + X − 3)(3X^{2} + 8X − 4)(3X^{2} + 23X − 9)

a_{4} −3X^{6} + 6X^{5} − 25X^{4} − 10X^{3} − 55X^{2} − 24X − 36

= −(X^{2} + X + 1)(X^{2} − 3X + 9)(3X^{2} + 4)

275 a_{1} −5(3X^{6} − 4X^{5} − 16X^{4} + 2X^{3} + 4X^{2} − 4X + 3)

= −5(X + 1)(X − 3)(X^{2} + X − 1)(3X^{2} − X + 1)

a_{2} −15(X^{6} − 6X^{5} + 27X^{4} + 2X^{3} − 27X^{2} − 36X − 16)

= −15(X^{2} − X − 1)(X^{4} − 5X^{3} + 23X^{2} + 20X + 16)

a_{3} −5(3X^{6} + 2X^{5} + 21X^{4} + 6X^{3} + 39X^{2} − 8X + 12)

= −5(X^{2} + 4)(X^{2} + X + 3)(3X^{2} − X + 1)

a_{4} −15(X^{6} − 18X^{5} + 63X^{4} + 154X^{3} − 27X^{2} − 48X − 4)

= −15(X^{2} + X − 1)(X^{2} − 8X − 4)(X^{2} − 11X − 1)

278 a_{1} 3(−8X^{6} − 81X^{4} + 62X^{3} − 45X^{2} + 12X − 4)

N C F_{6}
285 a_{1} −3X^{6} + 4X^{4} − 10X^{3} − 8X^{2} + 5

= −(X^{2} + X − 1)(3X^{4} − 3X^{3} + 2X^{2} + 5X + 5)

a_{2} −3(X^{6} + 18X^{5} + 3X^{4} − 54X^{3} − 27X^{2} − 72X − 4)

= −3(X − 2)(X + 2)(X^{4} + 18X^{3} + 7X^{2} + 18X + 1)

b_{1} 5X^{6} + 32X^{4} − 42X^{3} + 92X^{2} − 60X + 45

= (X^{2} − X + 1)(5X^{4} + 5X^{3} + 32X^{2} − 15X + 45)

b_{2} 5X^{6} − 28X^{4} + 42X^{3} − 40X^{2} − 24X + 45

= (X − 1)(X + 3)(5X^{4} − 10X^{3} + 7X^{2} − 2X − 15)

303 a_{1} X^{6} − 4X^{4} + 8X^{3} + 8X^{2} − 36X − 36
315 a_{1} X^{6} − 12X^{4} − 10X^{3} + 12X + 9

= (X − 1)(X + 3)(X^{2} + X + 1)(X^{2} − 3X − 3)

a_{2} X^{6} + 10X^{3} + 12X^{2} − 7

= (X^{2} + X − 1)(X^{4} − X^{3} + 2X^{2} + 7X + 7)

a_{3} X^{6} − 24X^{4} − 10X^{3} + 84X^{2} − 60X + 9

= (X − 1)(X + 3)(X^{2} + 3X − 3)(X^{2} − 5X + 1)

a_{4} 9X^{6} − 24X^{5} + 72X^{4} − 98X^{3} + 132X^{2} − 84X + 49

= (X^{2} − X + 1)(3X^{2} − 3X + 7)(3X^{2} − 2X + 7)

b_{1} −3(X^{6} − 12X^{4} − 10X^{3} + 12X + 9)

= −3(X − 1)(X + 3)(X^{2} + X + 1)(X^{2} − 3X − 3)

b_{2} −3(X^{6} + 10X^{3} + 12X^{2} − 7)

= −3(X^{2} + X − 1)(X^{4} − X^{3} + 2X^{2} + 7X + 7)

b_{3} −3(X^{6} − 24X^{4} − 10X^{3} + 84X^{2} − 60X + 9)

= −3(X − 1)(X + 3)(X^{2} + 3X − 3)(X^{2} − 5X + 1)

b_{4} −3(9X^{6} − 24X^{5} + 72X^{4} − 98X^{3} + 132X^{2} − 84X + 49)

= −3(X^{2} − X + 1)(3X^{2} − 3X + 7)(3X^{2} − 2X + 7)

N C F_{6}
320 a_{1} 2(X^{5} + 3X^{3} − 4X)

= 2X(X − 1)(X + 1)(X^{2} + 4)

a_{2} 3(2X^{6} + 24X^{5} + 117X^{4} + 301X^{3} + 441X^{2} + 360X + 130)

= 3(X^{2} + 2X + 2)(X^{2} + 5X + 5)(2X^{2} + 10X + 13)

a_{3} 2(2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3)

= 2(X − 1)(X − 3)(2X − 1)(X^{2} − 3X + 1)

a_{4} 6(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= 6(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

a_{5} −6(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= −6(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

325 a_{1} −15(5X^{6} − 12X^{4} − 10X^{3} − 24X^{2} + 13)

= −15(X^{2} + X − 1)(5X^{4} − 5X^{3} − 2X^{2} − 13X − 13)

a_{2} 5(X^{6} + 2X^{5} + 3X^{4} − 6X^{3} − 27X^{2} − 8X − 4)

= 5(X − 2)(X + 2)(X^{4} + 2X^{3} + 7X^{2} + 2X + 1)

b_{1} −3(X^{6} + 6X^{5} + 27X^{4} − 2X^{3} + 45X^{2} + 20)
c_{1} −15(X^{6} + 6X^{5} + 27X^{4} − 2X^{3} + 45X^{2} + 20)
335 a_{1} X^{6} + 4X^{5} − 48X^{2} + 20X − 4
339 a_{1} −3X^{6} + 6X^{5} − 25X^{4} + 36X^{3} − 61X^{2} + 42X − 31
345 a_{1} X^{6} + 12X^{5} + 32X^{4} − 24X^{3} + 8X^{2} + 12X + 4
357 a_{1} X^{6} + 8X^{4} − 8X^{3} + 20X^{2} − 12X + 12
369 a_{1} 3(3X^{6} − 6X^{5} + 17X^{4} − 20X^{3} + 29X^{2} − 18X + 15)
385 a_{1} −3(X^{6} + 6X^{5} + 99X^{4} − 74X^{3} − 171X^{2} + 252X − 88)

= −3(X^{2} + X − 1)(X^{4} + 5X^{3} + 95X^{2} − 164X + 88)

a_{2} −19X^{6} − 86X^{5} − 57X^{4} + 258X^{3} + 513X^{2} + 344X + 76

= −(X − 2)(X + 2)(19X^{4} + 86X^{3} + 133X^{2} + 86X + 19)

N C F_{6}
387 a_{1} 3(3X^{6} + 2X^{5} − 7X^{4} + 4X^{3} + 13X^{2} − 10X + 7)
a_{2} X^{6} + 8X^{3} − 36X^{2} + 12X − 4
b_{1} 3(3X^{6} + 12X^{5} + 16X^{4} + 10X^{3} + 4X^{2} − 12X + 3)
389 a_{1} X^{6} − 10X^{5} + 23X^{4} + 20X^{3} − 45X^{2} − 46X − 11
397 a_{1} −3(5X^{6} + 12X^{5} − 2X^{3} + 36X^{2} − 24X + 13)
425 a_{1} 5(X^{6} + 4X^{5} + 8X^{4} − 2X^{3} − 12X^{2} + 4X + 1)

= 5(X^{2} + X − 1)(X^{4} + 3X^{3} + 6X^{2} − 5X − 1)

a_{2} 5(X^{6} + 2X^{5} + 7X^{4} + 6X^{3} + 13X^{2} − 8X + 4)

= 5(X^{2} + 4)(X^{4} + 2X^{3} + 3X^{2} − 2X + 1)

429 a_{1} 9X^{6} + 30X^{5} + 79X^{4} + 98X^{3} + 109X^{2} + 24X + 36

= (X^{2} + X + 3)(9X^{4} + 21X^{3} + 31X^{2} + 4X + 12)

a_{2} X^{6} − 24X^{5} + 100X^{4} + 102X^{3} − 80X^{2} − 132X − 39

= (X^{2} − 6X − 3)(X^{4} − 18X^{3} − 5X^{2} + 18X + 13)

437 a_{1} X^{6} + 8X^{5} + 16X^{4} + 4X^{2} + 4X + 4
441 a_{1} −3(X^{6} − 4X^{4} + 2X^{3} + 8X^{2} − 12X + 9)

= −3(X^{2} − X + 1)(X^{4} + X^{3} − 4X^{2} − 3X + 9)

a_{2} −3(X^{6} − 16X^{4} − 2X^{3} + 44X^{2} − 36X + 9)

= −3(X − 1)(X + 3)(X^{4} − 2X^{3} − 9X^{2} + 10X − 3)

b_{1} 21(X^{6} − 4X^{4} + 2X^{3} + 8X^{2} − 12X + 9)

= 21(X^{2} − X + 1)(X^{4} + X^{3} − 4X^{2} − 3X + 9)

b_{2} 21(X^{6} − 16X^{4} − 2X^{3} + 44X^{2} − 36X + 9)

= 21(X − 1)(X + 3)(X^{4} − 2X^{3} − 9X^{2} + 10X − 3)

N C F_{6}
445 a_{1} −3(X^{6} + 6X^{5} − 9X^{4} + 34X^{3} + 45X^{2} − 72X + 20)

= −3(X^{2} + X − 1)(X^{4} + 5X^{3} − 13X^{2} + 52X − 20)

a_{2} 5X^{6} + 8X^{5} − 20X^{4} + 6X^{3} + 8X^{2} − 4X − 3

= (X − 1)(X + 3)(5X^{4} − 2X^{3} − X^{2} + 2X + 1)

465 a_{1} X^{6} + 28X^{4} − 30X^{3} + 40X^{2} − 12X + 9

= (X^{2} − X + 1)(X^{4} + X^{3} + 28X^{2} − 3X + 9)

a_{2} X^{6} + 16X^{4} + 30X^{3} − 116X^{2} + 60X + 9

= (X − 1)(X + 3)(X^{4} − 2X^{3} + 23X^{2} − 22X − 3)

495 a_{1} 3(3X^{6} − 4X^{5} − 16X^{4} + 2X^{3} + 4X^{2} − 4X + 3)

= 3(X + 1)(X − 3)(X^{2} + X − 1)(3X^{2} − X + 1)

a_{2} X^{6} − 6X^{5} + 27X^{4} + 18X^{3} − 27X^{2} − 36X − 16

= (X^{2} − X − 1)(X^{4} − 5X^{3} + 23X^{2} + 20X + 16)

a_{3} 3(3X^{6} + 2X^{5} + 21X^{4} + 6X^{3} + 39X^{2} − 8X + 12)

= 3(X^{2} + 4)(X^{2} + X + 3)(3X^{2} − X + 1)

a_{4} X^{6} − 18X^{5} + 63X^{4} + 154X^{3} − 27X^{2} − 48X − 4

= (X^{2} + X − 1)(X^{2} − 8X − 4)(X^{2} − 11X − 1)

b_{1} −3(X^{6} + 6X^{5} + 11X^{4} + 14X^{3} + 5X^{2} − 12X)

= −3X(X + 4)(X^{2} + X − 1)(X^{2} + X + 3)

b_{2} −3(X^{6} + 8X^{4} − 10X^{3} + 20X^{2} − 12X + 9)

= −3(X^{2} − X + 1)(X^{4} + X^{3} + 8X^{2} − 3X + 9)

b_{3} 5X^{6} − 18X^{5} + 15X^{4} + 54X^{3} − 135X^{2} + 72X − 20

= (X − 2)(X + 2)(X^{2} − 3X + 5)(5X^{2} − 3X + 1)

b_{4} −3(X^{6} − 12X^{5} + 32X^{4} + 34X^{3} − 52X^{2} − 60X − 15)

= −3(X^{2} − X − 1)(X^{2} − 5X − 5)(X^{2} − 6X − 3)

N C F_{6}
507 a_{1} 13(X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9)
= 13(X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)
a_{2} 13(X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9)
= 13(X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)
a_{3} 13(X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9)
= 13(X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)
a_{4} −65(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)
= −65(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)
517 a_{1} −3X^{6} + 16X^{5} − 4X^{4} − 34X^{3} − 56X^{2} − 44X − 11
529 a_{1} −3(X^{6} − 12X^{5} + 24X^{4} + 82X^{3} + 84X^{2} + 60X + 17)
a_{2} 23(3X^{6} − 2X^{5} + 21X^{4} + 2X^{3} + 111X^{2} − 100X + 24)
b_{1} 69(X^{6} − 12X^{5} + 24X^{4} + 82X^{3} + 84X^{2} + 60X + 17)
b_{2} −3X^{6} + 2X^{5} − 21X^{4} − 2X^{3} − 111X^{2} + 100X − 24
545 a_{1} 13X^{6} − 116X^{5} − 316X^{4} + 58X^{3} + 264X^{2} − 116X + 13
= (X^{2} + X − 1)(13X^{4} − 129X^{3} − 174X^{2} + 103X − 13)
a_{2} 13X^{6} + 58X^{5} + 91X^{4} + 174X^{3} + 169X^{2} − 232X + 52
= (X^{2} + 4)(13X^{4} + 58X^{3} + 39X^{2} − 58X + 13)
547 a_{1} −5(3X^{6} − 26X^{5} + 57X^{4} + 14X^{3} + 39X^{2} − 4X + 24)
549 a_{1} −3(X^{6} + 8X^{4} + 10X^{3} + 20X^{2} + 24X + 9)
561 a_{1} 9X^{6} − 6X^{5} + 19X^{4} + 4X^{3} + 7X^{2} + 6X + 5
583 a_{1} X^{6} + 4X^{5} + 8X^{4} − 16X^{2} − 4X − 4

N C F_{6}
585 a_{1} 5X^{6} − 12X^{4} − 10X^{3} − 24X^{2} + 13

= (X^{2} + X − 1)(5X^{4} − 5X^{3} − 2X^{2} − 13X − 13)

a_{2} −3(X^{6} + 2X^{5} + 3X^{4} − 6X^{3} − 27X^{2} − 8X − 4)

= −3(X − 2)(X + 2)(X^{4} + 2X^{3} + 7X^{2} + 2X + 1)

591 a_{1} 81X^{6} − 144X^{5} − 100X^{4} + 200X^{3} + 80X^{2} − 84X − 36
605 a_{1} 11(3X^{6} − 4X^{5} − 16X^{4} + 2X^{3} + 4X^{2} − 4X + 3)

= 11(X + 1)(X − 3)(X^{2} + X − 1)(3X^{2} − X + 1)

a_{2} −33(X^{6} − 6X^{5} + 27X^{4} + 2X^{3} − 27X^{2} − 36X − 16)

= −33(X^{2} − X − 1)(X^{4} − 5X^{3} + 23X^{2} + 20X + 16)

a_{3} 11(3X^{6} + 2X^{5} + 21X^{4} + 6X^{3} + 39X^{2} − 8X + 12)

= 11(X^{2} + 4)(X^{2} + X + 3)(3X^{2} − X + 1)

a_{4} −33(X^{6} − 18X^{5} + 63X^{4} + 154X^{3} − 27X^{2} − 48X − 4)

= −33(X^{2} + X − 1)(X^{2} − 8X − 4)(X^{2} − 11X − 1)

609 a_{1} X^{6} + 16X^{4} − 18X^{3} + 28X^{2} − 12X + 9

= (X^{2} − X + 1)(X^{4} + X^{3} + 16X^{2} − 3X + 9)

a_{2} X^{6} + 4X^{4} + 18X^{3} − 56X^{2} + 24X + 9

= (X − 1)(X + 3)(X^{4} − 2X^{3} + 11X^{2} − 10X − 3)

621 a_{1} X^{6} + 12X^{5} + 36X^{4} − 6X^{3} − 72X^{2} + 48X − 7

= (X^{3} + 6X^{2} + 6X − 7)(X^{3} + 6X^{2} − 6X + 1)

b_{1} −3(X^{6} + 12X^{5} + 36X^{4} − 6X^{3} − 72X^{2} + 48X − 7)

= −3(X^{3} + 6X^{2} + 6X − 7)(X^{3} + 6X^{2} − 6X + 1)

c_{1} 5X^{6} + 6X^{5} + 27X^{4} + 20X^{3} + 39X^{2} + 18X + 9
d_{1} −3(5X^{6} + 6X^{5} + 27X^{4} + 20X^{3} + 39X^{2} + 18X + 9)
637 a_{1} 7(3X^{6} + 2X^{5} + 13X^{4} + 12X^{3} + 17X^{2} + 14X + 7)

N C F_{6}
639 a_{1} X^{6} + 12X^{4} − 14X^{3} + 24X^{2} − 12X + 9
= (X^{2} − X + 1)(X^{4} + X^{3} + 12X^{2} − 3X + 9)
a_{2} X^{6} + 14X^{3} − 36X^{2} + 12X + 9
= (X − 1)(X + 3)(X^{4} − 2X^{3} + 7X^{2} − 6X − 3)
b_{1} −3(X^{6} + 12X^{4} − 14X^{3} + 24X^{2} − 12X + 9)
= −3(X^{2} − X + 1)(X^{4} + X^{3} + 12X^{2} − 3X + 9)
b_{2} −3(X^{6} + 14X^{3} − 36X^{2} + 12X + 9)
= −3(X − 1)(X + 3)(X^{4} − 2X^{3} + 7X^{2} − 6X − 3)
645 a_{1} X^{6} + 8X^{4} + 20X^{2} − 12X + 4
661 a_{1} −3(X^{6} + 6X^{5} + 27X^{4} + 14X^{3} + 117X^{2} + 60X + 8)
675 a_{1} X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3
a_{2} −3(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3)
b_{1} 5(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3)
b_{2} −15(X^{6} + 6X^{5} − 9X^{4} + 4X^{3} + 3X^{2} − 6X − 3)
703 a_{1} X^{6} + 2X^{5} − 13X^{4} − 4X^{3} + 15X^{2} + 14X + 5
705 a_{1} 9X^{6} + 12X^{5} − 32X^{4} + 6X^{3} + 76X^{2} − 60X + 25
711 a_{1} 3(27X^{6} + 72X^{5} + 100X^{4} + 70X^{3} + 40X^{2} + 12X + 3)
715 a_{1} 5X^{6} + 10X^{5} + 19X^{4} + 60X^{3} + 47X^{2} − 50X + 9

N C F_{6}
725 a_{1} 5(X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7)
a_{2} 5(X^{6} − 2X^{5} + 7X^{4} − 6X^{3} + 13X^{2} − 4X + 8)
b_{1} 5(X^{6} − 8X^{4} + 30X^{3} − 44X^{2} + 25)
= 5(X^{2} − X − 1)(X^{2} − 2X + 5)(X^{2} + 3X − 5)
b_{2} −15(13X^{6} + 24X^{5} − 36X^{4} + 38X^{3} + 24X^{2} − 36X − 27)
= −15(X − 1)(X + 3)(13X^{4} − 2X^{3} + 7X^{2} + 18X + 9)
b_{3} 5(X^{6} + 12X^{5} + 28X^{4} − 6X^{3} − 32X^{2} + 12X + 1)
= 5(X^{2} + X − 1)(X^{4} + 11X^{3} + 18X^{2} − 13X − 1)
b_{4} 15(27X^{6} + 84X^{4} − 50X^{3} + 72X^{2} − 60X + 35)
= 15(3X^{2} + 2X + 7)(9X^{4} − 6X^{3} + 11X^{2} − 10X + 5)

N C F_{6}
735 a_{1} −3X^{6} − 8X^{4} − 50X^{3} + 148X^{2} − 60X − 27

= −(X − 1)(X + 3)(3X^{2} − 3X − 1)(X^{2} − X + 9)

a_{2} −7(5X^{6} + 8X^{4} − 18X^{3} + 68X^{2} − 60X + 45)

= −7(X^{2} − X + 1)(5X^{4} + 5X^{3} + 8X^{2} − 15X + 45)

a_{3} 7(75X^{6} + 60X^{5} − 736X^{4} − 110X^{3} + 1844X^{2} − 1140X + 75)

= 7(3X^{2} − 9X + 5)(5X^{2} + 13X − 1)(5X^{2} + 6X − 15)

a_{4} −7(13X^{6} − 180X^{5} + 640X^{4} − 1770X^{3} + 796X^{2} + 60X − 99)

= −7(X^{2} − 10X − 3)(X^{2} − 3X + 11)(13X^{2} − 11X + 3)

b_{1} 7(3X^{6} + 8X^{4} + 50X^{3} − 148X^{2} + 60X + 27)

= 7(X − 1)(X + 3)(3X^{2} − 3X − 1)(X^{2} − X + 9)

b_{2} 5X^{6} + 8X^{4} − 18X^{3} + 68X^{2} − 60X + 45

= (X^{2} − X + 1)(5X^{4} + 5X^{3} + 8X^{2} − 15X + 45)

b_{3} −75X^{6} − 60X^{5} + 736X^{4} + 110X^{3} − 1844X^{2} + 1140X − 75

= −(3X^{2} − 9X + 5)(5X^{2} + 13X − 1)(5X^{2} + 6X − 15)

b_{4} 13X^{6} − 180X^{5} + 640X^{4} − 1770X^{3} + 796X^{2} + 60X − 99

= (X^{2} − 10X − 3)(X^{2} − 3X + 11)(13X^{2} − 11X + 3)

c_{1} −3X^{6} + 6X^{5} + 23X^{4} − 4X^{3} − 13X^{2} − 30X + 25
d_{1} 7(3X^{6} − 6X^{5} − 23X^{4} + 4X^{3} + 13X^{2} + 30X − 25)
737 a_{1} X^{6} − 4X^{5} + 16X^{4} − 24X^{3} + 40X^{2} − 20X + 4

