Modular Varieties and Visibility

A thesis presented by

Tomas Antonius Klenke

to

The Department of Mathematics in partial fulfillment of the requirements

for the degree of Doctor of Philosophy

in the subject of

Mathematics

Harvard University Cambridge, Massachusetts

May 2001 \begin{matrix}c \\ ◯ \end{matrix}2001 – Tomas Antonius Klenke

All rights reserved.

Thesis Advisor: Prof. Barry C. Mazur Author: Tomas Antonius Klenke

Modular Varieties and Visibility

ABSTRACT

It is well known that the moduli problem of classifying elliptic curves with full level-two-structure is not rigid and hence not representable in the category of schemes. An easy argument shows, however, that it is possible to represent the moduli problem of classifying elliptic curves with full level-two-structure and an invariant non-vanishing regular relative differential (outside characteristic two) by a smooth affine surface Y (2, ω) over Spec(ℤ[1/2]).

In this dissertation I find compactifications of the moduli scheme Y (2, ω) and of the universal elliptic curve over it which have a number of good arithmetic and geometric properties. In particular, the action of SL_{2}(𝔽_{2}) on the universal level structure and the translations by two-torsion sections, which together generate an action of S_{4} on the universal elliptic curve, extend to regular automorphisms of the compactified elliptic threefold E(2, ω) . We achieve our construction by desingularizing a certain “naive” compactification which we pick as a starting point. It turns out that we can also find a simple geometric description of E(2, ω) as a certain blow-up of a ℙ^{1}-bundle over ℙ^{2} blown up at four points, compatibly with the natural S_{4}-action.

We use our description of E(2, ω) to show that its twists by the S_{4}-action have many rational points; as a consequence, we deduce that two-torsion elements of the Weil-Châtelet group of any elliptic curve are visible in infinitely many abelian surfaces, in the sense put forward by B. Mazur. Finally, we use the representation of the universal generalized elliptic curve with full levelfour-structure as the desingularized Kummer surface of a certain abelian surface to show that certain two-torsion elements of the Weil-Châtelet group of elliptic curves are visible in abelian surfaces, a large part of whose fourtorsion Galois module we can specify.

Contents

1 Introduction 1

1.1 Classifying Elliptic Curves with

Level-Two-Structure . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.2 Visibility and Universal Elliptic Curves . . . . . . . . 5

1.3 Outline and Results . . . . . . . . . . . . . . . . . . . . . . . 6

2 The Universal Family F(2, ω)

and a Compactification 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 The Universal Family F(2, ω) . . . . . . . . . . . . . . 11 2.2.2 Initial Setup . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Preliminary Remarks . . . . . . . . . . . . . . . . . . . 14

2.3 A Note on Blowing Up, on Multiplicities, and on Regularity . 17 2.3.1 Blowing Up . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.2 Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3.3 Regularity and Smoothness . . . . . . . . . . . . . . . 19

2.4 Performing the Blow-ups . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 The Fibers over the Line t = 0 . . . . . . . . . . . . . 22 2.4.2 The Fibers over the Point r = s = 0 . . . . . . . . . . 29 2.4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Translation by Two-torsion Sections . . . . . . . . . . . . . . 36 2.5.1 Extension of Translations by Sections . . . . . . . . . 36 2.5.2 Proof of the Proposition . . . . . . . . . . . . . . . . . 37

2.6 Minimality and Twists of E(2, ω) . . . . . . . . . . . . . . . . 41 2.6.1 Minimality . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6.2 Summary of Results . . . . . . . . . . . . . . . . . . . 42

2.6.3 The Twisting Property of F(2, ω) . . . . . . . . . . . 43

3 Alternative Construction of E(2, ω) 52 3.1 Some Elementary Blow-ups . . . . . . . . . . . . . . . . . . . 52 3.2 Constructing E(2, ω) . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.1 Performing The Final Blow-up . . . . . . . . . . . . . 60 3.2.2 Comparison with the Previous Construction . . . . . . 68

3.3 Statement of Main Result . . . . . . . . . . . . . . . . . . . . 76 3.4 Applications to Visibility . . . . . . . . . . . . . . . . . . . . 77