N C F_{6}
738 a_{1} 3(12X^{6} + 4X^{5} + 87X^{4} + 100X^{3} + 69X^{2} + 26X + 6)

= 3(12X^{2} + 4X + 3)(X^{4} + 7X^{2} + 6X + 2)

a_{2} 4X^{6} − 36X^{5} + 81X^{4} + 50X^{3} − 207X^{2} − 204X − 52

= (X^{2} − 4X − 4)(4X^{4} − 20X^{3} + 17X^{2} + 38X + 13)

a_{3} 4X^{6} − 36X^{5} + 189X^{4} − 500X^{3} + 423X^{2} − 66X + 2

= (2X^{2} − 7X + 1)(2X^{4} − 11X^{3} + 55X^{2} − 52X + 2)

a_{4} −3(4X^{6} + 4X^{5} − 11X^{4} + 2X^{3} + 29X^{2} + 4X − 12)

= −3(X + 1)(2X + 3)(2X^{4} − 3X^{3} − X^{2} + 8X − 4)

741 a_{1} −3X^{6} − 6X^{5} − 17X^{4} − 76X^{3} − 53X^{2} − 18X + 9
747 a_{1} −3(9X^{6} + 30X^{5} + 19X^{4} − 38X^{3} − 47X^{2} + 12X + 24)
753 a_{1} 9X^{6} + 12X^{5} + 64X^{4} + 40X^{3} + 112X^{2} − 12X + 36
765 a_{1} −3(X^{6} + 4X^{5} + 8X^{4} − 2X^{3} − 12X^{2} + 4X + 1)

= −3(X^{2} + X − 1)(X^{4} + 3X^{3} + 6X^{2} − 5X − 1)

a_{2} −3(X^{6} + 2X^{5} + 7X^{4} + 6X^{3} + 13X^{2} − 8X + 4)

= −3(X^{2} + 4)(X^{4} + 2X^{3} + 3X^{2} − 2X + 1)

767 a_{1} X^{6} + 4X^{5} + 16X^{4} + 34X^{3} + 60X^{2} + 52X + 17
777 a_{1} −3X^{6} + 44X^{4} + 2X^{3} − 112X^{2} + 96X − 27

= −(X − 1)(X + 3)(3X^{2} − 3X + 1)(X^{2} − X − 9)

a_{2} −3X^{6} + 8X^{4} − 2X^{3} − 28X^{2} + 36X − 27

= −(X^{2} − X + 1)(3X^{4} + 3X^{3} − 8X^{2} − 9X + 27)

a_{3} 11(7X^{6} − 180X^{5} + 1120X^{4} − 2982X^{3} + 1348X^{2} − 36X − 57)

= 11(X^{2} − 18X − 3)(X^{2} − 7X + 19)(7X^{2} − 5X + 1)

a_{4} −5(63X^{6} + 36X^{5} − 448X^{4} − 14X^{3} + 884X^{2} − 396X − 81)

= −5(3X^{2} + 2X − 9)(3X^{2} + 7X + 1)(7X^{2} − 17X + 9)

779 a_{1} X^{6} + 2X^{5} − X^{4} − 12X^{3} + 19X^{2} − 10X − 3

N C F_{6}
800 a_{1} 5(X^{5} + 3X^{3} − 4X)

= 5X(X − 1)(X + 1)(X^{2} + 4)

a_{2} 30(2X^{6} + 24X^{5} + 117X^{4} + 301X^{3} + 441X^{2} + 360X + 130)

= 30(X^{2} + 2X + 2)(X^{2} + 5X + 5)(2X^{2} + 10X + 13)

a_{3} 5(2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3)

= 5(X − 1)(X − 3)(2X − 1)(X^{2} − 3X + 1)

a_{4} 15(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= 15(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

a_{5} −15(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= −15(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

819 a_{1} 3(3X^{6} + 2X^{5} + 13X^{4} + 12X^{3} + 17X^{2} + 14X + 7)
b_{1} −3(5X^{6} + 6X^{5} − 25X^{4} − 30X^{3} + 5X^{2} − 24X − 36)

= −3(X + 2)(5X + 6)(X^{2} − X + 1)(X^{2} − X − 3)

b_{2} −15(9X^{6} + 66X^{5} + 127X^{4} − 58X^{3} − 251X^{2} + 204X − 48)

= −15(X^{2} + 3X − 3)(9X^{4} + 39X^{3} + 37X^{2} − 52X + 16)

b_{3} 15(9X^{6} + 102X^{5} + 211X^{4} − 298X^{3} − 563X^{2} + 528X − 108)

= 15(X^{2} + X − 3)(3X^{2} + 8X − 4)(3X^{2} + 23X − 9)

b_{4} 3(3X^{6} − 6X^{5} + 25X^{4} + 10X^{3} + 55X^{2} + 24X + 36)

= 3(X^{2} + X + 1)(X^{2} − 3X + 9)(3X^{2} + 4)

825 a_{1} 5(X^{6} + 6X^{5} + 11X^{4} + 14X^{3} + 5X^{2} − 12X)

= 5X(X + 4)(X^{2} + X − 1)(X^{2} + X + 3)

a_{2} 5(X^{6} + 8X^{4} − 10X^{3} + 20X^{2} − 12X + 9)

= 5(X^{2} − X + 1)(X^{4} + X^{3} + 8X^{2} − 3X + 9)

a_{3} −15(5X^{6} − 18X^{5} + 15X^{4} + 54X^{3} − 135X^{2} + 72X − 20)

= −15(X − 2)(X + 2)(X^{2} − 3X + 5)(5X^{2} − 3X + 1)

a_{4} 5(X^{6} − 12X^{5} + 32X^{4} + 34X^{3} − 52X^{2} − 60X − 15)

= 5(X^{2} − X − 1)(X^{2} − 5X − 5)(X^{2} − 6X − 3)

N C F_{6}
837 a_{1} 5X^{6} + 36X^{5} + 60X^{4} − 14X^{3} − 48X^{2} + 24X − 3

= (X^{3} + 6X^{2} + 6X − 3)(5X^{3} + 6X^{2} − 6X + 1)

b_{1} −3(5X^{6} + 36X^{5} + 60X^{4} − 14X^{3} − 48X^{2} + 24X − 3)

= −3(X^{3} + 6X^{2} + 6X − 3)(5X^{3} + 6X^{2} − 6X + 1)

841 a_{1} 29(X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7)
a_{2} 29(X^{6} − 2X^{5} + 7X^{4} − 6X^{3} + 13X^{2} − 4X + 8)
845 a_{1} −39(5X^{6} − 12X^{4} − 10X^{3} − 24X^{2} + 13)

= −39(X^{2} + X − 1)(5X^{4} − 5X^{3} − 2X^{2} − 13X − 13)

a_{2} 13(X^{6} + 2X^{5} + 3X^{4} − 6X^{3} − 27X^{2} − 8X − 4)

= 13(X − 2)(X + 2)(X^{4} + 2X^{3} + 7X^{2} + 2X + 1)

855 a_{1} 3(3X^{6} − 4X^{4} + 10X^{3} + 8X^{2} − 5)

= 3(X^{2} + X − 1)(3X^{4} − 3X^{3} + 2X^{2} + 5X + 5)

a_{2} X^{6} + 18X^{5} + 3X^{4} − 54X^{3} − 27X^{2} − 72X − 4

= (X − 2)(X + 2)(X^{4} + 18X^{3} + 7X^{2} + 18X + 1)

b_{1} −3(5X^{6} + 32X^{4} − 42X^{3} + 92X^{2} − 60X + 45)

= −3(X^{2} − X + 1)(5X^{4} + 5X^{3} + 32X^{2} − 15X + 45)

b_{2} −3(5X^{6} − 28X^{4} + 42X^{3} − 40X^{2} − 24X + 45)

= −3(X − 1)(X + 3)(5X^{4} − 10X^{3} + 7X^{2} − 2X − 15)

882 a_{1} −3(2X^{6} + 4X^{5} + 7X^{4} + 48X^{3} + 20X^{2} − 20X + 3)

= −3(X + 1)(X + 3)(2X^{4} − 4X^{3} + 17X^{2} − 8X + 1)

a_{2} 21(2X^{6} + 4X^{5} + 7X^{4} + 48X^{3} + 20X^{2} − 20X + 3)

= 21(X + 1)(X + 3)(2X^{4} − 4X^{3} + 17X^{2} − 8X + 1)

a_{3} −4X^{6} + 36X^{5} − 63X^{4} − 86X^{3} + 9X^{2} − 12X − 20

= −(X^{2} − 4X − 4)(4X^{4} − 20X^{3} − X^{2} + 2X − 5)

a_{4} 7(4X^{6} − 36X^{5} + 63X^{4} + 86X^{3} − 9X^{2} + 12X + 20)

= 7(X^{2} − 4X − 4)(4X^{4} − 20X^{3} − X^{2} + 2X − 5)

885 a_{1} 9X^{6} − 78X^{5} + 163X^{4} + 116X^{3} − 353X^{2} − 330X − 75

N C F_{6}
909 a_{1} −3(X^{6} − 4X^{4} + 8X^{3} + 8X^{2} − 36X − 36)
931 a_{1} X^{6} − 4X^{5} + 16X^{4} + 34X^{3} + 116X^{2} + 152X + 57
b_{1} −7(X^{6} − 4X^{5} + 16X^{4} + 34X^{3} + 116X^{2} + 152X + 57)
975 a_{1} 5(X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9)

= 5(X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} 5(X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9)

= 5(X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)

a_{3} 5(X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9)

= 5(X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)

a_{4} −(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)

= −(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)

1017 a_{1} 3(3X^{6} − 6X^{5} + 25X^{4} − 36X^{3} + 61X^{2} − 42X + 31)
1035 a_{1} −3(X^{6} + 12X^{5} + 32X^{4} − 24X^{3} + 8X^{2} + 12X + 4)
b_{1} X^{6} − 12X^{4} + 96X^{2} − 180X + 100
c_{1} −3(X^{6} − 12X^{4} + 96X^{2} − 180X + 100)
1045 a_{1} 3(7X^{6} + 54X^{5} + 21X^{4} − 162X^{3} − 189X^{2} − 216X − 28)

= 3(X − 2)(X + 2)(X^{2} + 7X + 1)(7X^{2} + 5X + 7)

a_{2} 5X^{6} − 4X^{4} + 30X^{3} + 32X^{2} − 19

= (X^{2} + X − 1)(5X^{4} − 5X^{3} + 6X^{2} + 19X + 19)

a_{3} 21(81X^{6} + 252X^{5} − 2172X^{4} + 14X^{3} + 4968X^{2} + 3192X + 361)

= 21(X^{2} + 7X + 1)(9X^{2} − 14X − 19)(9X^{2} − 21X − 19)

a_{4} 3(175X^{6} + 300X^{5} + 972X^{4} + 970X^{3} + 1464X^{2} + 720X + 567)

= 3(5X^{2} + 2X + 9)(5X^{2} + 3X + 9)(7X^{2} + 5X + 7)

1071 a_{1} −3(X^{6} + 8X^{4} − 8X^{3} + 20X^{2} − 12X + 12)

N C F_{6}
1075 a_{1} −5(3X^{6} + 2X^{5} − 7X^{4} + 4X^{3} + 13X^{2} − 10X + 7)
a_{2} −15(X^{6} + 8X^{3} − 36X^{2} + 12X − 4)
1165 a_{1} −3(X^{6} − 60X^{4} − 290X^{3} − 408X^{2} + 233)

= −3(X^{2} + X − 1)(X^{4} − X^{3} − 58X^{2} − 233X − 233)

a_{2} 13X^{6} + 58X^{5} + 39X^{4} − 174X^{3} − 351X^{2} − 232X − 52

= (X − 2)(X + 2)(13X^{4} + 58X^{3} + 91X^{2} + 58X + 13)

1183 a_{1} −13(3X^{6} + 2X^{5} + 13X^{4} + 12X^{3} + 17X^{2} + 14X + 7)
1185 a_{1} −3X^{6} + 16X^{4} − 10X^{3} − 20X^{2} + 36X − 27

= (X^{2} − X + 1)(−3X^{4} − 3X^{3} + 16X^{2} + 9X − 27)

a_{2} −3X^{6} + 52X^{4} + 10X^{3} − 152X^{2} + 120X − 27

= −(X − 1)(X + 3)(3X^{4} − 6X^{3} − 31X^{2} + 34X − 9)

1197 a_{1} 13X^{6} − 108X^{5} + 36X^{4} + 458X^{3} + 768X^{2} + 624X + 189

= (X − 7)(13X + 9)(X^{2} + X + 1)(X^{2} − 3X − 3)

a_{2} −5(259X^{6} + 792X^{5} + 588X^{4} + 126X^{3} + 264X^{2} − 72X + 43)

= −5(7X^{2} − 3X + 1)(37X^{4} + 129X^{3} + 134X^{2} + 57X + 43)

a_{3} 5X^{6} + 204X^{5} + 2544X^{4} + 9150X^{3} + 5484X^{2} + 1176X + 85

= (X^{2} + 5X + 1)(X^{2} + 18X + 5)(5X^{2} + 89X + 17)

a_{4} 13X^{6} + 84X^{5} + 108X^{4} + 66X^{3} − 24X + 5

= (X + 1)(X + 5)(X^{2} + X + 1)(13X^{2} − 7X + 1)

b_{1} −3(13X^{6} − 108X^{5} + 36X^{4} + 458X^{3} + 768X^{2} + 624X + 189)

= −3(X − 7)(13X + 9)(X^{2} + X + 1)(X^{2} − 3X − 3)

b_{2} 15(259X^{6} + 792X^{5} + 588X^{4} + 126X^{3} + 264X^{2} − 72X + 43)

= 15(7X^{2} − 3X + 1)(37X^{4} + 129X^{3} + 134X^{2} + 57X + 43)

b_{3} −3(5X^{6} + 204X^{5} + 2544X^{4} + 9150X^{3} + 5484X^{2} + 1176X + 85)

−3(X^{2} + 5X + 1)(X^{2} + 18X + 5)(5X^{2} + 89X + 17)

b_{4} −3(13X^{6} + 84X^{5} + 108X^{4} + 66X^{3} − 24X + 5)

= −3(X + 1)(X + 5)(X^{2} + X + 1)(13X^{2} − 7X + 1)

N C F_{6}
1203 a_{1} 9X^{6} + 12X^{5} + 52X^{4} + 62X^{3} + 88X^{2} + 72X + 33
1209 a_{1} 9X^{6} + 48X^{5} + 112X^{4} + 144X^{3} + 124X^{2} + 60X + 36
1225 a_{1} 5(X^{6} − 2X^{5} + 7X^{4} + 4X^{3} + 3X^{2} + 10X + 5)
b_{1} −35(X^{6} − 2X^{5} + 7X^{4} + 4X^{3} + 3X^{2} + 10X + 5)
c_{1} 5(5X^{6} − 10X^{5} − X^{4} + 2X^{3} + 45X^{2} − 76X + 40)
d_{1} −35(5X^{6} − 10X^{5} − X^{4} + 2X^{3} + 45X^{2} − 76X + 40)
1275 a_{1} 9X^{6} + 60X^{5} + 88X^{4} − 40X^{3} − 8X^{2} − 60X + 36
b_{1} 5(9X^{6} + 60X^{5} + 88X^{4} − 40X^{3} − 8X^{2} − 60X + 36)
1287 a_{1} −3(9X^{6} + 30X^{5} + 79X^{4} + 98X^{3} + 109X^{2} + 24X + 36)
= −3(X^{2} + X + 3)(9X^{4} + 21X^{3} + 31X^{2} + 4X + 12)
a_{2} −3(X^{6} − 24X^{5} + 100X^{4} + 102X^{3} − 80X^{2} − 132X − 39)
= −3(X^{2} − 6X − 3)(X^{4} − 18X^{3} − 5X^{2} + 18X + 13)
1295 a_{1} X^{6} + 2X^{5} − 17X^{4} + 4X^{3} + 3X^{2} + 62X − 59
1305 a_{1} −3(X^{6} − 8X^{4} + 30X^{3} − 44X^{2} + 25)
= −3(X^{2} − X − 1)(X^{2} − 2X + 5)(X^{2} + 3X − 5)
a_{2} 13X^{6} + 24X^{5} − 36X^{4} + 38X^{3} + 24X^{2} − 36X − 27
= (X − 1)(X + 3)(13X^{4} − 2X^{3} + 7X^{2} + 18X + 9)
a_{3} −3(X^{6} + 12X^{5} + 28X^{4} − 6X^{3} − 32X^{2} + 12X + 1)
= −3(X^{2} + X − 1)(X^{4} + 11X^{3} + 18X^{2} − 13X − 1)
a_{4} −27X^{6} − 84X^{4} + 50X^{3} − 72X^{2} + 60X − 35
= −(3X^{2} + 2X + 7)(9X^{4} − 6X^{3} + 11X^{2} − 10X + 5)

N C F_{6}
1325 a_{1} 5(X^{6} + 10X^{5} + 23X^{4} − 10X^{3} + 5X^{2} + 8X + 4)
b_{1} −3(X^{6} + 6X^{5} − 57X^{4} + 12X^{3} + 171X^{2} + 114X + 13)
c_{1} −15(X^{6} + 6X^{5} − 57X^{4} + 12X^{3} + 171X^{2} + 114X + 13)
1395 a_{1} −3(X^{6} + 28X^{4} − 30X^{3} + 40X^{2} − 12X + 9)
= −3(X^{2} − X + 1)(X^{4} + X^{3} + 28X^{2} − 3X + 9)
a_{2} −3(X^{6} + 16X^{4} + 30X^{3} − 116X^{2} + 60X + 9)
= −3(X − 1)(X + 3)(X^{4} − 2X^{3} + 23X^{2} − 22X − 3)
1409 a_{1} −3(X^{6} + 18X^{5} + 87X^{4} + 54X^{3} + 93X^{2} + 36X + 8)
1421 a_{1} −7(X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7)
a_{2} −7(X^{6} − 2X^{5} + 7X^{4} − 6X^{3} + 13X^{2} − 4X + 8)
1425 a_{1} −5(3X^{6} − 4X^{4} + 10X^{3} + 8X^{2} − 5)
= −5(X^{2} + X − 1)(3X^{4} − 3X^{3} + 2X^{2} + 5X + 5)
a_{2} −15(X^{6} + 18X^{5} + 3X^{4} − 54X^{3} − 27X^{2} − 72X − 4)
= −15(X − 2)(X + 2)(X^{4} + 18X^{3} + 7X^{2} + 18X + 1)
b_{1} 5(5X^{6} + 32X^{4} − 42X^{3} + 92X^{2} − 60X + 45)
= 5(X^{2} − X + 1)(5X^{4} + 5X^{3} + 32X^{2} − 15X + 45)
b_{2} 5(5X^{6} − 28X^{4} + 42X^{3} − 40X^{2} − 24X + 45)
= 5(X − 1)(X + 3)(5X^{4} − 10X^{3} + 7X^{2} − 2X − 15)
1433 a_{1} −3(X^{6} − 12X^{4} − 106X^{3} + 144X^{2} − 108X − 783)

N C F_{6}
1440 a_{1} −3(X^{5} + 3X^{3} − 4X)

= −3X(X − 1)(X + 1)(X^{2} + 4)

a_{2} −2(2X^{6} + 24X^{5} + 117X^{4} + 301X^{3} + 441X^{2} + 360X + 130)

= −2(X^{2} + 2X + 2)(X^{2} + 5X + 5)(2X^{2} + 10X + 13)

a_{3} −3(2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3)

= −3(X − 1)(X − 3)(2X − 1)(X^{2} − 3X + 1)

a_{4} −X^{6} − 12X^{5} − 27X^{4} + 34X^{3} + 72X^{2} − 72X + 16

= −(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

a_{5} X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16

= (X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

1445 a_{1} 17(X^{6} + 4X^{5} + 8X^{4} − 2X^{3} − 12X^{2} + 4X + 1)

= 17(X^{2} + X − 1)(X^{4} + 3X^{3} + 6X^{2} − 5X − 1)

a_{2} 17(X^{6} + 2X^{5} + 7X^{4} + 6X^{3} + 13X^{2} − 8X + 4)

= 17(X^{2} + 4)(X^{4} + 2X^{3} + 3X^{2} − 2X + 1)

1455 a_{1} 9X^{6} + 24X^{5} + 88X^{4} + 432X^{3} + 412X^{2} − 276X + 36
1485 a_{1} −15X^{6} − 12X^{5} − 60X^{4} − 34X^{3} − 72X^{2} − 24X − 23
b_{1} 3(15X^{6} + 12X^{5} + 60X^{4} + 34X^{3} + 72X^{2} + 24X + 23)
1521 a_{1} −39(X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9)

= −39(X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} −39(X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9)

= −39(X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)

a_{3} −39(X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9)

= −39(X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)

a_{4} 195(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)

= 195(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)

1533 a_{1} 5(9X^{6} + 30X^{5} + 199X^{4} − 430X^{3} + 613X^{2} − 480X + 252)

N C F_{6}
1575 a_{1} 5(X^{6} − 12X^{4} − 10X^{3} + 12X + 9)

= 5(X − 1)(X + 3)(X^{2} + X + 1)(X^{2} − 3X − 3)

a_{2} 5(X^{6} + 10X^{3} + 12X^{2} − 7)

= 5(X^{2} + X − 1)(X^{4} − X^{3} + 2X^{2} + 7X + 7)

a_{3} 5(X^{6} − 24X^{4} − 10X^{3} + 84X^{2} − 60X + 9)

= 5(X − 1)(X + 3)(X^{2} + 3X − 3)(X^{2} − 5X + 1)

a_{4} 5(9X^{6} − 24X^{5} + 72X^{4} − 98X^{3} + 132X^{2} − 84X + 49)

= 5(X^{2} − X + 1)(3X^{2} − 3X + 7)(3X^{2} − 2X + 7)

b_{1} −15(X^{6} − 12X^{4} − 10X^{3} + 12X + 9)

= −15(X − 1)(X + 3)(X^{2} + X + 1)(X^{2} − 3X − 3)

b_{2} −15(X^{6} + 10X^{3} + 12X^{2} − 7)

= −15(X^{2} + X − 1)(X^{4} − X^{3} + 2X^{2} + 7X + 7)

b_{3} −15(X^{6} − 24X^{4} − 10X^{3} + 84X^{2} − 60X + 9)

= −15(X − 1)(X + 3)(X^{2} + 3X − 3)(X^{2} − 5X + 1)

b_{4} −15(9X^{6} − 24X^{5} + 72X^{4} − 98X^{3} + 132X^{2} − 84X + 49)

= −15(X^{2} − X + 1)(3X^{2} − 3X + 7)(3X^{2} − 2X + 7)

1600 a_{1} 10(X^{5} + 3X^{3} − 4X)

= 10X(X − 1)(X + 1)(X^{2} + 4)

a_{2} 15(2X^{6} + 24X^{5} + 117X^{4} + 301X^{3} + 441X^{2} + 360X + 130)

= 15(X^{2} + 2X + 2)(X^{2} + 5X + 5)(2X^{2} + 10X + 13)

a_{3} 10(2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3)

= 10(X − 1)(X − 3)(2X − 1)(X^{2} − 3X + 1)

a_{4} 30(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= 30(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

a_{5} −30(X^{6} + 12X^{5} + 27X^{4} − 34X^{3} − 72X^{2} + 72X − 16)

= −30(X^{2} + 2X − 1)(X^{2} + 2X − 4)(X^{2} + 8X − 4)

1602 a_{1} −3(4X^{6} + 4X^{5} − 3X^{4} − 24X^{3} + 37X^{2} − 20X + 4)

N C F_{6}
1675 a_{1} 5(X^{6} + 4X^{5} − 48X^{2} + 20X − 4)
1683 a_{1} −3(9X^{6} − 6X^{5} + 19X^{4} + 4X^{3} + 7X^{2} + 6X + 5)
1713 a_{1} 5X^{6} − 18X^{5} − 5X^{4} + 20X^{3} + 55X^{2} + 42X + 9
1725 a_{1} 5(X^{6} + 12X^{5} + 32X^{4} − 24X^{3} + 8X^{2} + 12X + 4)
1743 a_{1} X^{6} − 68X^{4} + 66X^{3} − 56X^{2} − 12X + 9
= (X^{2} − X + 1)(X^{4} + X^{3} − 68X^{2} − 3X + 9)
a_{2} X^{6} − 80X^{4} − 66X^{3} + 364X^{2} − 228X + 9
= (X − 1)(X + 3)(X^{4} − 2X^{3} − 73X^{2} + 74X − 3)
1764 a_{1} −6(3X^{6} + 26X^{5} + 69X^{4} + 108X^{3} + 33X^{2} − 50X − 18)
= −6(X^{2} + 6X + 2)(3X^{4} + 8X^{3} + 15X^{2} + 2X − 9)
1773 a_{1} −3(81X^{6} − 144X^{5} − 100X^{4} + 200X^{3} + 80X^{2} − 84X − 36)
1815 a_{1} −11(X^{6} + 6X^{5} + 11X^{4} + 14X^{3} + 5X^{2} − 12X)
= −11X(X + 4)(X^{2} + X − 1)(X^{2} + X + 3)
a_{2} −11(X^{6} + 8X^{4} − 10X^{3} + 20X^{2} − 12X + 9)
= −11(X^{2} − X + 1)(X^{4} + X^{3} + 8X^{2} − 3X + 9)
a_{3} 33(5X^{6} − 18X^{5} + 15X^{4} + 54X^{3} − 135X^{2} + 72X − 20)
= 33(X − 2)(X + 2)(X^{2} − 3X + 5)(5X^{2} − 3X + 1)
a_{4} −11(X^{6} − 12X^{5} + 32X^{4} + 34X^{3} − 52X^{2} − 60X − 15)
= −11(X^{2} − X − 1)(X^{2} − 5X − 5)(X^{2} − 6X − 3)
1827 a_{1} −3(X^{6} + 16X^{4} − 18X^{3} + 28X^{2} − 12X + 9)
= −3(X^{2} − X + 1)(X^{4} + X^{3} + 16X^{2} − 3X + 9)
a_{2} −3(X^{6} + 4X^{4} + 18X^{3} − 56X^{2} + 24X + 9)
= −3(X − 1)(X + 3)(X^{4} − 2X^{3} + 11X^{2} − 10X − 3)
1849 a_{1} −43(−3X^{6} − 2X^{5} + 7X^{4} − 4X^{3} − 13X^{2} + 10X − 7)
a_{2} 129(X^{6} + 8X^{3} − 36X^{2} + 12X − 4)

N C F_{6}
1881 a_{1} −3(X^{6} + 4X^{5} + 8X^{4} + 8X^{3} + 8X^{2} − 4X + 4)
1885 a_{1} 5X^{6} − 12X^{5} − 40X^{4} + 6X^{3} + 20X^{2} − 12X + 5

= (X^{2} + X − 1)(X^{2} − 3X − 5)(5X^{2} − 2X + 1)

a_{2} 5X^{6} + 6X^{5} + 35X^{4} + 18X^{3} + 65X^{2} − 24X + 20

= (X^{2} + 4)(5X^{4} + 6X^{3} + 15X^{2} − 6X + 5)

a_{3} −27X^{6} − 162X^{5} − 261X^{4} + 106X^{3} + 729X^{2} − 432X + 108

= −(3X^{2} + 8X − 12)(9X^{4} + 30X^{3} + 43X^{2} − 30X + 9)

a_{4} −3(X^{6} + 18X^{5} + 27X^{4} − 126X^{3} − 147X^{2} + 108X − 16)

= −3(X^{2} + 3X − 1)(X^{4} + 15X^{3} − 17X^{2} − 60X + 16)

b_{1} 21X^{6} + 32X^{5} − 92X^{4} + 14X^{3} + 32X^{2} − 4X − 3

= (X − 1)(X + 3)(3X^{2} − X − 1)(7X^{2} − X − 1)

b_{2} −3(X^{6} + 6X^{5} − 81X^{4} + 106X^{3} + 189X^{2} − 288X + 92)

= −3(X^{2} + X − 1)(X^{4} + 5X^{3} − 85X^{2} + 196X − 92)

b_{3} 3(7X^{6} + 78X^{5} + 297X^{4} + 562X^{3} + 315X^{2} − 36X − 40)

= 3(X^{2} + 4X + 8)(X^{2} + 7X + 5)(7X^{2} + X − 1)

b_{4} 7(3X^{6} + 94X^{5} + 789X^{4} + 1458X^{3} + 591X^{2} − 268X − 144)

= 7(3X^{2} + X − 1)(X^{2} + 12X + 16)(X^{2} + 19X + 9)

N C F_{6}
1911 a_{1} −7(X^{6} − 8X^{4} + 6X^{3} + 4X^{2} − 12X + 9)

= −7(X − 1)(X + 3)(X^{2} − X + 1)(X^{2} − X − 3)

a_{2} −7(X^{6} + 4X^{4} − 6X^{3} + 16X^{2} − 12X + 9)

= −7(X^{2} − X + 1)(X^{4} + X^{3} + 4X^{2} − 3X + 9)

a_{3} −7(X^{6} − 20X^{4} − 6X^{3} + 64X^{2} − 48X + 9)

= −7(X − 1)(X + 3)(X^{2} + 3X − 1)(X^{2} − 5X + 3)

a_{4} 35(3X^{6} − 6X^{5} − 23X^{4} − 126X^{3} − 281X^{2} − 204X − 56)

= 35(X^{2} + X + 7)(X^{2} − 4X − 8)(3X^{2} + 3X + 1)

b_{1} −7(5X^{6} + 6X^{5} − 25X^{4} − 30X^{3} + 5X^{2} − 24X − 36)

= −7(X + 2)(5X + 6)(X^{2} − X + 1)(X^{2} − X − 3)

b_{2} −35(9X^{6} + 66X^{5} + 127X^{4} − 58X^{3} − 251X^{2} + 204X − 48)

= −35(X^{2} + 3X − 3)(9X^{4} + 39X^{3} + 37X^{2} − 52X + 16)

b_{3} 35(9X^{6} + 102X^{5} + 211X^{4} − 298X^{3} − 563X^{2} + 528X − 108)

= 35(X^{2} + X − 3)(3X^{2} + 8X − 4)(3X^{2} + 23X − 9)

b_{4} 7(3X^{6} − 6X^{5} + 25X^{4} + 10X^{3} + 55X^{2} + 24X + 36)

= 7(X^{2} + X + 1)(X^{2} − 3X + 9)(3X^{2} + 4)

c_{1} 13X^{6} − 78X^{5} + 119X^{4} + 84X^{3} − 301X^{2} + 126X + 49
d_{1} −7(13X^{6} − 78X^{5} + 119X^{4} + 84X^{3} − 301X^{2} + 126X + 49)
1925 a_{1} −15(X^{6} + 6X^{5} + 99X^{4} − 74X^{3} − 171X^{2} + 252X − 88)

= −15(X^{2} + X − 1)(X^{4} + 5X^{3} + 95X^{2} − 164X + 88)

a_{2} −5(19X^{6} + 86X^{5} + 57X^{4} − 258X^{3} − 513X^{2} − 344X − 76)

= −5(X − 2)(X + 2)(19X^{4} + 86X^{3} + 133X^{2} + 86X + 19)

1935 a_{1} −3(X^{6} + 8X^{4} + 20X^{2} − 12X + 4)
1983 a_{1} X^{6} + 6X^{5} − 25X^{4} + 42X^{3} − 139X^{2} + 60X − 8

A.2 Discriminants and Galois Groups

The table below gives, for each curve in the table in Section A.1, the discriminant of F_{6} w.r.t. X (computed as a sextic) and a permutation group S on 6 letters with the following property: there exists a labelling x_{0}, …, x_{5} of the zeros of F_{6} such that S is compatibly isomorphic to the Galois group of F_{6} over ℚ and that, defining P_{i}: = (x_{i}, 0), (3.11) holds. Note that, if F_{6} is a quintic, the discriminant of F_{6} w.r.t. X (as a quintic) is not always the discriminant of F_{6} w.r.t. X (as a sextic); more precisely, the latter is the former multiplied by the square of the leading coefficient of F_{6}. For example, the discriminant of

F_{6}(X): = 2X^{5} − 15X^{4} + 39X^{3} − 42X^{2} + 19X − 3

(corresponding to the curve 160a_{3}) w.r.t. X is 2^{2} · 5^{3} (as a quintic) and 2^{4} · 5^{3} (as a sextic). In our table, the latter discriminant is always used.

The permutation groups are labelled as follows:

Label  Permutation group
C_{2}^{A}  ⟨(25)⟩ ≅ C_{2}
C_{2}^{B}  ⟨(12)(45)⟩ ≅ C_{2}
C_{2}^{C}  ⟨(03)(12)(45)⟩ ≅ C_{2}
V_{4}^{A}  ⟨(14), (25)⟩ ≅ V_{4}
V_{4}^{B}  ⟨(03), (12)(45)⟩ ≅ V_{4}
V_{4}^{C}  ⟨(14)(25), (12)(45)⟩ ≅ V_{4}
V_{4}^{D}  ⟨(14)(25), (03)(12)(45)⟩ ≅ V_{4}
V_{4}^{E}  ⟨(12)(45), (03)(14)(25)⟩ ≅ V_{4}
S_{3}  ⟨(01)(34), (012)(345)⟩ ≅ S_{3}
C_{2}^{3A}  ⟨(03), (14), (25)⟩ ≅ C_{2}^{3}
C_{2}^{3B} ⟨(03), (12)(45), (14)(25)⟩ ≅ C_{2}^{3}
D_{8}^{A}  ⟨(1245), (14)⟩ ≅ D_{8}
D_{8}^{B}  ⟨(03)(1245), (14)⟩ ≅ D_{8}
D_{12}  ⟨(15)(24), (012345)⟩ ≅ D_{12}
D_{8} × C_{2} ⟨(03), (1245), (14) ≅ D_{8} × C_{2}
S_{4}^{+}  ⟨(05)(23), (012)(345)⟩ ≅ S_{4}
S_{4}^{−}  ⟨(0134), (02)(35)⟩ ≅ S_{4}
S_{4} × C_{2} ⟨(0134), (02)(35)⟩ ≅ S_{4} × C_{2}

Note that S_{4}^{+} is a subgroup of A_{6}, and S_{4}^{−} is not. Whenever ker ε is cannot be determined just by considering S, extra information is given so that ker ε can be determined, as follows.

If every element in S sends the set of letters {1, 2} to the set of letters {1, 2} and the set of letters {4, 5} to the set of letters {4, 5} (which implies that it sends the set of letters {0, 3} to the set of letters {0, 3}), then the roots of F_{6} which are not x_{0}, x_{3} all lie in one quadratic extension of ℚ, resulting in 6 possibilities for the set {x_{1}, x_{4}}, 6 possibilities for the set {x_{2}, x_{5}}, and so 3 possibilities for ker ε (only one of which, of course, is correct). The extra information gives a possibility for the set {x_{1}, x_{4}} which corresponds to ker ε. For example, consider the curve 39a_{3}. We have

\{x_{1}, x_{2}, x_{4}, x_{5}\} = \{\frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(−3 − \sqrt{13}), \frac{1}{2}(5 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})\} .

Taking

x_{1} = \frac{1}{2}(−3 + \sqrt{13}), x_{4} = \frac{1}{2}(5 − \sqrt{13})

(the first, second entries in the extra information) gives ker ε.

If F_{6} has a quartic factor F_{4} which is irreducible over ℚ and whose Galois group over ℚ is isomorphic to V_{4}, then there exist 3 quadratic extensions of ℚ over which F_{6} factors into the product of two quadratics, resulting in 6 possibilities for the set {x_{1}, x_{4}}, 6 possibilities for the set {x_{2}, x_{5}}, and so 3 possibilities for ker ε (only one of which, of course, is correct). The extra information gives a possibility for the set {x_{1}, x_{4}} which corresponds to ker ε (the field of definition of (X − x_{1})(X − x_{4})). For example, consider the curve 39a_{2}. We have

F_{4} = X^{4} + X^{3} + 4X^{2} − 3X + 9
= \frac{1}{4} ∏_{s∈\{±1\}}(2X^{2} + X(1 + s\sqrt{−3}) + 3(1 − s\sqrt{−3}))

= \frac{1}{4} ∏_{s∈\{±1\}}(2X^{2} + X(1 − s\sqrt{13}) + 7 − s\sqrt{13})

= \frac{1}{4} ∏_{s∈\{±1\}}(2X^{2} + X(1 + s\sqrt{−39}) − 6).

Choosing x_{1}, x_{4} so that

(X − x_{1})(X − x_{4}) = \frac{1}{2}(2X^{2} + X(1 + \sqrt{−39}) − 6)

(the field of definition of (X −x_{1})(X −x_{4}) is the entry in the extra information) gives ker ε.

Curve Discriminant of F_{6}  Galois Group of F_{6}
29a_{1}  2^{12} · 29^{5}  D_{12}
29a_{2}  −2^{12} · 29^{2}  D_{12}
39a_{1}  −2^{12} · 3^{7} · 13^{3}  V_{4}^{A}
39a_{2}  2^{12} · 3^{5} · 13^{2}  V_{4}^{E}; ℚ(\sqrt{−39})
39a_{3}  2^{12} · 3^{10} · 13^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
39a_{4} 2^{12} · 3^{5} · 5^{22} · 13^{4} V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})
43a_{1}  −2^{12} · 43^{5}  S_{4}^{−}
43a_{2}  2^{12} · 3^{22} · 43^{2}  S_{4}^{+}
55a_{1}  −2^{12} · 5^{7} · 11^{3}  V_{4}^{A}
55a_{2} 2^{12} · 3^{22} · 5^{5} · 11^{2}  V_{4}^{E}; ℚ(\sqrt{−55})
55a_{3} −2^{12} · 5^{10} · 11^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{−11}), \frac{1}{6}(1 + \sqrt{−11})
55a_{4} 2^{12} · 3^{22} · 5^{5} · 11^{4}  C_{2}^{C}; \frac{1}{2}(−1 + \sqrt{5}), \frac{1}{2}(11 − 5\sqrt{5})
65a_{1} −2^{12} · 3^{22} · 5^{5} · 13^{2}  D_{8} × C_{2}
65a_{2}  2^{12} · 5^{7} · 13^{3}  D_{8}^{A}
82a_{1}  −2^{17} · 41^{7}  D_{8} × C_{2}
82a_{2}  2^{19} · 3^{22} · 41^{3}  D_{8} × C_{2}
82a_{3}  −2^{18} · 3^{22} · 41^{5}  D_{8}^{B}
82a_{4}  −2^{22} · 41^{2}  D_{8}^{A}
85a_{1}  −2^{12} · 5^{3} · 17^{2}  D_{8} × C_{2}
85a_{2}  −2^{12} · 5^{4} · 17^{3}  D_{8}^{B}
91a_{1}  −2^{12} · 7^{5} · 13^{2}  S_{4} × C_{2}
98a_{1}  2^{29} · 7^{5}  D_{8}^{A}
98a_{2}  2^{29} · 7^{15}  D_{8}^{A}
98a_{3}  −2^{17} · 3^{22} · 7^{5}  D_{8} × C_{2}
98a_{4}  −2^{17} · 3^{22} · 7^{15}  D_{8} × C_{2}
117a_{1}  −2^{12} · 3^{17} · 13^{3}  V_{4}^{A}
117a_{2}  2^{12} · 3^{15} · 13^{2}  V_{4}^{E}; ℚ(\sqrt{−39})
117a_{3}  2^{12} · 3^{20} · 13^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
117a_{4} 2^{12} · 3^{15} · 5^{22} · 13^{4} V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})

Curve Discriminant of F_{6}  Galois Group of F_{6}
129a_{1}  −2^{12} · 3^{5} · 43^{2}  D_{12}
145a_{1}  −2^{12} · 5^{7} · 29^{3}  C_{2}^{3A}
145a_{2} 2^{12} · 3^{22} · 5^{10} · 29^{2}  V_{4}^{C}; ℚ(\sqrt{−29})
145a_{3} 2^{12} · 5^{5} · 29^{2} V_{4}^{E}; ℚ(\sqrt{145})
145a_{4} −2^{12} · 3^{22} · 5^{5} · 29^{4}  V_{4}^{D}; ℚ(\sqrt{−5})
147a_{1}  −2^{12} · 3^{3} · 7^{5}  D_{8} × C_{2}
147a_{2}  −2^{12} · 3^{4} · 7^{5}  D_{8}^{A}
147b_{1}  −2^{12} · 3^{3} · 7^{15}  D_{8} × C_{2}
147b_{2}  −2^{12} · 3^{4} · 7^{15}  D_{8}^{A}
160a_{1}  −2^{10} · 5^{4}  C_{2}^{A}; 0, ∞
160a_{2}  2^{14} · 3^{22} · 5^{3}  V_{4}^{B}; −1 + \sqrt{−1}, \frac{1}{2}(−5 − \sqrt{−1})
160a_{3}  2^{4} · 5^{3}  C_{2}^{A}; 3, ∞
160a_{4}  2^{17} · 3^{22} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
160a_{5}  2^{17} · 3^{22} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
165a_{1} −2^{12} · 3^{4} · 5^{3} · 11^{3}  V_{4}^{A}
165a_{2} −2^{12} · 3^{3} · 5^{2} · 11^{2}  C_{2}^{3B}; ℚ(\sqrt{−55})
165a_{3} −2^{12} · 3^{19} · 5^{4} · 11^{2} C_{2}^{B}; \frac{1}{2}(3 + \sqrt{−11}), \frac{1}{10}(3 − \sqrt{−11})
165a_{4} 2^{12} · 3^{3} · 5^{2} · 11^{4}  V_{4}^{B}; \frac{1}{2}(1 + \sqrt{5}), \frac{1}{2}(5 + 3\sqrt{5})
178a_{1}  −2^{20} · 89^{2}  D_{12}
183a_{1}  −2^{12} · 3^{6} · 61^{2}  D_{12}
196a_{1}  −2^{16} · 7^{18}  D_{8}^{B}
207a_{1}  −2^{12} · 3^{6} · 23^{2}  D_{8} × C_{2}
207a_{2}  −2^{12} · 3^{6} · 23^{3}  D_{8}^{A}
207b_{1}  −2^{12} · 3^{16} · 23^{2}  D_{8} × C_{2}
207b_{2}  −2^{12} · 3^{16} · 23^{3}  D_{8}^{A}
209a_{1}  −2^{12} · 11^{2} · 19^{3}  S_{4} × C_{2}
223a_{1}  −2^{12} · 223^{2}  S_{4} × C_{2}
227a_{1}  2^{12} · 3^{12} · 227^{2}  S_{4}^{+}

Curve Discriminant of F_{6}  Galois Group of F_{6}
232a_{1}  −2^{8} · 29^{2}  D_{12}
237a_{1}  −2^{12} · 3^{11} · 79^{2}  D_{12}
245a_{1}  −2^{12} · 5^{2} · 7^{5}  S_{4} × C_{2}
245b_{1}  −2^{12} · 5^{2} · 7^{15}  S_{4} × C_{2}
245c_{1}  −2^{12} · 5^{2} · 7^{10}  D_{12}
245d_{1}  −2^{12} · 5^{2} · 7^{20}  D_{12}
249a_{1}  2^{12} · 3^{11} · 83^{2}  D_{12}
256a_{1}  −2^{18}  V_{4}^{A}
256a_{2}  2^{43}  V_{4}^{B}; 1 + \sqrt{−2}, \frac{1}{3}(−1 − \sqrt{−2})
261a_{1}  2^{12} · 3^{10} · 29^{5}  D_{12}
261a_{2}  −2^{12} · 3^{10} · 29^{2}  D_{12}
265a_{1}  −2^{12} · 5^{6} · 53^{2}  D_{12}
273a_{1} −2^{12} · 3^{7} · 7^{4} · 13^{3}  V_{4}^{A}
273a_{2} 2^{12} · 3^{5} · 5^{22} · 7^{3} · 13^{2}  C_{2}^{3B}; ℚ(\sqrt{−39})
273a_{3} 2^{12} · 3^{10} · 5^{22} · 7^{3} · 13^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{13}), \frac{1}{6}(−23 − 7\sqrt{13})
273a_{4} −2^{12} · 3^{5} · 7^{6} · 13^{4}  C_{2}^{C}; \frac{1}{2}(3 + 3\sqrt{−3}), \frac{1}{2}(−1 + \sqrt{−3})
275a_{1}  −2^{12} · 5^{17} · 11^{3}  V_{4}^{A}
275a_{2} 2^{12} · 3^{22} · 5^{15} · 11^{2}  V_{4}^{E}; ℚ(\sqrt{−55})
275a_{3} −2^{12} · 5^{20} · 11^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{−11}), \frac{1}{6}(1 + \sqrt{−11})
275a_{4} 2^{12} · 3^{22} · 5^{15} · 11^{4}  C_{2}^{C}; \frac{1}{2}(−1 + \sqrt{5}), \frac{1}{2}(11 − 5\sqrt{5})
278a_{1}  −2^{19} · 3^{22} · 139^{2}  S_{4} × C_{2}
285a_{1}  2^{12} · 3^{2} · 5^{5} · 19^{3}  D_{8} × C_{2}
285a_{2} −2^{12} · 3^{25} · 5^{7} · 19^{2}  D_{8}^{A}
285b_{1} −2^{12} · 3^{13} · 5^{2} · 19^{3}  D_{8} × C_{2}
285b_{2} −2^{12} · 3^{19} · 5^{3} · 19^{2}  D_{8}^{A}

Curve Discriminant of F_{6}  Galois Group of F_{6}
315a_{1} −2^{12} · 3^{6} · 5^{4} · 7^{3}  V_{4}^{A}
315a_{2}  2^{12} · 3^{6} · 5^{3} · 7^{2}  C_{2}^{3B}; ℚ(\sqrt{−7})
315a_{3}  2^{12} · 3^{6} · 5^{6} · 7^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{21}), \frac{1}{2}(5 − \sqrt{21})
315a_{4} −2^{12} · 3^{6} · 5^{3} · 7^{4} V_{4}^{B}; \frac{1}{2}(1 + \sqrt{−3}), \frac{1}{6}(3 + 5\sqrt{−3})
315b_{1} −2^{12} · 3^{16} · 5^{4} · 7^{3}  V_{4}^{A}
315b_{2}  2^{12} · 3^{16} · 5^{3} · 7^{2}  C_{2}^{3B}; ℚ(\sqrt{−7})
315b_{3}  2^{12} · 3^{16} · 5^{6} · 7^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{21}), \frac{1}{2}(5 − \sqrt{21})
315b_{4} −2^{12} · 3^{16} · 5^{3} · 7^{4} V_{4}^{B}; \frac{1}{2}(1 + \sqrt{−3}), \frac{1}{6}(3 + 5\sqrt{−3})
320a_{1}  −2^{20} · 5^{4}  C_{2}^{A}; 0, ∞
320a_{2}  2^{4} · 3^{22} · 5^{3}  V_{4}^{B}; −1 + \sqrt{−1}, \frac{1}{2}(−5 − \sqrt{−1})
320a_{3}  2^{14} · 5^{3}  C_{2}^{A}; 3, ∞
320a_{4}  2^{27} · 3^{22} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
320a_{5}  2^{27} · 3^{22} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
325a_{1} −2^{12} · 3^{22} · 5^{15} · 13^{2}  D_{8} × C_{2}
325a_{2} 2^{12} · 5^{17} · 13^{3} D_{8}^{A}
325b_{1} −2^{12} · 3^{22} · 5^{4} · 13^{2}  D_{12}
325c_{1} −2^{12} · 3^{22} · 5^{14} · 13^{2}  D_{12}
335a_{1}  2^{12} · 5^{9} · 67^{2}  S_{4} × C_{2}
339a_{1}  −2^{12} · 3^{5} · 113^{2}  S_{4} × C_{2}
345a_{1} −2^{12} · 3^{9} · 5^{2} · 23^{2}  S_{4} × C_{2}
357a_{1} −2^{12} · 3^{2} · 7^{3} · 17^{2}  S_{4} × C_{2}
369a_{1}  −2^{12} · 3^{15} · 41^{2}  S_{4} × C_{2}
385a_{1} 2^{12} · 3^{12} · 5^{3} · 7^{6} · 11^{3}  D_{8} × C_{2}
385a_{2} −2^{12} · 5^{4} · 7^{9} · 11^{2}  D_{8}^{A}
387a_{1}  −2^{12} · 3^{10} · 43^{5}  S_{4}^{−}
387a_{2}  2^{12} · 3^{12} · 43^{2}  S_{4}^{+}
387b_{1}  −2^{12} · 3^{15} · 43^{2}  D_{12}
389a_{1}  2^{12} · 389^{2}  S_{4}^{+}

Curve Discriminant of F_{6}  Galois Group of F_{6}
425a_{1}  −2^{12} · 5^{13} · 17^{2}  D_{8} × C_{2}
425a_{2}  −2^{12} · 5^{14} · 17^{3}  D_{8}^{B}
429a_{1} −2^{12} · 3^{13} · 11^{5} · 13^{2}  D_{8} × C_{2}
429a_{2} −2^{12} · 3^{9} · 11^{7} · 13^{3}  D_{8}^{B}
437a_{1}  −2^{12} · 19^{2} · 23^{3}  S_{4} × C_{2}
441a_{1}  −2^{12} · 3^{13} · 7^{5}  D_{8} × C_{2}
441a_{2}  −2^{12} · 3^{14} · 7^{5}  D_{8}^{A}
441b_{1}  −2^{12} · 3^{13} · 7^{15}  D_{8} × C_{2}
441b_{2}  −2^{12} · 3^{14} · 7^{15}  D_{8}^{A}
445a_{1} −2^{12} · 3^{22} · 5^{3} · 89^{2}  D_{8} × C_{2}
445a_{2}  2^{12} · 5^{4} · 89^{3}  D_{8}^{A}
465a_{1} −2^{12} · 3^{9} · 5^{2} · 31^{3}  D_{8} × C_{2}
465a_{2} −2^{12} · 3^{13} · 5^{3} · 31^{2}  D_{8}^{A}
495a_{1} −2^{12} · 3^{10} · 5^{7} · 11^{3}  V_{4}^{A}
495a_{2} 2^{12} · 3^{12} · 5^{5} · 11^{2}  V_{4}^{E}; ℚ(\sqrt{−55})
495a_{3} −2^{12} · 3^{10} · 5^{10} · 11^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{−11}), \frac{1}{6}(1 + \sqrt{−11})
495a_{4} 2^{12} · 3^{12} · 5^{5} · 11^{4}  C_{2}^{C}; \frac{1}{2}(−1 + \sqrt{5}), \frac{1}{2}(11 − 5\sqrt{5})
495b_{1} −2^{12} · 3^{14} · 5^{3} · 11^{3}  V_{4}^{A}
495b_{2} −2^{12} · 3^{13} · 5^{2} · 11^{2}  C_{2}^{3B}; ℚ(\sqrt{−55})
495b_{3} −2^{12} · 3^{9} · 5^{4} · 11^{2} C_{2}^{B}; \frac{1}{2}(3 + \sqrt{−11}), \frac{1}{10}(3 − \sqrt{−11})
495b_{4} 2^{12} · 3^{13} · 5^{2} · 11^{4}  V_{4}^{B}; \frac{1}{2}(1 + \sqrt{5}), \frac{1}{2}(5 + 3\sqrt{5})
507a_{1}  −2^{12} · 3^{7} · 13^{13}  V_{4}^{A}
507a_{2}  2^{12} · 3^{5} · 13^{12}  V_{4}^{E}; ℚ(\sqrt{−39})
507a_{3}  2^{12} · 3^{10} · 13^{12}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
507a_{4} 2^{12} · 3^{5} · 5^{22} · 13^{14} V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})
517a_{1}  2^{12} · 11^{5} · 47^{2}  S_{4} × C_{2}
529a_{1}  2^{12} · 3^{22} · 23^{4}  S_{4}^{+}
529a_{2}  −2^{12} · 23^{20}  S_{4}^{−}
529b_{1}  2^{12} · 3^{22} · 23^{14}  S_{4}^{+}
529b_{2}  −2^{12} · 23^{10}  S_{4}^{−}

Curve Discriminant of F_{6}  Galois Group of F_{6}
545a_{1} −2^{12} · 5^{9} · 109^{2} D_{8} × C_{2}
545a_{2} −2^{12} · 5^{13} · 109^{3}  D_{8}^{B}
547a_{1} −2^{12} · 5^{22} · 547^{2}  S_{4} × C_{2}
549a_{1}  −2^{12} · 3^{16} · 61^{2}  D_{12}
561a_{1} −2^{12} · 3^{9} · 11^{2} · 17^{2}  S_{4} × C_{2}
583a_{1}  2^{12} · 11^{2} · 53^{3}  S_{4} × C_{2}
585a_{1} −2^{12} · 3^{12} · 5^{5} · 13^{2}  D_{8} × C_{2}
585a_{2}  2^{12} · 5^{7} · 13^{3}  D_{8}^{A}
591a_{1} −2^{12} · 3^{11} · 197^{2}  S_{4} × C_{2}
605a_{1}  −2^{12} · 5^{7} · 11^{13}  V_{4}^{A}
605a_{2} 2^{12} · 3^{22} · 5^{5} · 11^{12}  V_{4}^{E}; ℚ(\sqrt{−55})
605a_{3} −2^{12} · 5^{10} · 11^{12} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{−11}), \frac{1}{6}(1 + \sqrt{−11})
605a_{4} 2^{12} · 3^{22} · 5^{5} · 11^{14}  C_{2}^{C}; \frac{1}{2}(−1 + \sqrt{5}), \frac{1}{2}(11 − 5\sqrt{5})
609a_{1} −2^{12} · 3^{5} · 7^{3} · 29^{2}  D_{8} × C_{2}
609a_{2} −2^{12} · 3^{7} · 7^{2} · 29^{3}  D_{8}^{A}
621a_{1}  2^{12} · 3^{6} · 23^{2}  S_{3}
621b_{1}  2^{12} · 3^{16} · 23^{2}  S_{3}
621c_{1}  −2^{12} · 3^{10} · 23^{3}  S_{4} × C_{2}
621d_{1}  −2^{12} · 3^{20} · 23^{3}  S_{4} × C_{2}
637a_{1}  −2^{12} · 7^{15} · 13^{2}  S_{4} × C_{2}
639a_{1}  −2^{12} · 3^{6} · 71^{2}  D_{8} × C_{2}
639a_{2}  −2^{12} · 3^{6} · 71^{3}  D_{8}^{A}
639b_{1}  −2^{12} · 3^{16} · 71^{2}  D_{8} × C_{2}
639b_{2}  −2^{12} · 3^{16} · 71^{3}  D_{8}^{A}
645a_{1} −2^{12} · 3^{3} · 5^{3} · 43^{2}  S_{4} × C_{2}

Curve Discriminant of F_{6}  Galois Group of F_{6}
675a_{1}  2^{12} · 3^{6} · 5^{6}  S_{4}^{+}
675a_{2}  2^{12} · 3^{16} · 5^{6}  S_{4}^{+}
675b_{1}  2^{12} · 3^{6} · 5^{16}  S_{4}^{+}
675b_{2}  2^{12} · 3^{16} · 5^{16}  S_{4}^{+}
703a_{1}  −2^{12} · 19^{3} · 37^{2}  S_{4} × C_{2}
705a_{1} −2^{12} · 3^{14} · 5^{5} · 47^{2}  D_{12}
711a_{1}  −2^{12} · 3^{21} · 79^{2}  D_{12}
715a_{1} −2^{12} · 5^{5} · 11^{5} · 13^{2}  S_{4} × C_{2}
725a_{1}  2^{12} · 5^{10} · 29^{5}  D_{12}
725a_{2}  −2^{12} · 5^{10} · 29^{2}  D_{12}
725b_{1}  −2^{12} · 5^{17} · 29^{3}  C_{2}^{3A}
725b_{2} 2^{12} · 3^{22} · 5^{20} · 29^{2}  V_{4}^{C}; ℚ(\sqrt{−29})
725b_{3} 2^{12} · 5^{15} · 29^{2} V_{4}^{E}; ℚ(\sqrt{145})
725b_{4} −2^{12} · 3^{22} · 5^{15} · 29^{4}  V_{4}^{D}; ℚ(\sqrt{−5})
735a_{1} −2^{12} · 3^{7} · 5^{3} · 7^{10}  V_{4}^{A}
735a_{2} −2^{12} · 3^{5} · 5^{2} · 7^{20}  C_{2}^{3B}; ℚ(\sqrt{−15})
735a_{3} 2^{12} · 3^{5} · 5^{4} · 7^{31}  C_{2}^{C}; \frac{1}{6}(9 + \sqrt{21}), \frac{1}{10}(−13 − 3\sqrt{21})
735a_{4} 2^{12} · 3^{10} · 5^{2} · 7^{31} V_{4}^{B}; \frac{1}{2}(3 + \sqrt{−35}), \frac{1}{26}(11 − \sqrt{−35})
735b_{1} −2^{12} · 3^{7} · 5^{3} · 7^{20}  V_{4}^{A}
735b_{2} −2^{12} · 3^{5} · 5^{2} · 7^{10}  C_{2}^{3B}; ℚ(\sqrt{−15})
735b_{3} 2^{12} · 3^{5} · 5^{4} · 7^{21}  C_{2}^{C}; \frac{1}{6}(9 + \sqrt{21}), \frac{1}{10}(−13 − 3\sqrt{21})
735b_{4} 2^{12} · 3^{10} · 5^{2} · 7^{21} V_{4}^{B}; \frac{1}{2}(3 + \sqrt{−35}), \frac{1}{26}(11 − \sqrt{−35})
735c_{1} 2^{12} · 3^{2} · 5^{3} · 7^{10}  S_{4} × C_{2}
735d_{1} 2^{12} · 3^{2} · 5^{3} · 7^{20}  S_{4} × C_{2}
737a_{1}  −2^{12} · 11^{3} · 67^{2}  S_{4} × C_{2}
738a_{1}  −2^{17} · 3^{10} · 41^{7}  D_{8} × C_{2}
738a_{2}  2^{19} · 3^{12} · 41^{3}  D_{8} × C_{2}
738a_{3}  −2^{18} · 3^{12} · 41^{5}  D_{8}^{B}
738a_{4}  −2^{22} · 3^{10} · 41^{2}  D_{8}^{A}
741a_{1} 2^{12} · 3^{7} · 13^{3} · 19^{5}  S_{4} × C_{2}

Curve Discriminant of F_{6}  Galois Group of F_{6}
747a_{1}  2^{12} · 3^{21} · 83^{2}  D_{12}
753a_{1}  −2^{12} · 3^{19} · 251^{2}  S_{4} × C_{2}
765a_{1} −2^{12} · 3^{10} · 5^{3} · 17^{2}  D_{8} × C_{2}
765a_{2} −2^{12} · 3^{10} · 5^{4} · 17^{3}  D_{8}^{B}
767a_{1}  −2^{12} · 13^{3} · 59^{2}  S_{4} × C_{2}
777a_{1} −2^{12} · 3^{7} · 5^{10} · 7^{4} · 37^{3}  V_{4}^{A}
777a_{2} −2^{12} · 3^{5} · 7^{6} · 37^{2} V_{4}^{E}; ℚ(\sqrt{−111})
777a_{3} 2^{12} · 3^{5} · 7^{3} · 11^{22} · 37^{4} V_{4}^{B}; \frac{1}{2}(7 + 3\sqrt{−3}), \frac{1}{14}(5 − \sqrt{−3})
777a_{4} 2^{12} · 3^{10} · 5^{22} · 7^{3} · 37^{2} V_{4}^{B}; \frac{1}{6}(−7 + \sqrt{37}), \frac{1}{14}(17 − \sqrt{37})
779a_{1}  2^{12} · 19^{2} · 41^{3}  S_{4} × C_{2}
800a_{1}  −2^{10} · 5^{14}  C_{2}^{A}; 0, ∞
800a_{2}  2^{14} · 3^{22} · 5^{13}  V_{4}^{B}; −1 + \sqrt{−1}, \frac{1}{2}(−5 − \sqrt{−1})
800a_{3}  2^{4} · 5^{13}  C_{2}^{A}; 3, ∞
800a_{4}  2^{17} · 3^{22} · 5^{12}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
800a_{5}  2^{17} · 3^{22} · 5^{12}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
819a_{1} −2^{12} · 3^{10} · 7^{5} · 13^{2}  S_{4} × C_{2}
819b_{1} −2^{12} · 3^{17} · 7^{4} · 13^{3}  V_{4}^{A}
819b_{2} 2^{12} · 3^{15} · 5^{22} · 7^{3} · 13^{2}  C_{2}^{3B}; ℚ(\sqrt{−39})
819b_{3} 2^{12} · 3^{20} · 5^{22} · 7^{3} · 13^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{13}), \frac{1}{6}(−23 − 7\sqrt{13})
819b_{4} −2^{12} · 3^{15} · 7^{6} · 13^{4}  C_{2}^{C}; \frac{1}{2}(3 + 3\sqrt{−3}), \frac{1}{2}(−1 + \sqrt{−3})
825a_{1} −2^{12} · 3^{4} · 5^{13} · 11^{3}  V_{4}^{A}
825a_{2} −2^{12} · 3^{3} · 5^{12} · 11^{2}  C_{2}^{3B}; ℚ(\sqrt{−55})
825a_{3} −2^{12} · 3^{19} · 5^{14} · 11^{2} C_{2}^{B}; \frac{1}{2}(3 + \sqrt{−11}), \frac{1}{10}(3 − \sqrt{−11})
825a_{4}  2^{12} · 3^{3} · 5^{12} · 11^{4}  V_{4}^{B}; \frac{1}{2}(1 + \sqrt{5}), \frac{1}{2}(5 + 3\sqrt{5})
837a_{1}  2^{12} · 3^{6} · 31^{2}  S_{3}
837b_{1}  2^{12} · 3^{16} · 31^{2}  S_{3}
841a_{1}  2^{12} · 29^{15}  D_{12}
841a_{2}  −2^{12} · 29^{12}  D_{12}

Curve  Discriminant of F_{6}  Galois Group of F_{6}
845a_{1}  −2^{12} · 3^{22} · 5^{15} · 13^{2}  D_{8} × C_{2}
845a_{2}  2^{12} · 5^{7} · 13^{13}  D_{8}^{A}
855a_{1}  2^{12} · 3^{12} · 5^{5} · 19^{3}  D_{8} × C_{2}
855a_{2}  −2^{12} · 3^{15} · 5^{7} · 19^{2}  D_{8}^{A}
855b_{1}  −2^{12} · 3^{23} · 5^{2} · 19^{3}  D_{8} × C_{2}
855b_{2}  −2^{12} · 3^{29} · 5^{3} · 19^{2}  D_{8}^{A}
882a_{1}  2^{29} · 3^{10} · 7^{5}  D_{8}^{A}
882a_{2}  2^{29} · 3^{10} · 7^{15}  D_{8}^{A}
882a_{3}  −2^{17} · 3^{12} · 7^{5}  D_{8} × C_{2}
882a_{4}  −2^{17} · 3^{12} · 7^{15}  D_{8} × C_{2}
885a_{1}  −2^{12} · 3^{11} · 5^{6} · 59^{2}  S_{4} × C_{2}
909a_{1}  −2^{12} · 3^{13} · 101^{2}  S_{4} × C_{2}
931a_{1}  −2^{12} · 7^{10} · 19^{2}  S_{4} × C_{2}
931b_{1}  −2^{12} · 7^{20} · 19^{2}  S_{4} × C_{2}
975a_{1}  −2^{12} · 3^{7} · 5^{10} · 13^{3}  V_{4}^{A}
975a_{2}  2^{12} · 3^{5} · 5^{10} · 13^{2}  V_{4}^{E}; ℚ(\sqrt{−39})
975a_{3}  2^{12} · 3^{10} · 5^{10} · 13^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
975a_{4}  2^{12} · 3^{5} · 5^{12} · 13^{4}  V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})
1017a_{1}  −2^{12} · 3^{15} · 113^{2}  S_{4} × C_{2}
1035a_{1}  −2^{12} · 3^{19} · 5^{2} · 23^{2}  S_{4} × C_{2}
1035b_{1}  2^{12} · 3^{6} · 5^{5} · 23^{2}  S_{4} × C_{2}
1035c_{1}  2^{12} · 3^{16} · 5^{5} · 23^{2}  S_{4} × C_{2}
1045a_{1} −2^{12} · 3^{22} · 5^{7} · 11^{4} · 19^{3}  V_{4}^{A}
1045a_{2} 2^{12} · 5^{5} · 11^{6} · 19^{2} V_{4}^{E}; ℚ(\sqrt{−95})
1045a_{3} 2^{12} · 3^{22} · 5^{5} · 7^{22} · 11^{3} · 19^{4}  V_{4}^{B}; \frac{1}{2}(−7 + 3\sqrt{5}), \frac{1}{6}(7 − 5\sqrt{5})
1045a_{4} −2^{12} · 3^{22} · 5^{10} · 11^{3} · 19^{2} V_{4}^{B}; \frac{1}{10}(3\sqrt{−19} − 3), \frac{1}{14}(3\sqrt{−19} − 5)
1071a_{1}  −2^{12} · 3^{12} · 7^{3} · 17^{2}  S_{4} × C_{2}
1075a_{1}  −2^{12} · 5^{10} · 43^{5}  S_{4}^{−}
1075a_{2}  2^{12} · 3^{22} · 5^{10} · 43^{2}  S_{4}^{+}

Curve Discriminant of F_{6}  Galois Group of F_{6}
1165a_{1} −2^{12} · 3^{22} · 5^{3} · 233^{2}  D_{8} × C_{2}
1165a_{2}  2^{12} · 5^{4} · 233^{3}  D_{8}^{A}
1183a_{1}  −2^{12} · 7^{5} · 13^{12}  S_{4} × C_{2}
1185a_{1}  −2^{12} · 3^{5} · 5^{2} · 79^{3}  D_{8} × C_{2}
1185a_{2}  −2^{12} · 3^{7} · 5^{3} · 79^{2}  D_{8}^{A}
1197a_{1} −2^{12} · 3^{6} · 5^{12} · 7^{3} · 19^{4}  V_{4}^{A}
1197a_{2} −2^{12} · 3^{18} · 5^{22} · 7^{2} · 19^{3}  C_{2}^{3B}; ℚ(\sqrt{−7})
1197a_{3} 2^{12} · 3^{6} · 5^{12} · 7^{2} · 19^{3} V_{4}^{B}; \frac{1}{2}(\sqrt{21} − 5), \frac{1}{10}(19\sqrt{21} − 89)
1197a_{4} 2^{12} · 3^{6} · 7^{4} · 19^{6} C_{2}^{B}; \frac{1}{2}(−1 + \sqrt{−3}), \frac{1}{26}(7 − \sqrt{−3})
1197b_{1} −2^{12} · 3^{16} · 5^{12} · 7^{3} · 19^{4}  V_{4}^{A}
1197b_{2} −2^{12} · 3^{28} · 5^{22} · 7^{2} · 19^{3}  C_{2}^{3B}; ℚ(\sqrt{−7})
1197b_{3} 2^{12} · 3^{16} · 5^{12} · 7^{2} · 19^{3} V_{4}^{B}; \frac{1}{2}(\sqrt{21} − 5), \frac{1}{10}(19\sqrt{21} − 89)
1197b_{4}  2^{12} · 3^{16} · 7^{4} · 19^{6}  C_{2}^{B}; \frac{1}{2}(−1 + \sqrt{−3}), \frac{1}{26}(7 − \sqrt{−3})
1203a_{1}  −2^{12} · 3^{14} · 401^{2}  S_{4} × C_{2}
1209a_{1} −2^{12} · 3^{9} · 13^{5} · 31^{2}  S_{4} × C_{2}
1225a_{1}  −2^{12} · 5^{12} · 7^{5}  S_{4} × C_{2}
1225b_{1}  −2^{12} · 5^{12} · 7^{15}  S_{4} × C_{2}
1225c_{1}  −2^{12} · 5^{12} · 7^{10}  D_{12}
1225d_{1}  −2^{12} · 5^{12} · 7^{20}  D_{12}
1275a_{1}  −2^{12} · 3^{17} · 5^{4} · 17^{3}  S_{4} × C_{2}
1275b_{1} −2^{12} · 3^{17} · 5^{14} · 17^{3}  S_{4} × C_{2}
1287a_{1} −2^{12} · 3^{23} · 11^{5} · 13^{2}  D_{8} × C_{2}
1287a_{2} −2^{12} · 3^{19} · 11^{7} · 13^{3}  D_{8}^{B}
1295a_{1}  2^{12} · 5^{5} · 7^{2} · 37^{3}  S_{4} × C_{2}
1305a_{1}  −2^{12} · 3^{10} · 5^{7} · 29^{3}  C_{2}^{3A}
1305a_{2}  2^{12} · 3^{12} · 5^{10} · 29^{2}  V_{4}^{C}; ℚ(\sqrt{−29})
1305a_{3}  2^{12} · 3^{10} · 5^{5} · 29^{2}  V_{4}^{E}; ℚ(\sqrt{145})
1305a_{4}  −2^{12} · 3^{12} · 5^{5} · 29^{4}  V_{4}^{D}; ℚ(\sqrt{−5})

Curve Discriminant of F_{6}  Galois Group of F_{6}
1325a_{1}  −2^{12} · 5^{16} · 53^{2}  D_{12}
1325b_{1}  2^{12} · 3^{22} · 5^{4} · 53^{2}  S_{4}^{+}
1325c_{1}  2^{12} · 3^{22} · 5^{14} · 53^{2}  S_{4}^{+}
1395a_{1}  −2^{12} · 3^{19} · 5^{2} · 31^{3}  D_{8} × C_{2}
1395a_{2}  −2^{12} · 3^{23} · 5^{3} · 31^{2}  D_{8}^{A}
1409a_{1}  −2^{12} · 3^{22} · 1409^{2}  S_{4} × C_{2}
1421a_{1}  2^{12} · 7^{10} · 29^{5}  D_{12}
1421a_{2}  −2^{12} · 7^{10} · 29^{2}  D_{12}
1425a_{1}  2^{12} · 3^{2} · 5^{15} · 19^{3}  D_{8} × C_{2}
1425a_{2} −2^{12} · 3^{25} · 5^{17} · 19^{2}  D_{8}^{A}
1425b_{1} −2^{12} · 3^{13} · 5^{12} · 19^{3}  D_{8} × C_{2}
1425b_{2} −2^{12} · 3^{19} · 5^{13} · 19^{2}  D_{8}^{A}
1433a_{1}  2^{12} · 3^{24} · 1433^{2}  S_{4}^{+}
1440a_{1}  −2^{10} · 3^{10} · 5^{4}  C_{2}^{A}; 0, ∞
1440a_{2}  2^{14} · 3^{12} · 5^{3}  V_{4}^{B}; −1 + \sqrt{−1}, \frac{1}{2}(−5 − \sqrt{−1})
1440a_{3}  2^{4} · 3^{10} · 5^{3}  C_{2}^{A}; 3, ∞
1440a_{4}  2^{17} · 3^{12} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
1440a_{5}  2^{17} · 3^{12} · 5^{2}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
1445a_{1}  −2^{12} · 5^{3} · 17^{12}  D_{8} × C_{2}
1445a_{2}  −2^{12} · 5^{4} · 17^{13}  D_{8}^{B}
1455a_{1}  −2^{12} · 3^{21} · 5^{5} · 97^{2}  S_{4} × C_{2}
1485a_{1}  −2^{12} · 3^{6} · 5^{2} · 11^{5}  D_{12}
1485b_{1}  −2^{12} · 3^{16} · 5^{2} · 11^{5}  D_{12}
1521a_{1}  −2^{12} · 3^{17} · 13^{13}  V_{4}^{A}
1521a_{2}  2^{12} · 3^{15} · 13^{12}  V_{4}^{E}; ℚ(\sqrt{−39})
1521a_{3}  2^{12} · 3^{20} · 13^{12}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
1521a_{4}  2^{12} · 3^{15} · 5^{22} · 13^{4}  V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})
1533a_{1} −2^{12} · 3^{15} · 5^{22} · 7^{2} · 73^{2}  D_{12}

Curve Discriminant of F_{6}  Galois Group of F_{6}
1575a_{1} −2^{12} · 3^{6} · 5^{14} · 7^{3}  V_{4}^{A}
1575a_{2} 2^{12} · 3^{6} · 5^{13} · 7^{2}  C_{2}^{3B}; ℚ(\sqrt{−7})
1575a_{3} 2^{12} · 3^{6} · 5^{16} · 7^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{21}), \frac{1}{2}(5 − \sqrt{21})
1575a_{4} −2^{12} · 3^{6} · 5^{13} · 7^{4} V_{4}^{B}; \frac{1}{2}(1 + \sqrt{−3}), \frac{1}{6}(3 + 5\sqrt{−3})
1575b_{1} −2^{12} · 3^{16} · 5^{14} · 7^{3}  V_{4}^{A}
1575b_{2} 2^{12} · 3^{16} · 5^{13} · 7^{2}  C_{2}^{3B}; ℚ(\sqrt{−7})
1575b_{3} 2^{12} · 3^{16} · 5^{16} · 7^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{21}), \frac{1}{2}(5 − \sqrt{21})
1575b_{4} −2^{12} · 3^{16} · 5^{13} · 7^{4} V_{4}^{B}; \frac{1}{2}(1 + \sqrt{−3}), \frac{1}{6}(3 + 5\sqrt{−3})
1600a_{1}  −2^{20} · 5^{14}  C_{2}^{A}; 0, ∞
1600a_{2}  2^{4} · 3^{22} · 5^{13}  V_{4}^{B}; −1 + \sqrt{−1}, \frac{1}{2}(−5 − \sqrt{−1})
1600a_{3}  2^{14} · 5^{13}  C_{2}^{A}; 3, ∞
1600a_{4}  2^{27} · 3^{22} · 5^{12}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
1600a_{5}  2^{27} · 3^{22} · 5^{12}  V_{4}^{B}; 1 + \sqrt{5}, 4 − 2\sqrt{5}
1602a_{1}  −2^{20} · 3^{10} · 89^{2}  D_{12}
1675a_{1}  2^{12} · 5^{19} · 67^{2}  S_{4} × C_{2}
1683a_{1} −2^{12} · 3^{19} · 11^{2} · 17^{2}  S_{4} × C_{2}
1713a_{1}  2^{12} · 3^{6} · 571^{2}  S_{4}^{+}
1725a_{1} −2^{12} · 3^{9} · 5^{12} · 23^{2}  S_{4} × C_{2}
1743a_{1} −2^{12} · 3^{5} · 7^{9} · 83^{2}  D_{8} × C_{2}
1743a_{2} 2^{12} · 3^{7} · 7^{6} · 83^{3}  D_{8}^{A}
1764a_{1}  −2^{16} · 3^{10} · 7^{18}  D_{8}^{B}
1773a_{1} −2^{12} · 3^{21} · 197^{2}  S_{4} × C_{2}
1815a_{1} −2^{12} · 3^{4} · 5^{3} · 11^{13}  V_{4}^{A}
1815a_{2} −2^{12} · 3^{3} · 5^{2} · 11^{12}  C_{2}^{3B}; ℚ(\sqrt{−55})
1815a_{3} −2^{12} · 3^{19} · 5^{4} · 11^{12} C_{2}^{B}; \frac{1}{2}(3 + \sqrt{−11}), \frac{1}{10}(3 − \sqrt{−11})
1815a_{4} 2^{12} · 3^{3} · 5^{2} · 11^{14}  V_{4}^{B}; \frac{1}{2}(1 + \sqrt{5}), \frac{1}{2}(5 + 3\sqrt{5})
1827a_{1} −2^{12} · 3^{15} · 7^{3} · 29^{2}  D_{8} × C_{2}
1827a_{2} −2^{12} · 3^{17} · 7^{2} · 29^{3}  D_{8}^{A}

Curve  Discriminant of F_{6}  Galois Group of F_{6}
1849a_{1}  −2^{12} · 43^{15}  S_{4}^{−}
1849a_{2}  2^{12} · 3^{22} · 43^{12}  S_{4}^{+}
1881a_{1} −2^{12} · 3^{10} · 11^{2} · 19^{3}  S_{4} × C_{2}
1885a_{1}  −2^{12} · 5^{3} · 13^{4} · 29^{3}  C_{2}^{3A}
1885a_{2}  −2^{12} · 5^{4} · 13^{6} · 29^{2}  V_{4}^{E}; ℚ(\sqrt{−29})
1885a_{3} 2^{12} · 3^{24} · 5^{2} · 13^{3} · 29^{4}  C_{2}^{3B}; ℚ(\sqrt{−5})
1885a_{4} 2^{12} · 3^{12} · 5^{2} · 13^{3} · 29^{2}  C_{2}^{3B}; ℚ(\sqrt{145})
1885b_{1} 2^{12} · 5^{4} · 13^{3} · 29^{3} V_{4}^{A}
1885b_{2} 2^{12} · 3^{22} · 5^{3} · 13^{2} · 29^{2}  C_{2}^{3B}; ℚ(\sqrt{377})
1885b_{3} −2^{12} · 3^{22} · 5^{6} · 13^{4} · 29^{2}  V_{4}^{B}; \frac{1}{2}(\sqrt{29} − 7), \frac{1}{14}(\sqrt{29} − 1)
1885b_{4} 2^{12} · 5^{3} · 7^{22} · 13^{2} · 29^{4} V_{4}^{B}; \frac{1}{2}(−19 + 5\sqrt{13}), \frac{1}{6}(−1 + \sqrt{13})
1911a_{1}  −2^{12} · 3^{7} · 7^{10} · 13^{3}  V_{4}^{A}
1911a_{2}  2^{12} · 3^{5} · 7^{10} · 13^{2}  V_{4}^{E}; ℚ(\sqrt{−39})
1911a_{3}  2^{12} · 3^{10} · 7^{10} · 13^{2}  C_{2}^{B}; \frac{1}{2}(−3 + \sqrt{13}), \frac{1}{2}(5 − \sqrt{13})
1911a_{4} 2^{12} · 3^{5} · 7^{10} · 5^{22} · 13^{4} V_{4}^{B}; \frac{1}{2}(−1 + 3\sqrt{−3}), \frac{1}{6}(−3 − \sqrt{−3})
1911b_{1} −2^{12} · 3^{7} · 7^{14} · 13^{3} V_{4}^{A}
1911b_{2} 2^{12} · 3^{5} · 5^{22} · 7^{13} · 13^{2}  C_{2}^{3B}; ℚ(\sqrt{−39})
1911b_{3} 2^{12} · 3^{10} · 5^{22} · 7^{13} · 13^{2} V_{4}^{B}; \frac{1}{2}(−1 + \sqrt{13}), \frac{1}{6}(−23 − 7\sqrt{13})
1911b_{4}  −2^{12} · 3^{5} · 7^{16} · 13^{4}  C_{2}^{C}; \frac{1}{2}(3 + 3\sqrt{−3}), \frac{1}{2}(−1 + \sqrt{−3})
1911c_{1}  2^{12} · 3^{3} · 7^{5} · 13^{2}  S_{4} × C_{2}
1911d_{1}  2^{12} · 3^{3} · 7^{15} · 13^{2}  S_{4} × C_{2}
1925a_{1} 2^{12} · 3^{12} · 5^{13} · 7^{6} · 11^{3}  D_{8} × C_{2}
1925a_{2}  −2^{12} · 5^{14} · 7^{9} · 11^{2}  D_{8}^{A}
1935a_{1}  −2^{12} · 3^{13} · 5^{3} · 43^{2}  S_{4} × C_{2}
1983a_{1}  2^{12} · 3^{15} · 661^{2}  S_{4} × C_{2}

A.3 Parameterisations for Theorem 3.2.2

The table below gives, for each curve C_{1} in the table in Section A.1, the values of A, P, Q such that it is ℚ-isomorphic to a curve C_{2} from our \sqrt{2} multiplication family, and M_{11}, M_{12}, M_{21}, M_{22} such that the map

(x, y) ↦ (\frac{M_{11}x + M_{12}}{M_{21}x + M_{22}} , \frac{y}{(M_{21}x + M_{22})^{3}} )

induces an isomorphism from C_{2} to C_{1}; Δ can always be chosen so that the isomorphism is of the above form, and A can always be chosen to be nonnegative. For example, consider the curve 29a_{1}, for which

A = 2, P = −1, Q = −4, Δ = 1, M_{11} = 1, M_{12} = 1, M_{21} = 0, M_{22} = 1.

Then, C_{1} is the curve

C_{1}: Y^{2} = X^{6} − 4X^{5} − 12X^{4} + 2X^{3} + 8X^{2} + 8X − 7,

C_{2} is the curve

C_{2}: Y^{2} = X^{6} + 2X^{5} − 17X^{4} − 66X^{3} − 83X^{2} − 32X − 4,

and the map

(x, y) ↦ (\frac{M_{11}x + M_{12}}{M_{21}x + M_{22}} , \frac{y}{(M_{21}x + M_{22})^{3}} ) = (x + 1, y)

induces an isomorphism from C_{2} to C_{1}.

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 29a_{1} & 2 & −1 & −4 & 1 & 1 & 1 & 0 & 1 \\ 29a_{2} & 1 & \frac{1}{2} & 0 & 2^{3} & 0 & 2 & −1 & 0 \\ 39a_{1} & \frac{22}{3} & 3 & 12 & 3^{2} & 0 & 1 & −1 & −1 \\ 39a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & 1 & 1 & 1 & 0 & 1 \\ 39a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & 3^{2} & 0 & 1 & −1 & −1 \\ 39a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & 2^{3} · 5 · 7 & 0 & 2 & 1 & 0 \\ 43a_{1} & \frac{20}{3} & 1 & \frac{11}{3} & −3 & 1 & 1 & 0 & 1 \\ 43a_{2} & 0 & −1 & 1 & −3 & 1 & 0 & 0 & 1 \\ 55a_{1} & \frac{14}{3} & −1 & −\frac{14}{3} & −3 & 1 & 1 & 0 & 1 \\ 55a_{2} & 6 & −\frac{1}{4} & −\frac{1}{4} & −3 & 1 & 0 & 0 & −1 \\ 55a_{3} & \frac{2}{3} & 1 & \frac{2}{3} & −3 & 1 & 0 & 0 & 1 \\ 55a_{4} & 18 & −1 & −8 & −3 & 1 & 0 & 0 & −1 \\ 65a_{1} & 6 & −\frac{5}{2} & −\frac{23}{2} & −3 · 5 & 1 & 1 & 0 & −1 \\ 65a_{2} & 2 & −1 & −2 & 1 & 1 & 0 & 0 & 1 \\ 82a_{1} & \frac{2}{3} & \frac{1}{8} & −\frac{2}{3} & −2^{2} · 3 & 1 & 0 & 0 & 2 \\ 82a_{2} & 9 & −\frac{4}{13} & −\frac{7}{13} & −2^{2} · 3 & 1 & 0 & 0 & −1 \\ 82a_{3} & 9 & 8 & 37 & −2^{2} · 3 & 1 & 0 & 0 & −1 \\ 82a_{4} & 11 & \frac{1}{2} & \frac{7}{4} & 2^{2} & 1 & 2 & 0 & −1 \\ 85a_{1} & 10 & 1 & 4 & 1 & 1 & 1 & 0 & 1 \\ 85a_{2} & 2 & 1 & 2 & 1 & 1 & 0 & 0 & 1 \\ 91a_{1} & \frac{16}{3} & 1 & \frac{7}{3} & −3 & 1 & 1 & 0 & −1 \\ 98a_{1} & 8 & \frac{1}{8} & −\frac{1}{2} & 2 & 1 & 1 & 0 & 1 \\ 98a_{2} & 8 & \frac{1}{8} & −\frac{1}{2} & −2 · 7 & 1 & 1 & 0 & 1 \\ 98a_{3} & \frac{6}{5} & \frac{5}{16} & \frac{7}{4} & 2^{2} · 3 · 5 & 0 & 2 & 1 & 0 \\ 98a_{4} & \frac{6}{5} & \frac{5}{16} & \frac{7}{4} & −2^{2} · 3 · 5 · 7 & 0 & 2 & 1 & 0 \\ 117a_{1} & \frac{22}{3} & 3 & 12 & −3^{3} & 0 & 1 & −1 & −1 \\ 117a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & −3 & 1 & 1 & 0 & 1 \\ 117a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & −3^{3} & 0 & 1 & −1 & −1 \\ 117a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & −2^{3} · 3 · 5 · 7 & 0 & 2 & 1 & 0 \\ 123a_{1} & 4 & \frac{3}{5} & 1 & −3 & 1 & 1 & 0 & 1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 129a_{1} & 10 & \frac{1}{3} & \frac{2}{3} & −3 & 1 & 1 & 0 & 1 \\ 145a_{1} & 6 & 1 & 6 & 1 & 1 & 1 & 0 & 1 \\ 145a_{2} & \frac{54}{13} & −1 & −\frac{54}{13} & −3 · 13 & 1 & 1 & 0 & −1 \\ 145a_{3} & \frac{27}{2} & 1 & 6 & 2^{4} & 1 & 2 & 1 & 0 \\ 145a_{4} & 6 & 1 & \frac{8}{3} & 3^{4} & 1 & 1 & 0 & 1 \\ 147a_{1} & 6 & 1 & 4 & 1 & 1 & 1 & 0 & 1 \\ 147a_{2} & 10 & \frac{9}{19} & \frac{28}{19} & 3^{2} & 0 & 1 & −1 & −1 \\ 147b_{1} & 6 & 1 & 4 & −7 & 1 & 1 & 0 & 1 \\ 147b_{2} & 10 & \frac{9}{19} & \frac{28}{19} & −3^{2} · 7 & 0 & 1 & −1 & −1 \\ 160a_{1} & \frac{1121}{156} & −\frac{26}{25} & −\frac{412}{75} & 2^{3} · 3 · 13 & 3 & 2 & 1 & 3 \\ 160a_{2} & \frac{18}{13} & −\frac{104}{275} & \frac{17}{55} & 2^{2} · 3 · 5 · 13 & 0 & 7 & 1 & −3 \\ 160a_{3} & \frac{101}{12} & \frac{32}{55} & \frac{59}{33} & −2^{3} · 3 · 5 & 1 & 2 & −1 & −1 \\ 160a_{4} & 12 & −\frac{1}{4} & −\frac{1}{2} & 3 & 1 & 0 & 0 & 1 \\ 160a_{5} & 12 & −\frac{1}{4} & −\frac{1}{2} & −3 & 1 & 0 & 0 & 1 \\ 165a_{1} & 6 & −\frac{9}{5} & −8 & 3^{2} & 1 & 2 & −1 & −1 \\ 165a_{2} & 6 & \frac{1}{4} & \frac{1}{4} & 1 & 1 & 1 & 0 & 1 \\ 165a_{3} & \frac{18}{5} & −1 & −\frac{18}{5} & −3 · 5 & 1 & 0 & 0 & −1 \\ 165a_{4} & 18 & 1 & 8 & 1 & 1 & 1 & 0 & −1 \\ 178a_{1} & 7 & 1 & \frac{7}{2} & 2^{2} & 1 & 1 & 0 & −1 \\ 183a_{1} & 6 & 1 & 2 & 1 & 1 & 1 & 0 & −1 \\ 196a_{1} & \frac{26}{3} & −\frac{2}{3} & −\frac{43}{9} & 2 · 3 & 1 & 0 & 0 & 1 \\ 207a_{1} & 6 & \frac{1}{2} & \frac{3}{2} & 1 & 1 & 1 & 0 & 1 \\ 207a_{2} & \frac{26}{3} & \frac{9}{11} & \frac{32}{11} & 3^{2} & 0 & 1 & −1 & −1 \\ 207b_{1} & 6 & \frac{1}{2} & \frac{3}{2} & −3 & 1 & 1 & 0 & 1 \\ 207b_{2} & \frac{26}{3} & \frac{9}{11} & \frac{32}{11} & −3^{3} & 0 & 1 & −1 & −1 \\ 209a_{1} & 4 & 1 & 3 & 1 & 1 & 0 & 0 & 1 \\ 223a_{1} & 14 & 1 & 6 & 1 & 1 & 3 & 0 & −1 \\ 227a_{1} & 8 & 1 & \frac{11}{3} & 3^{6} & 3 & 4 & 0 & −1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 232a_{1} & 10 & \frac{1}{2} & \frac{3}{2} & 1 & 1 & 1 & 0 & −1 \\ 237a_{1} & \frac{26}{3} & \frac{1}{3} & \frac{2}{3} & −3^{3} & 1 & 1 & 0 & 1 \\ 245a_{1} & 8 & 1 & 3 & 1 & 1 & 1 & 0 & −1 \\ 245b_{1} & 8 & 1 & 3 & −7 & 1 & 1 & 0 & −1 \\ 245c_{1} & \frac{41}{5} & \frac{1}{2} & \frac{8}{5} & 2^{3} · 5 & 0 & 2 & 1 & 2 \\ 245d_{1} & \frac{41}{5} & \frac{1}{2} & \frac{8}{5} & −2^{3} · 5 · 7 & 0 & 2 & 1 & 2 \\ 249a_{1} & \frac{26}{3} & 1 & 4 & 3^{2} & 1 & 2 & 0 & −1 \\ 256a_{1} & \frac{17}{3} & −\frac{1}{2} & −\frac{17}{6} & −3 & 1 & 2 & 1 & 1 \\ 256a_{2} & \frac{16}{3} & −\frac{1}{2} & −\frac{8}{3} & 3 & 1 & 1 & 0 & −1 \\ 261a_{1} & 2 & −1 & −4 & −3 & 1 & 1 & 0 & 1 \\ 261a_{2} & 1 & \frac{1}{2} & 0 & −2^{3} · 3 & 0 & 2 & −1 & 0 \\ 265a_{1} & 10 & 1 & 6 & 1 & 1 & 0 & 0 & 1 \\ 273a_{1} & \frac{32}{3} & \frac{3}{4} & 3 & −2^{2} · 3^{2} & 0 & 2 & −1 & −2 \\ 273a_{2} & \frac{22}{3} & −\frac{3}{4} & −\frac{9}{4} & 3^{2} · 5 & 1 & 0 & 0 & 1 \\ 273a_{3} & \frac{34}{3} & −\frac{1}{3} & −\frac{8}{9} & 3^{2} · 5 & 1 & 0 & 0 & 1 \\ 273a_{4} & 2 & \frac{1}{3} & \frac{2}{3} & −3 & 1 & 0 & 0 & −1 \\ 275a_{1} & \frac{14}{3} & −1 & −\frac{14}{3} & −3 · 5 & 1 & 1 & 0 & 1 \\ 275a_{2} & 6 & −\frac{1}{4} & −\frac{1}{4} & −3 · 5 & 1 & 0 & 0 & −1 \\ 275a_{3} & \frac{2}{3} & 1 & \frac{2}{3} & −3 · 5 & 1 & 0 & 0 & 1 \\ 275a_{4} & 18 & −1 & −8 & −3 · 5 & 1 & 0 & 0 & −1 \\ 278a_{1} & 0 & \frac{1}{8} & \frac{1}{2} & −2^{3} · 3 & 1 & 0 & 0 & 2 \\ 285a_{1} & 6 & −\frac{3}{2} & −\frac{13}{2} & −3 & 1 & 1 & 0 & −1 \\ 285a_{2} & 18 & −1 & −18 & −3 & 1 & 0 & 0 & 1 \\ 285b_{1} & 6 & \frac{5}{18} & \frac{7}{18} & 5 & 1 & 1 & 0 & 1 \\ 285b_{2} & \frac{98}{15} & −5 & −\frac{64}{3} & 3^{2} · 5 & 0 & 1 & −1 & −1 \\ 303a_{1} & 12 & 1 & 5 & 1 & 1 & 2 & 0 & −1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 315a_{1} & 6 & −1 & −6 & 1 & 1 & 1 & 0 & −1 \\ 315a_{2} & 6 & −1 & −4 & 1 & 1 & 1 & 0 & −1 \\ 315a_{3} & \frac{42}{5} & \frac{7}{9} & \frac{122}{45} & −5^{2} · 7 & 2 & 3 & 1 & 1 \\ 315a_{4} & \frac{10}{3} & \frac{9}{14} & \frac{11}{14} & 3^{2} & 1 & 1 & 0 & 1 \\ 315b_{1} & 6 & −1 & −6 & −3 & 1 & 1 & 0 & −1 \\ 315b_{2} & 6 & −1 & −4 & −3 & 1 & 1 & 0 & −1 \\ 315b_{3} & \frac{42}{5} & \frac{7}{9} & \frac{122}{45} & 3 · 5^{2} · 7 & 2 & 3 & 1 & 1 \\ 315b_{4} & \frac{10}{3} & \frac{9}{14} & \frac{11}{14} & −3^{3} & 1 & 1 & 0 & 1 \\ 320a_{1} & \frac{1121}{156} & −\frac{26}{25} & −\frac{412}{75} & 2^{4} · 3 · 13 & 3 & 2 & 1 & 3 \\ 320a_{2} & \frac{18}{13} & −\frac{104}{275} & \frac{17}{55} & 2 · 3 · 5 · 13 & 0 & 7 & 1 & −3 \\ 320a_{3} & \frac{101}{12} & \frac{32}{55} & \frac{59}{33} & −2^{4} · 3 · 5 & 1 & 2 & −1 & −1 \\ 320a_{4} & 12 & −\frac{1}{4} & −\frac{1}{2} & 2 · 3 & 1 & 0 & 0 & 1 \\ 320a_{5} & 12 & −\frac{1}{4} & −\frac{1}{2} & −2 · 3 & 1 & 0 & 0 & 1 \\ 325a_{1} & 6 & −\frac{5}{2} & −\frac{23}{2} & −3 · 5^{2} & 1 & 1 & 0 & −1 \\ 325a_{2} & 2 & −1 & −2 & 5 & 1 & 0 & 0 & 1 \\ 325b_{1} & 6 & \frac{1}{5} & \frac{4}{5} & −3 & 1 & 0 & 0 & 1 \\ 325c_{1} & 6 & \frac{1}{5} & \frac{4}{5} & −3 · 5 & 1 & 0 & 0 & 1 \\ 335a_{1} & 4 & −1 & −1 & 1 & 1 & 0 & 0 & 1 \\ 339a_{1} & 4 & \frac{1}{3} & \frac{1}{3} & −3 & 1 & 1 & 0 & 1 \\ 345a_{1} & 12 & 1 & 7 & 1 & 1 & 0 & 0 & 1 \\ 357a_{1} & 0 & \frac{1}{3} & \frac{1}{3} & 1 & 1 & 0 & 0 & 1 \\ 369a_{1} & 4 & \frac{3}{5} & 1 & 3^{2} & 1 & 1 & 0 & 1 \\ 385a_{1} & 6 & −\frac{1}{22} & \frac{17}{22} & −3 & 1 & 0 & 0 & 1 \\ 385a_{2} & \frac{86}{19} & −1 & −\frac{86}{19} & −19 & 1 & 0 & 0 & 1 \\ 387a_{1} & \frac{20}{3} & 1 & \frac{11}{3} & 3^{2} & 1 & 1 & 0 & 1 \\ 387a_{2} & 0 & −1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 387b_{1} & 10 & \frac{1}{3} & \frac{2}{3} & 3^{2} & 1 & 1 & 0 & 1 \\ 389a_{1} & 16 & 1 & 7 & 1 & 1 & 1 & 0 & −1 \\ 397a_{1} & \frac{42}{5} & \frac{1}{2} & \frac{17}{10} & −3 · 5 & 1 & 1 & 0 & 1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 425a_{1} & 10 & 1 & 4 & 5 & 1 & 1 & 0 & 1 \\ 425a_{2} & 2 & 1 & 2 & 5 & 1 & 0 & 0 & 1 \\ 429a_{1} & \frac{10}{3} & 1 & 2 & 3^{2} & 1 & 0 & 0 & 1 \\ 429a_{2} & 30 & \frac{1}{9} & \frac{4}{9} & 1 & 1 & 1 & 0 & −1 \\ 437a_{1} & 8 & 1 & 5 & 1 & 1 & 0 & 0 & 1 \\ 441a_{1} & 6 & 1 & 4 & −3 & 1 & 1 & 0 & 1 \\ 441a_{2} & 10 & \frac{9}{19} & \frac{28}{19} & −3^{3} & 0 & 1 & −1 & −1 \\ 441b_{1} & 6 & 1 & 4 & 3 · 7 & 1 & 1 & 0 & 1 \\ 441b_{2} & 10 & \frac{9}{19} & \frac{28}{19} & 3^{3} · 7 & 0 & 1 & −1 & −1 \\ 445a_{1} & 6 & \frac{1}{5} & 2 & −3 & 1 & 0 & 0 & 1 \\ 445a_{2} & \frac{22}{5} & −1 & −\frac{22}{5} & 5 & 1 & 1 & 0 & −1 \\ 465a_{1} & 6 & \frac{1}{9} & −\frac{4}{9} & 1 & 1 & 1 & 0 & 1 \\ 465a_{2} & \frac{2}{3} & −\frac{1}{5} & \frac{4}{3} & 3^{2} & 0 & 1 & 1 & −1 \\ 495a_{1} & \frac{14}{3} & −1 & −\frac{14}{3} & 3^{2} & 1 & 1 & 0 & 1 \\ 495a_{2} & 6 & −\frac{1}{4} & −\frac{1}{4} & 1 & 1 & 0 & 0 & −1 \\ 495a_{3} & \frac{2}{3} & 1 & \frac{2}{3} & 3^{2} & 1 & 0 & 0 & 1 \\ 495a_{4} & 18 & −1 & −8 & −3 & 1 & 0 & 0 & 1 \\ 495b_{1} & 6 & −\frac{9}{5} & −8 & −3^{3} & 1 & 2 & −1 & −1 \\ 495b_{2} & 6 & \frac{1}{4} & \frac{1}{4} & −3 & 1 & 1 & 0 & 1 \\ 495b_{3} & \frac{18}{5} & −1 & −\frac{18}{5} & 5 & 1 & 0 & 0 & −1 \\ 495b_{4} & 18 & 1 & 8 & −3 & 1 & 1 & 0 & −1 \\ 507a_{1} & \frac{22}{3} & 3 & 12 & 3^{2} · 13 & 0 & 1 & −1 & −1 \\ 507a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & 13 & 1 & 1 & 0 & 1 \\ 507a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & 3^{2} · 13 & 0 & 1 & −1 & −1 \\ 507a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & 2^{3} · 5 · 7 · 13 & 0 & 2 & 1 & 0 \\ 517a_{1} & \frac{34}{3} & 1 & \frac{16}{3} & −3 & 1 & 1 & 0 & −1 \\ 529a_{1} & 18 & −1 & −10 & −3 & 1 & 1 & 0 & −1 \\ 529a_{2} & \frac{2}{3} & \frac{1}{2} & −\frac{7}{6} & 3 · 23 & 1 & 0 & 0 & −1 \\ 529b_{1} & 18 & −1 & −10 & 3 · 23 & 1 & 1 & 0 & −1 \\ 529b_{2} & \frac{2}{3} & \frac{1}{2} & −\frac{7}{6} & −3 & 1 & 0 & 0 & −1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 545a_{1} & \frac{38}{13} & −\frac{13}{50} & \frac{73}{50} & 13 & 1 & −1 & 0 & −1 \\ 545a_{2} & \frac{58}{13} & 1 & \frac{58}{13} & 13 & 1 & 0 & 0 & 1 \\ 547a_{1} & \frac{1}{3} & \frac{1}{2} & −\frac{4}{3} & −2^{3} · 3 · 5 & 0 & 2 & −1 & 0 \\ 549a_{1} & 6 & 1 & 2 & −3 & 1 & 1 & 0 & −1 \\ 561a_{1} & −\frac{26}{3} & \frac{1}{4} & −\frac{1}{3} & 2^{2} · 3^{2} & 1 & −2 & −1 & 0 \\ 583a_{1} & 4 & −1 & −3 & 1 & 1 & 0 & 0 & 1 \\ 585a_{1} & 6 & −\frac{5}{2} & −\frac{23}{2} & 5 & 1 & 1 & 0 & −1 \\ 585a_{2} & 2 & −1 & −2 & −3 & 1 & 0 & 0 & 1 \\ 591a_{1} & 20 & \frac{1}{3} & \frac{5}{3} & −3 & 1 & 0 & 1 & 1 \\ 605a_{1} & \frac{14}{3} & −1 & −\frac{14}{3} & 3 · 11 & 1 & 1 & 0 & 1 \\ 605a_{2} & 6 & −\frac{1}{4} & −\frac{1}{4} & 3 · 11 & 1 & 0 & 0 & −1 \\ 605a_{3} & \frac{2}{3} & 1 & \frac{2}{3} & 3 · 11 & 1 & 0 & 0 & 1 \\ 605a_{4} & 18 & −1 & −8 & 3 · 11 & 1 & 0 & 0 & −1 \\ 609a_{1} & 6 & \frac{1}{6} & −\frac{1}{6} & 1 & 1 & 1 & 0 & 1 \\ 609a_{2} & \frac{10}{3} & −\frac{3}{7} & −\frac{16}{7} & 3^{2} & 0 & 1 & −1 & −1 \\ 621a_{1} & 18 & \frac{1}{3} & \frac{4}{3} & 1 & 1 & 1 & 0 & 1 \\ 621b_{1} & 18 & \frac{1}{3} & \frac{4}{3} & −3 & 1 & 1 & 0 & 1 \\ 621c_{1} & \frac{24}{5} & \frac{5}{9} & \frac{7}{9} & 5 & 1 & 1 & 0 & −1 \\ 621d_{1} & \frac{24}{5} & \frac{5}{9} & \frac{7}{9} & −3 · 5 & 1 & 1 & 0 & −1 \\ 637a_{1} & \frac{16}{3} & 1 & \frac{7}{3} & 3 · 7 & 1 & 1 & 0 & −1 \\ 639a_{1} & 6 & \frac{1}{5} & 0 & 1 & 1 & 1 & 0 & 1 \\ 639a_{2} & \frac{14}{3} & −\frac{9}{13} & −\frac{44}{13} & 3^{2} & 0 & 1 & −1 & −1 \\ 639b_{1} & 6 & \frac{1}{5} & 0 & −3 & 1 & 1 & 0 & 1 \\ 639b_{2} & \frac{14}{3} & −\frac{9}{13} & −\frac{44}{13} & −3^{3} & 0 & 1 & −1 & −1 \\ 645a_{1} & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 1 \\ 661a_{1} & 15 & \frac{1}{2} & 4 & −2^{3} · 3 & 0 & 2 & 1 & 0 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 675a_{1} & 0 & −\frac{1}{3} & \frac{5}{3} & 1 & 1 & −1 & 0 & 1 \\ 675a_{2} & 0 & −\frac{1}{3} & \frac{5}{3} & −3 & 1 & −1 & 0 & 1 \\ 675b_{1} & 0 & −\frac{1}{3} & \frac{5}{3} & 5 & 1 & −1 & 0 & 1 \\ 675b_{2} & 0 & −\frac{1}{3} & \frac{5}{3} & −3 · 5 & 1 & −1 & 0 & 1 \\ 703a_{1} & 4 & −1 & −5 & 1 & 1 & 1 & 0 & −1 \\ 705a_{1} & \frac{22}{3} & 1 & 4 & 3^{2} & 1 & 1 & 0 & 1 \\ 711a_{1} & \frac{26}{3} & \frac{1}{3} & \frac{2}{3} & 3^{4} & 1 & 1 & 0 & 1 \\ 715a_{1} & 8 & \frac{1}{5} & −\frac{1}{5} & 5 & 1 & 1 & 0 & 1 \\ 725a_{1} & 2 & −1 & −4 & 5 & 1 & 1 & 0 & 1 \\ 725a_{2} & 1 & \frac{1}{2} & 0 & 2^{3} · 5 & 0 & 2 & −1 & 0 \\ 725b_{1} & 6 & 1 & 6 & 5 & 1 & 1 & 0 & 1 \\ 725b_{2} & \frac{54}{13} & −1 & −\frac{54}{13} & −3 · 5 · 13 & 1 & 1 & 0 & −1 \\ 725b_{3} & \frac{27}{2} & 1 & 6 & 2^{4} · 5 & 1 & 2 & 1 & 0 \\ 725b_{4} & 6 & 1 & \frac{8}{3} & 3^{4} · 5 & 1 & 1 & 0 & 1 \\ 735a_{1} & \frac{34}{9} & −\frac{27}{55} & −\frac{28}{11} & −3^{3} & 0 & 1 & −1 & −1 \\ 735a_{2} & 6 & \frac{5}{12} & \frac{13}{12} & −5 · 7 & 1 & 1 & 0 & 1 \\ 735a_{3} & \frac{34}{5} & \frac{75}{17} & \frac{338}{17} & 3 · 5^{2} · 7 & 1 & 1 & 0 & 1 \\ 735a_{4} & \frac{102}{13} & −\frac{13}{135} & \frac{92}{135} & −7 · 13 & 1 & −1 & 0 & −1 \\ 735b_{1} & \frac{34}{9} & −\frac{27}{55} & −\frac{28}{11} & 3^{3} · 7 & 0 & 1 & −1 & −1 \\ 735b_{2} & 6 & \frac{5}{12} & \frac{13}{12} & 5 & 1 & 1 & 0 & 1 \\ 735b_{3} & \frac{34}{5} & \frac{75}{17} & \frac{338}{17} & −3 · 5^{2} & 1 & 1 & 0 & 1 \\ 735b_{4} & \frac{102}{13} & −\frac{13}{135} & \frac{92}{135} & 13 & 1 & −1 & 0 & −1 \\ 735c_{1} & \frac{36}{5} & \frac{5}{3} & \frac{19}{3} & 5^{2} & 0 & 1 & −1 & −1 \\ 735d_{1} & \frac{36}{5} & \frac{5}{3} & \frac{19}{3} & −5^{2} · 7 & 0 & 1 & −1 & −1 \\ 737a_{1} & 4 & 1 & 1 & 1 & 1 & 0 & 0 & −1 \\ 738a_{1} & \frac{2}{3} & \frac{1}{8} & −\frac{2}{3} & 2^{2} · 3^{2} & 1 & 0 & 0 & 2 \\ 738a_{2} & 9 & −\frac{4}{13} & −\frac{7}{13} & 2^{2} & 1 & 0 & 0 & −1 \\ 738a_{3} & 9 & 8 & 37 & 2^{2} & 1 & 0 & 0 & −1 \\ 738a_{4} & 11 & \frac{1}{2} & \frac{7}{4} & −2^{2} · 3 & 1 & 2 & 0 & −1 \\ 741a_{1} & 4 & −\frac{1}{3} & −\frac{5}{3} & −3 & 1 & 1 & 0 & −1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 747a_{1} & \frac{26}{3} & 1 & 4 & −3^{3} & 1 & 2 & 0 & −1 \\ 753a_{1} & \frac{4}{3} & 1 & 1 & 3^{2} & 1 & 0 & 0 & 1 \\ 765a_{1} & 10 & 1 & 4 & −3 & 1 & 1 & 0 & 1 \\ 765a_{2} & 2 & 1 & 2 & −3 & 1 & 0 & 0 & 1 \\ 767a_{1} & 2 & 1 & 0 & 1 & 1 & 1 & 0 & −1 \\ 777a_{1} & \frac{86}{9} & \frac{27}{49} & \frac{88}{49} & −3^{3} & 0 & 1 & −1 & −1 \\ 777a_{2} & 6 & \frac{3}{4} & \frac{11}{4} & −3 & 1 & 1 & 0 & 1 \\ 777a_{3} & \frac{138}{7} & −\frac{7}{195} & \frac{2}{3} & 7 · 11 & 1 & −1 & 0 & −1 \\ 777a_{4} & \frac{46}{7} & \frac{63}{11} & \frac{268}{11} & −3^{2} · 5 · 7 & 1 & 1 & 0 & 1 \\ 779a_{1} & 8 & −1 & −5 & 1 & 1 & 1 & 0 & 1 \\ 800a_{1} & \frac{1121}{156} & −\frac{26}{25} & −\frac{412}{75} & 2^{3} · 3 · 5 · 13 & 3 & 2 & 1 & 3 \\ 800a_{2} & \frac{18}{13} & −\frac{104}{275} & \frac{17}{55} & 2^{2} · 3 · 5^{2} · 13 & 0 & 7 & 1 & −3 \\ 800a_{3} & \frac{101}{12} & \frac{32}{55} & \frac{59}{33} & −2^{3} · 3 · 5^{2} & 1 & 2 & −1 & −1 \\ 800a_{4} & 12 & −\frac{1}{4} & −\frac{1}{2} & 3 · 5 & 1 & 0 & 0 & 1 \\ 800a_{5} & 12 & −\frac{1}{4} & −\frac{1}{2} & −3 · 5 & 1 & 0 & 0 & 1 \\ 819a_{1} & \frac{16}{3} & 1 & \frac{7}{3} & 3^{2} & 1 & 1 & 0 & −1 \\ 819b_{1} & \frac{32}{3} & \frac{3}{4} & 3 & 2^{2} · 3^{3} & 0 & 2 & −1 & −2 \\ 819b_{2} & \frac{22}{3} & −\frac{3}{4} & −\frac{9}{4} & −3^{3} · 5 & 1 & 0 & 0 & 1 \\ 819b_{3} & \frac{34}{3} & −\frac{1}{3} & −\frac{8}{9} & −3^{3} · 5 & 1 & 0 & 0 & 1 \\ 819b_{4} & 2 & \frac{1}{3} & \frac{2}{3} & 3^{2} & 1 & 0 & 0 & −1 \\ 825a_{1} & 6 & −\frac{9}{5} & −8 & 3^{2} · 5 & 1 & 2 & −1 & −1 \\ 825a_{2} & 6 & \frac{1}{4} & \frac{1}{4} & 5 & 1 & 1 & 0 & 1 \\ 825a_{3} & \frac{18}{5} & −1 & −\frac{18}{5} & −3 · 5^{2} & 1 & 0 & 0 & −1 \\ 825a_{4} & 18 & 1 & 8 & 5 & 1 & 1 & 0 & −1 \\ 837a_{1} & \frac{66}{5} & \frac{1}{3} & \frac{16}{15} & 5 & 1 & 1 & 0 & 1 \\ 837b_{1} & \frac{66}{5} & \frac{1}{3} & \frac{16}{15} & −3 · 5 & 1 & 1 & 0 & 1 \\ 841a_{1} & 2 & −1 & −4 & 29 & 1 & 1 & 0 & 1 \\ 841a_{2} & 1 & \frac{1}{2} & 0 & 2^{3} · 29 & 0 & 2 & −1 & 0 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 845a_{1} & 6 & −\frac{5}{2} & −\frac{23}{2} & −3 · 5 · 13 & 1 & 1 & 0 & −1 \\ 845a_{2} & 2 & −1 & −2 & 13 & 1 & 0 & 0 & 1 \\ 855a_{1} & 6 & −\frac{3}{2} & −\frac{13}{2} & 3^{2} & 1 & 1 & 0 & −1 \\ 855a_{2} & 18 & −1 & −18 & 1 & 1 & 0 & 0 & 1 \\ 855b_{1} & 6 & \frac{5}{18} & \frac{7}{18} & −3 · 5 & 1 & 1 & 0 & 1 \\ 855b_{2} & \frac{98}{15} & −5 & −\frac{64}{3} & −3^{3} · 5 & 0 & 1 & −1 & −1 \\ 882a_{1} & 8 & \frac{1}{8} & −\frac{1}{2} & −2 · 3 & 1 & 1 & 0 & 1 \\ 882a_{2} & 8 & \frac{1}{8} & −\frac{1}{2} & 2 · 3 · 7 & 1 & 1 & 0 & 1 \\ 882a_{3} & \frac{6}{5} & \frac{5}{16} & \frac{7}{4} & −2^{2} · 3^{2} · 5 & 0 & 2 & 1 & 0 \\ 882a_{4} & \frac{6}{5} & \frac{5}{16} & \frac{7}{4} & 2^{2} · 3^{2} · 5 · 7 & 0 & 2 & 1 & 0 \\ 885a_{1} & \frac{44}{3} & 1 & \frac{19}{3} & 3^{2} & 1 & 1 & 0 & −1 \\ 909a_{1} & 12 & 1 & 5 & −3 & 1 & 2 & 0 & −1 \\ 931a_{1} & 10 & \frac{1}{2} & \frac{1}{2} & 1 & 1 & 1 & 0 & −1 \\ 931a_{1} & 10 & \frac{1}{2} & \frac{1}{2} & −7 & 1 & 1 & 0 & −1 \\ 975a_{1} & \frac{22}{3} & 3 & 12 & 3^{2} · 5 & 0 & 1 & −1 & −1 \\ 975a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & 5 & 1 & 1 & 0 & 1 \\ 975a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & 3^{2} · 5 & 0 & 1 & −1 & −1 \\ 975a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & 2^{3} · 7 & 0 & 2 & 1 & 0 \\ 1017a_{1} & 4 & \frac{1}{3} & \frac{1}{3} & 3^{2} & 1 & 1 & 0 & 1 \\ 1035a_{1} & 12 & 1 & 7 & −3 & 1 & 0 & 0 & 1 \\ 1035b_{1} & 12 & −1 & −7 & 1 & 1 & 2 & 0 & 1 \\ 1035c_{1} & 12 & −1 & −7 & −3 & 1 & 2 & 0 & 1 \\ 1045a_{1} & \frac{54}{7} & −1 & −\frac{54}{7} & 3 · 7 & 1 & 0 & 0 & 1 \\ 1045a_{2} & 6 & −\frac{5}{4} & −\frac{21}{4} & 5 & 1 & 1 & 0 & −1 \\ 1045a_{3} & \frac{26}{9} & −\frac{81}{55} & −\frac{386}{55} & 3^{5} · 7 & 1 & 1 & 0 & −1 \\ 1045a_{4} & \frac{30}{7} & \frac{175}{297} & \frac{272}{297} & 3 · 5^{2} · 7 & 1 & 1 & 0 & −1 \\ 1071a_{1} & 0 & \frac{1}{3} & \frac{1}{3} & −3 & 1 & 0 & 0 & 1 \\ 1075a_{1} & \frac{20}{3} & 1 & \frac{11}{3} & −3 · 5 & 1 & 1 & 0 & 1 \\ 1075a_{2} & 0 & −1 & 1 & −3 · 5 & 1 & 0 & 0 & 1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 1165a_{1} & 6 & \frac{1}{14} & \frac{19}{14} & −3 & 1 & 1 & 0 & −1 \\ 1165a_{2} & \frac{58}{13} & −1 & −\frac{58}{13} & 13 & 1 & 0 & 0 & 1 \\ 1183a_{1} & \frac{16}{3} & 1 & \frac{7}{3} & −3 · 13 & 1 & 1 & 0 & −1 \\ 1185a_{1} & 6 & \frac{3}{2} & \frac{13}{2} & −3 & 1 & 1 & 0 & 1 \\ 1185a_{2} & \frac{94}{9} & \frac{27}{65} & \frac{16}{13} & −3^{3} & 0 & 1 & −1 & −1 \\ 1197a_{1} & \frac{834}{65} & \frac{13}{215} & \frac{1524}{1075} & 5^{6} · 13 & 5 & 1 & 0 & −7 \\ 1197a_{2} & \frac{87}{10} & \frac{125}{259} & \frac{334}{259} & −2^{4} · 5^{4} & 1 & 2 & 1 & 0 \\ 1197a_{3} & \frac{50}{3} & \frac{63}{82} & \frac{425}{82} & −3^{2} · 5^{3} · 7 & 1 & 3 & −2 & −1 \\ 1197a_{4} & \frac{162}{13} & \frac{13}{63} & \frac{16}{63} & 13 & 1 & 1 & 0 & 1 \\ 1197b_{1} & \frac{834}{65} & \frac{13}{215} & \frac{1524}{1075} & −3 · 5^{6} · 13 & 5 & 1 & 0 & −7 \\ 1197b_{2} & \frac{87}{10} & \frac{125}{259} & \frac{334}{259} & 2^{4} · 3 · 5^{4} & 1 & 2 & 1 & 0 \\ 1197b_{3} & \frac{50}{3} & \frac{63}{82} & \frac{425}{82} & 3^{3} · 5^{3} · 7 & 1 & 3 & −2 & −1 \\ 1197b_{4} & \frac{162}{13} & \frac{13}{63} & \frac{16}{63} & −3 · 13 & 1 & 1 & 0 & 1 \\ 1203a_{1} & \frac{14}{3} & 1 & 2 & 3^{2} & 1 & 1 & 0 & −1 \\ 1209a_{1} & \frac{16}{3} & 1 & 3 & 3^{2} & 1 & 0 & 0 & 1 \\ 1225a_{1} & 8 & 1 & 3 & 5 & 1 & 1 & 0 & −1 \\ 1225b_{1} & 8 & 1 & 3 & −5 · 7 & 1 & 1 & 0 & −1 \\ 1225c_{1} & \frac{41}{5} & \frac{1}{2} & \frac{8}{5} & 2^{3} · 5^{2} & 0 & 2 & 1 & 2 \\ 1225d_{1} & \frac{41}{5} & \frac{1}{2} & \frac{8}{5} & −2^{3} · 5^{2} · 7 & 0 & 2 & 1 & 2 \\ 1275a_{1} & \frac{20}{3} & 1 & 5 & 3^{2} & 1 & 0 & 0 & 1 \\ 1275b_{1} & \frac{20}{3} & 1 & 5 & 3^{2} · 5 & 1 & 0 & 0 & 1 \\ 1287a_{1} & \frac{10}{3} & 1 & 2 & −3^{3} & 1 & 0 & 0 & 1 \\ 1287a_{2} & 30 & \frac{1}{9} & \frac{4}{9} & 3 & 1 & 1 & 0 & −1 \\ 1295a_{1} & 8 & −1 & −3 & 1 & 1 & 1 & 0 & 1 \\ 1305a_{1} & 6 & 1 & 6 & −3 & 1 & 1 & 0 & 1 \\ 1305a_{2} & \frac{54}{13} & −1 & −\frac{54}{13} & 13 & 1 & 1 & 0 & −1 \\ 1305a_{3} & \frac{27}{2} & 1 & 6 & −2^{4} · 3 & 1 & 2 & 1 & 0 \\ 1305a_{4} & 6 & 1 & \frac{8}{3} & −3^{3} & 1 & 1 & 0 & 1 \end{matrix}

APPENDIX A. TABLES 208

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 1325a_{1} & 10 & 1 & 6 & 5 & 1 & 0 & 0 & 1 \\ 1325b_{1} & 0 & −1 & 5 & −3 & 1 & −1 & 0 & 1 \\ 1325c_{1} & 0 & −1 & 5 & −3 · 15 & 1 & −1 & 0 & 1 \\ 1395a_{1} & 6 & \frac{1}{9} & −\frac{4}{9} & −3 & 1 & 1 & 0 & 1 \\ 1395a_{2} & \frac{2}{3} & −\frac{1}{5} & \frac{4}{3} & −3^{3} & 0 & 1 & 1 & −1 \\ 1409a_{1} & 18 & \frac{1}{2} & \frac{9}{2} & −3 & 1 & 0 & 0 & 1 \\ 1421a_{1} & 2 & −1 & −4 & −7 & 1 & 1 & 0 & 1 \\ 1421a_{2} & 1 & \frac{1}{2} & 0 & −2^{3} · 7 & 0 & 2 & −1 & 0 \\ 1425a_{1} & 6 & −\frac{3}{2} & −\frac{13}{2} & −3 · 5 & 1 & 1 & 0 & −1 \\ 1425a_{2} & 18 & −1 & −18 & −3 · 5 & 1 & 0 & 0 & 1 \\ 1425b_{1} & 6 & \frac{5}{18} & \frac{7}{18} & 5^{2} & 1 & 1 & 0 & 1 \\ 1425b_{2} & \frac{98}{15} & −5 & −\frac{64}{3} & 3^{2} · 5^{2} & 0 & 1 & −1 & −1 \\ 1433a_{1} & 6 & −1 & −\frac{14}{3} & −3^{7} & 3 & 3 & 0 & 1 \\ 1440a_{1} & \frac{1121}{156} & −\frac{26}{25} & −\frac{412}{75} & −2^{3} · 3^{2} · 13 & 3 & 2 & 1 & 3 \\ 1440a_{2} & \frac{18}{13} & −\frac{104}{275} & \frac{17}{55} & −2^{2} · 5 · 13 & 0 & 7 & 1 & −3 \\ 1440a_{3} & \frac{101}{12} & \frac{32}{55} & \frac{59}{33} & 2^{3} · 3^{2} · 5 & 1 & 2 & −1 & −1 \\ 1440a_{4} & 12 & −\frac{1}{4} & −\frac{1}{2} & −1 & 1 & 0 & 0 & 1 \\ 1440a_{5} & 12 & −\frac{1}{4} & −\frac{1}{2} & 1 & 1 & 0 & 0 & 1 \\ 1445a_{1} & 10 & 1 & 4 & 17 & 1 & 1 & 0 & 1 \\ 1445a_{2} & 2 & 1 & 2 & 17 & 1 & 0 & 0 & 1 \\ 1455a_{1} & \frac{8}{3} & 1 & \frac{13}{3} & 3^{2} & 1 & 0 & 0 & 1 \\ 1485a_{1} & \frac{26}{5} & \frac{3}{5} & \frac{28}{25} & −3 · 5 & 1 & 1 & 0 & −1 \\ 1485b_{1} & \frac{26}{5} & \frac{3}{5} & \frac{28}{25} & 3^{2} · 5 & 1 & 1 & 0 & −1 \\ 1521a_{1} & \frac{22}{3} & 3 & 12 & −3^{3} · 13 & 0 & 1 & −1 & −1 \\ 1521a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & −3 · 13 & 1 & 1 & 0 & 1 \\ 1521a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & −3^{3} · 13 & 0 & 1 & −1 & −1 \\ 1521a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & −2^{3} · 3 · 5 · 7 · 13 & 0 & 2 & 1 & 0 \\ 1533a_{1} & \frac{10}{3} & \frac{1}{7} & \frac{20}{21} & 3^{2} · 5 & 1 & 0 & 0 & 1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 1575a_{1} & 6 & −1 & −6 & 5 & 1 & 1 & 0 & −1 \\ 1575a_{2} & 6 & −1 & −4 & 5 & 1 & 1 & 0 & −1 \\ 1575a_{3} & \frac{42}{5} & \frac{7}{9} & \frac{122}{45} & −5^{3} · 7 & 2 & 3 & 1 & 1 \\ 1575a_{4} & \frac{10}{3} & \frac{9}{14} & \frac{11}{14} & 3^{2} · 5 & 1 & 1 & 0 & 1 \\ 1575b_{1} & 6 & −1 & −6 & −3 · 5 & 1 & 1 & 0 & −1 \\ 1575b_{2} & 6 & −1 & −4 & −3 · 5 & 1 & 1 & 0 & −1 \\ 1575b_{3} & \frac{42}{5} & \frac{7}{9} & \frac{122}{45} & 3 · 5^{3} · 7 & 2 & 3 & 1 & 1 \\ 1575b_{4} & \frac{10}{3} & \frac{9}{14} & \frac{11}{14} & −3^{3} · 5 & 1 & 1 & 0 & 1 \\ 1600a_{1} & \frac{1121}{156} & −\frac{26}{25} & −\frac{412}{75} & 2^{4} · 3 · 5 · 13 & 3 & 2 & 1 & 3 \\ 1600a_{2} & \frac{18}{13} & −\frac{104}{275} & \frac{17}{55} & 2 · 3 · 5^{2} · 13 & 0 & 7 & 1 & −3 \\ 1600a_{3} & \frac{101}{12} & \frac{32}{55} & \frac{59}{33} & −2^{4} · 3 · 5^{2} & 1 & 2 & −1 & −1 \\ 1600a_{4} & 12 & −\frac{1}{4} & −\frac{1}{2} & 2 · 3 · 5 & 1 & 0 & 0 & 1 \\ 1600a_{5} & 12 & −\frac{1}{4} & −\frac{1}{2} & −2 · 3 · 5 & 1 & 0 & 0 & 1 \\ 1602a_{1} & 7 & 1 & \frac{7}{2} & −2^{2} · 3 & 1 & 1 & 0 & −1 \\ 1675a_{1} & 4 & −1 & −1 & 5 & 1 & 0 & 0 & 1 \\ 1683a_{1} & −\frac{26}{3} & \frac{1}{4} & −\frac{1}{3} & −2^{2} · 3^{3} & 1 & −2 & −1 & 0 \\ 1713a_{1} & \frac{48}{5} & 1 & \frac{23}{5} & 5 & 1 & 1 & 0 & −1 \\ 1725a_{1} & 12 & 1 & 7 & 5 & 1 & 0 & 0 & 1 \\ 1743a_{1} & 6 & −\frac{1}{15} & −\frac{4}{3} & 1 & 1 & 1 & 0 & 1 \\ 1743a_{2} & \frac{94}{3} & \frac{3}{49} & −\frac{12}{49} & 3^{2} & 0 & 1 & −1 & −1 \\ 1764a_{1} & \frac{26}{3} & −\frac{2}{3} & −\frac{43}{9} & −2 · 3^{2} & 1 & 0 & 0 & 1 \\ 1773a_{1} & 20 & \frac{1}{3} & \frac{5}{3} & 3^{2} & 1 & 0 & 1 & 1 \\ 1815a_{1} & 6 & −\frac{9}{5} & −8 & −3^{2} · 11 & 1 & 2 & −1 & −1 \\ 1815a_{2} & 6 & \frac{1}{4} & \frac{1}{4} & −11 & 1 & 1 & 0 & 1 \\ 1815a_{3} & \frac{18}{5} & −1 & −\frac{18}{5} & 3 · 5 · 11 & 1 & 0 & 0 & −1 \\ 1815a_{4} & 18 & 1 & 8 & −11 & 1 & 1 & 0 & −1 \\ 1827a_{1} & 6 & \frac{1}{6} & −\frac{1}{6} & −3 & 1 & 1 & 0 & 1 \\ 1827a_{2} & \frac{10}{3} & −\frac{3}{7} & −\frac{16}{7} & −3^{3} & 0 & 1 & −1 & −1 \end{matrix}

\begin{matrix}Curve & A & P & Q & Δ & M_{11} & M_{12} & M_{21} & M_{22} \\ 1849a_{1} & \frac{20}{3} & 1 & \frac{11}{3} & 3 · 43 & 1 & 1 & 0 & 1 \\ 1849a_{2} & 0 & −1 & 1 & 3 · 43 & 1 & 0 & 0 & 1 \\ 1881a_{1} & 4 & 1 & 3 & −3 & 1 & 0 & 0 & 1 \\ 1885a_{1} & \frac{18}{5} & −\frac{5}{7} & −\frac{24}{7} & 5 & 1 & 1 & 0 & 1 \\ 1885a_{2} & \frac{6}{5} & 1 & \frac{6}{5} & 5 & 1 & 0 & 0 & 1 \\ 1885a_{3} & 6 & −1 & −\frac{8}{3} & −3^{3} & 1 & 0 & 0 & 1 \\ 1885a_{4} & 18 & −\frac{1}{4} & −\frac{3}{4} & −3 & 1 & 0 & 0 & 1 \\ 1885b_{1} & \frac{94}{21} & −1 & −\frac{94}{21} & 3 · 7 & 1 & 1 & 0 & −1 \\ 1885b_{2} & 6 & \frac{1}{23} & \frac{28}{23} & −3 & 1 & 0 & 0 & 1 \\ 1885b_{3} & \frac{78}{7} & −\frac{7}{10} & −\frac{11}{2} & 3 · 7 & 1 & 0 & 0 & 1 \\ 1885b_{4} & \frac{94}{3} & −\frac{1}{12} & −\frac{85}{36} & 3 · 7 & 1 & 0 & 0 & 1 \\ 1911a_{1} & \frac{22}{3} & 3 & 12 & −3^{2} · 7 & 0 & 1 & −1 & −1 \\ 1911a_{2} & 6 & \frac{1}{3} & \frac{2}{3} & −7 & 1 & 1 & 0 & 1 \\ 1911a_{3} & \frac{34}{3} & \frac{1}{3} & \frac{8}{9} & −3^{2} · 7 & 0 & 1 & −1 & −1 \\ 1911a_{4} & \frac{51}{7} & −\frac{7}{6} & −\frac{16}{3} & −2^{3} · 5 · 7^{2} & 0 & 2 & 1 & 0 \\ 1911b_{1} & \frac{32}{3} & \frac{3}{4} & 3 & 2^{2} · 3^{2} · 7 & 0 & 2 & −1 & −2 \\ 1911b_{2} & \frac{22}{3} & −\frac{3}{4} & −\frac{9}{4} & −3^{2} · 5 · 7 & 1 & 0 & 0 & 1 \\ 1911b_{3} & \frac{34}{3} & −\frac{1}{3} & −\frac{8}{9} & −3^{2} · 5 · 7 & 1 & 0 & 0 & 1 \\ 1911b_{4} & 2 & \frac{1}{3} & \frac{2}{3} & 3 · 7 & 1 & 0 & 0 & −1 \\ 1911c_{1} & 36 & \frac{1}{7} & −\frac{5}{7} & 7^{2} & 0 & 7 & 1 & 3 \\ 1911d_{1} & 36 & \frac{1}{7} & −\frac{5}{7} & −7^{3} & 0 & 7 & 1 & 3 \\ 1925a_{1} & 6 & −\frac{1}{22} & \frac{17}{22} & −3 · 5 & 1 & 0 & 0 & 1 \\ 1925a_{2} & \frac{86}{19} & −1 & −\frac{86}{19} & −19 · 5 & 1 & 0 & 0 & 1 \\ 1935a_{1} & 0 & 1 & 1 & −3 & 1 & 0 & 0 & 1 \\ 1983a_{1} & 6 & −\frac{1}{2} & \frac{1}{2} & 1 & 1 & 0 & 0 & 1 \end{matrix}

A.4 Parameterisations for Theorem 4.3.1

The table below shows the pairs of curves in the table in Section A.1 with the property that there are ℚ-isogenies of degree 2 between their jacobians, these isogenies coming from Theorem 4.3.1.

To see how to interpret the table, consider the part of it corresponding to the putative ℚ-isogeny class 39a:

ℚ-isogeny class  kernel  curves U V W Δ
39a  (X − 1)(X + 3) 1, 2 4 3 20 3
39a  X^{2} − X + 1  1, 3 1 1 4 1
39a  X^{2} − X − 3  1, 4 1 −3 4 1

The quadratic under the ‘kernel’ column in the first row is (X − 1)(X + 3), and the numbers under the ‘curves’ column are 1, 2, so there is a ℚ-isogeny of degree 2 from the jacobian of 39a_{1} to that of 39a_{2}, with kernel

[(1, 0) − (−3, 0)]

(1, −3 being the zeros of (X − 1)(X + 3)). Moreover, the numbers under the U, V, W, Δ columns indicate that there is a ℚ-isomorphism from the curve 39a_{1} to C (using the notation of Section 4.1) such that the points

(\frac{−U + \sqrt{U^{2} − 4V}}{2} , 0) , (\frac{−U − \sqrt{U^{2} − 4V}}{2} , 0)

(with U = 4, V = 3) are sent to the points

(1, 0), (−3, 0)

(not necessarily in that order), and that there is a ℚ-isomorphism from the curve 39a_{2} to C_{4,3,20}.

The quadratic under the ‘kernel’ column in the second row is X^{2} − X + 1, and the numbers under the ‘curves’ column are 1, 3, so there is a ℚ-isogeny of degree 2 from the jacobian of 39a_{1} to that of 39a_{3}, with kernel

[(\frac{1 + \sqrt{−3}}{2}, 0) − (\frac{1 − \sqrt{−3}}{2}, 0)]

(\frac{1+\sqrt{−3}}{2}, \frac{1−\sqrt{−3}}{2} being the zeros of X^{2} − X + 1). Moreover, the numbers under the U, V, W, Δ columns indicate that there is a ℚ-isomorphism from the curve 39a_{1} to C (using the notation of Section 4.1) such that the points

(\frac{−U + \sqrt{U^{2} − 4V}}{2} , 0) , (\frac{−U − \sqrt{U^{2} − 4V}}{2} , 0)

(with U = 1, V = 1) are sent to the points

(\frac{1 + \sqrt{−3}}{2}, 0) , (\frac{1 − \sqrt{−3}}{2}, 0)

(not necessarily in that order), and that there is a ℚ-isomorphism from the curve 39a_{3} to C_{1,1,4}.

The quadratic under the ‘kernel’ column in the third row is X^{2} − X − 3, and the numbers under the ‘curves’ column are 1, 4, so there is a ℚ-isogeny of degree 2 from the jacobian of 39a_{1} to that of 39a_{4}, with kernel

[(\frac{1 + \sqrt{13}}{2}, 0) − (\frac{1 − \sqrt{13}}{2}, 0)]

(\frac{1+\sqrt{13}}{2}, \frac{1−\sqrt{13}}{2} being the zeros of X^{2} − X − 3). Moreover, the numbers under the U, V, W, Δ columns indicate that there is a ℚ-isomorphism from the curve 39a_{1} to C (using the notation of Section 4.1) such that the points

(\frac{−U + \sqrt{U^{2} − 4V}}{2} , 0) , (\frac{−U − \sqrt{U^{2} − 4V}}{2} , 0)

(with U = 1, V = −3) are sent to the points

(\frac{1 + \sqrt{13}}{2}, 0) , (\frac{1 − \sqrt{13}}{2}, 0)

(not necessarily in that order), and that there is a ℚ-isomorphism defined over ℚ from the curve 39a_{4} to C_{1,−3,4}.

All the other rows in the table are interpreted in a similar way. Note that we may assume, w.l.o.g., that U is positive in each case.

ℚ-isogeny class  kernel  curves U V W  Δ
39a  (X − 1)(X + 3) 1, 2 4 3 20  3
39a  X^{2} − X + 1  1, 3 1 1  4  1
39a  X^{2} − X − 3  1, 4 1 −3 4  1
55a  (X + 1)(X − 3) 1, 2 \frac{12}{5} −\frac{32}{5} 1  −1
55a  X^{2} + X − 1  1, 3 3 1  0  −3
55a  3X^{2} − X + 1  1, 4 \frac{5}{3} 1  0  −3
65a  X^{2} + X − 1  1, 2 1 −1 \frac{28}{5} −3 · 5
82a  12X^{2} + 4X + 3 1, 3 \frac{2}{3} 1 36  −3
82a  X^{2} − 4X − 4  2, 4 4 −4 −\frac{29}{4}  −3
85a  X^{2} + X − 1  1, 2 3 1  8  1
98a  (X + 1)(X + 3) 1, 3 6 8  8  −1
98a  (X + 1)(X + 3) 2, 4 6 8  8  7
117a  (X − 1)(X + 3) 1, 2 4 3 20  −1
117a  X^{2} − X + 1  1, 3 1 1  4  −3
117a  X^{2} − X − 3  1, 4 1 −3 4  −3
145a  X^{2} − X − 1  1, 2 1 −1 0  1
145a  X^{2} − 2X + 5  1, 3 \frac{12}{5}  \frac{8}{5}  \frac{13}{5}  1
145a  X^{2} + 3X − 5  1, 4 5 −1 0  1
147a  X^{2} − X + 1  1, 2 1 1  8  1
147b  X^{2} − X + 1  1, 2 1 1  8  −7
160a  (X − 1)(X + 1) 1, 2 \frac{3}{4} −\frac{5}{8}  \frac{132}{13} 2 · 3 · 13
160a  X^{2} + 4  1, 3 \frac{36}{13}  \frac{40}{13}  \frac{11}{4} 2 · 3 · 13
165a  X(X + 4)  1, 2 4 3 −\frac{28}{5}  −5
165a  X^{2} + X − 1  1, 3 1 −1 4  1
165a  X^{2} + X + 3  1, 4 1 3  4  1
207a  X^{2} − X + 1  1, 2 1 1 12  1
207b  X^{2} − X + 1  1, 2 1 1 12  −3

ℚ-isogeny class  kernel  curves U V  W  Δ
273a  (X + 2)(5X + 6) 1, 2 \frac{4}{3}  \frac{1}{3}  20 −1
273a  X^{2} − X + 1  1, 3 5 7  4  5
273a  X^{2} − X − 3  1, 4 5 3  4  5
275a  (X + 1)(X − 3) 1, 2 \frac{12}{5} −\frac{32}{5} 1  −5
275a  X^{2} + X − 1  1, 3 3 1  0 −3 · 5
275a  3X^{2} − X + 1  1, 4 \frac{5}{3} 1  0 −3 · 5
285a  X^{2} + X − 1  1, 2 1 −1 \frac{20}{3}  −3
285b  X^{2} − X + 1  1, 2 1 1  \frac{92}{5}  5
315a  (X − 1)(X + 3) 1, 2 \frac{12}{5} −\frac{32}{5}  \frac{37}{5}  −7
315a  X^{2} + X + 1  1, 3 1 1  0  1
315a  X^{2} − 3X − 3  1, 4 5 1  0  1
315b  (X − 1)(X + 3) 1, 2 \frac{12}{5} −\frac{32}{5}  \frac{37}{5} 3 · 7
315b  X^{2} + X + 1  1, 3 1 1  0  −3
315b  X^{2} − 3X − 3  1, 4 5 1  0  −3
320a  (X − 1)(X + 1) 1, 2 \frac{3}{4} −\frac{5}{8}  \frac{132}{13} 3 · 13
320a  X^{2} + 4  1, 3 \frac{36}{13}  \frac{40}{13}  \frac{11}{4} 3 · 13
325a  X^{2} + X − 1  1, 2 1 −1 \frac{28}{5}  −3
385a  X^{2} − X − 1  1, 2 1 −1 92 −3
425a  X^{2} + X − 1  1, 2 3 1  8  5
429a  X^{2} + X + 3  1, 2 1 3  \frac{16}{3}  1
441a  X^{2} − X + 1  1, 2 1 1  8  −3
441b  X^{2} − X + 1  1, 2 1 1  8  3 · 7
445a  X^{2} − X + 1  1, 2 1 −1 −16 −3
465a  X^{2} − X + 1  1, 2 1 1 40  1

ℚ-isogeny class  kernel  curves U V  W  Δ
495a  (X + 1)(X − 3) 1, 2 \frac{12}{5} −\frac{32}{5} 1  3
495a  X^{2} + X − 1  1, 3 3 1  0  1
495a  3X^{2} − X + 1  1, 4 \frac{5}{3} 1  0  1
495b  X(X + 4)  1, 2 4 3 −\frac{28}{5}  −3 · 5
495b  X^{2} + X − 1  1, 3 1 −1 4  −3
495b  X^{2} + X + 3  1, 4 1 3  4  −4
507a  (X − 1)(X + 3) 1, 2 4 3  20  3 · 13
507a  X^{2} − X + 1  1, 3 1 1  4  13
507a  X^{2} − X − 3  1, 4 1 −3 4  13
545a  X^{2} + X − 1  1, 2 3 1 −\frac{148}{13}  13
585a  X^{2} + X − 1  1, 2 1 1  \frac{28}{5}  ·5
605a  (X + 1)(X − 3) 1, 2 \frac{12}{5} −\frac{32}{5} 1  11
605a  X^{2} + X − 1  1, 3 3 1  0  3 · 11
605a  3X^{2} − X + 1  1, 4 \frac{5}{3} 1  0  3 · 11
609a  X^{2} − X + 1  1, 2 1 1  28  1
639a  X^{2} − X + 1  1, 2 1 1  24  1
639b  X^{2} − X + 1  1, 2 1 1  24  −3
725b  X^{2} − X − 1  1, 2 1 −1 0  5
725b  X^{2} − 2X + 5  1, 3 \frac{12}{5}  \frac{8}{5}  \frac{13}{5}  5
725b  X^{2} + 3X − 5  1, 4 5 −1 0  5
735a  (X − 1)(X + 3) 1, 2 4 3  \frac{76}{55}  5 · 11
735a  3X^{2} − 3X − 1 1, 3 1 −\frac{1}{3}  4  −3
735a  X^{2} − X + 9  1, 4 1 9  4  −3
735b  (X − 1)(X + 3) 1, 2 4 3  \frac{76}{55} −5 · 7 · 11
735b  3X^{2} − 3X − 1 1, 3 1 −\frac{1}{3}  4  3 · 7
735b  X^{2} − X + 9  1, 4 1 9  4  3 · 7
738a  12X^{2} + 4X + 3 1, 3 \frac{2}{3} 1  36  1
738a  X^{2} − 4X − 4  2, 4 4 −4 −\frac{29}{4}  1
765a  X^{2} + X − 1  1, 2 3 1  8  −3

ℚ-isogeny class  kernel  curves U V W  Δ
777a  (X − 1)(X + 3) 1, 2 4 3 \frac{340}{49}  −1
777a  3X^{2} − 3X + 1  1, 3 5 7 \frac{136}{27}  −3
777a  X^{2} − X − 9  1, 4 \frac{17}{9}  \frac{7}{9}  \frac{40}{3}  −3
800a  (X − 1)(X + 1) 1, 2 \frac{3}{4} −\frac{5}{8}  \frac{132}{13} 2 · 3 · 5 · 13
800a  X^{2} + 4  1, 3 \frac{36}{13}  \frac{40}{13}  \frac{11}{4} 2 · 3 · 5 · 13
819b  (X + 2)(5X + 6) 1, 2 \frac{4}{3}  \frac{1}{3} 20  3
819b  X^{2} − X + 1  1, 3 5 7  4  −3 · 5
819b  X^{2} − X − 3  1, 4 5 3  4  −3 · 5
825a  X(X + 4)  1, 2 4 3 −\frac{28}{5}  −1
825a  X^{2} + X − 1  1, 3 1 −1 4  5
825a  X^{2} + X + 3  1, 4 1 3  4  5
845a  X^{2} + X − 1  1, 2 1 −1 \frac{28}{5} −3 · 5 · 13
855a  X^{2} + X − 1  1, 2 1 −1 \frac{20}{3}  1
855b  X^{2} − X + 1  1, 2 1 1 \frac{92}{5}  −3 · 5
882a  (X + 1)(X + 3) 1, 3 6 8  8  3
882a  (X + 1)(X + 3) 2, 4 6 8  8  −3 · 7
975a  (X − 1)(X + 3) 1, 2 4 3 20  3 · 5
975a  X^{2} − X + 1  1, 3 1 1  4  5
975a  X^{2} − X − 3  1, 4 1 −3 4  5
1045a  (X − 2)(X + 2) 1, 2 \frac{8}{15} −\frac{4}{5}  \frac{215}{19}  −3 · 19
1045a  X^{2} + 7X + 1  1, 3 7 1  0  3 · 7
1045a  7X^{2} + 5X + 7  1, 4 \frac{5}{7} 1  0  3 · 7
1165a  X^{2} + X − 1  1, 2 1 −1 −52  −3
1185a  X^{2} − X + 1  1, 2 1 1 \frac{20}{3}  −3

ℚ-isogeny class  kernel  curves U V  W  Δ
1197a  (X − 7)(13X + 9) 1, 2 \frac{16}{3} −4 −3  5
1197a  X^{2} + X + 1  1, 3 1 \frac{7}{3}  16  5
1197a  X^{2} − 3X − 3  1, 4 3 −3 −\frac{16}{3}  5
1197b  (X − 7)(13X + 9) 1, 2 \frac{16}{3} −4 −3 −3 · 5
1197b  X^{2} + X + 1  1, 3 1 \frac{7}{3}  16 −3 · 5
1197b  X^{2} − 3X − 3  1, 4 3 −3 −\frac{16}{3} −3 · 5
1287a  X^{2} + X + 3  1, 2 1 3  \frac{16}{3}  −3
1305a  X^{2} − X − 1  1, 2 1 −1 0  −3
1305a  X^{2} − 2X + 5  1, 3 \frac{12}{5}  \frac{8}{5}  \frac{13}{5}  −3
1305a  X^{2} + 3X − 5  1, 4 5 −1 0  −3
1395a  X^{2} − X + 1  1, 2 1 1 40  −3
1425a  X^{2} + X − 1  1, 2 1 −1 \frac{20}{3} −3 · 5
1425b  X^{2} − X + 1  1, 2 1 1  \frac{92}{5}  1
1440a  (X − 1)(X + 1)  1, 2 \frac{3}{4} −\frac{5}{8}  \frac{132}{13} 2 · 13
1440a  X^{2} + 4  1, 3 \frac{36}{13}  \frac{40}{13}  \frac{11}{4}  2 · 13
1445a  X^{2} + X − 1  1, 2 3 1  8  17
1521a  X^{2} − X + 1  1, 2 1 1  4 −3 · 13
1521a  X^{2} − X + 1  1, 3 1 1  4  13
1521a  X^{2} − X − 3  1, 4 1 −3 4  13
1575a  (X − 1)(X + 3)  1, 2 \frac{12}{5} −\frac{32}{5}  \frac{37}{5} −5 · 7
1575a  X^{2} + X + 1  1, 3 1 1  0  5
1575a  X^{2} − 3X − 3  1, 4 5 1  0  5
1575b  (X − 1)(X + 3)  1, 2 \frac{12}{5} −\frac{32}{5}  \frac{37}{5} −5 · 7
1575b  X^{2} + X + 1  1, 3 1 1  0  −3 · 5
1575b  X^{2} − 3X − 3  1, 4 5 1  0  −3 · 5
1600a  (X − 1)(X + 1)  1, 2 \frac{3}{4} −\frac{5}{8}  \frac{132}{13} 3 · 5 · 13
1600a  X^{2} + 4  1, 3 \frac{36}{13}  \frac{40}{13}  \frac{11}{4} 3 · 5 · 13
1743a  X^{2} − X + 1  1, 2 1 1 −56  1

ℚ-isogeny class  kernel  curves U V W  Δ
1815a  X(X + 4)  1, 2 4 3 −\frac{28}{5} 5 · 11
1815a  X^{2} + X − 1  1, 3 1 −1 4 −11
1815a  X^{2} + X + 3  1, 4 1 3  4 −11
1827a  X^{2} − X + 1  1, 2 1 1 28 −3
1885a  X^{2} + X − 1  1, 2 3 1 −\frac{8}{5}  5
1885a  X^{2} − 3X − 5  1, 3 \frac{7}{5}  \frac{1}{5} 12  5
1885a  5X^{2} − 2X + 1  1, 4 4 8 \frac{21}{5}  5
1885b  (X − 1)(X + 3) 1, 2 \frac{24}{7}  \frac{20}{7} −11 3 · 7
1885b  3X^{2} − X − 1  1, 3 \frac{11}{3} 3 −\frac{72}{7} 3 · 7
1885b  7X^{2} − X − 1  1, 4 3 −5 \frac{88}{7} 3 · 7
1911a  (X − 1)(X + 3) 1, 2 4 3 20 −3 · 7
1911a  X^{2} − X + 1  1, 3 1 1  4  7
1911a  X^{2} − X − 3  1, 4 1 −3 4  7
1911b  (X + 2)(5X + 6) 1, 2 \frac{4}{3}  \frac{1}{3} 20  7
1911b  X^{2} − X + 1  1, 3 5 7  4 −5 · 7
1911b  X^{2} − X − 3  1, 4 5 3  4 −5 · 7
1925a  X^{2} − X − 1  1, 2 1 −1 92 −3 · 5

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