4 Visualizing Certain Elements of H^{1}(G_{K}, E(K̄))[2] 82 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2 Explicit Examples . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.1 Two-torsion of the Weil-Châtelet Group . . . . . . . . 84 4.2.2 Three-torsion of the Weil-Châtelet Group . . . . . . . 85

4.3 Statement of Visibility Results . . . . . . . . . . . . . . . . . 87 4.4 The Universal Elliptic Curve E(4) . . . . . . . . . . . . . . . . 88

4.4.1 Explicit Equations and Formulae . . . . . . . . . . . . 88 4.4.2 E(4) as a Desingularized Kummer Surface . . . . . . . 91 4.4.3 Lifting Automorphisms . . . . . . . . . . . . . . . . . 92

4.5 Proof of Theorem ?? . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Acknowledgements

First and foremost, I would like to thank Barry Mazur, my advisor, who so generously gave his time to listen to my ideas and to provide new perspectives, and who supported my work so enthusiastically over four years. I am also particularly grateful to Joe Harris, who more than once gave valuable feedback and advice, and to Brian Conrad, whose constant willingness to answer every question has been truly amazing. Mike Szydlo was very helpful in introducing me to the topic of desingularizing higher dimensional elliptic schemes; Alexei Skorobogatov pointed out to me that it was known that any twist of projective two-space blown up at four points in general position has a rational point, which allowed me to remove the local conditions on cocycles in the visibility type results in Chapter 3 of this dissertation.

The Harvard Mathematics Department has provided an extremely stimulating and comfortable work environment throughout my time in graduate school. Special thanks go to the graduate program coordinators Donna D’Fini and Irene Minder for putting up with countless requests, many of them last minute, and to my fellow number theory graduate students Cathy O’Neill, Mira Bernstein (who isn’t really a number theorist, because she likes complex numbers), Mark Dickinson, Sam Williams, Elena Mantovan, and Russell Mann. Particular mention goes to my fellow Matsumura experts Rob Pollack, David Savitt and Tom Weston, the latter of whom helped me out on many occasions and also bears full responsibility for my less than perfect understanding of the Hauptidealsatz.

Last but not least, I am grateful for the personal support which many people have generously given over the last four and a half years, thereby helping make my time at Harvard such a rewarding and wonderful experience: My parents, who have always supported me in all my plans; János, who so often helped me make those plans; Angelika, who enriched my life over two years and beyond; Christine and Hagi, who gave so much when I asked them; Jayanthi, who has been a wonderful friend; my housemates Sam and Brian, who were responsible for many a feast; Mira, who introduced me to the joys of Mathcamp; Susan, who put up with me when that wasn’t quite so easy; Quenby, who cheered me up so often; Jan Hendrik, who provided crucial help when I made my first steps exploring the world outside the ivory tower; and Serra, who was always there when I needed someone to listen.

Chapter 1

Introduction

1.1 Classifying Elliptic Curves with Level-Two-Structure

The problem of classifying elliptic curves with full level-three-structure was solved by Hesse already in the late 19th century. Hesse constructed the corresponding (compactified) universal elliptic curve as a certain surface E(3), whose geometry can be described in very simple terms: E(3) is easily seen to be ℙ^{2} blown up at nine points. E(3) can be explicitly given inside ℙ_{(X:Y:Z)}^{2} × ℙ_{(u:v)}^{1} over X(3) = ℙ_{(u:v)} by the equation

u(X^{3} + Y^{3} + Z^{3}) = 3vXY Z,

and the blow-down map to ℙ^{2} is the projection onto the first factor ℙ_{(X:Y:Z)}^{2} of the ambient space. Moreover, the nine exceptional fibers are the nine three-torsion sections of E(3) over X(3).

In contrast, the equally basic task of classifying elliptic curves with full level-two-structure is made complicated by the fact that the moduli problem here is not rigid. Over ℂ, there is at least the coarse moduli space ℙ_{λ}^{1}\{0, 1, ∞} (the “λ-line”), over which we have the Legendre family

y^{2} = x(x − 1)(x − λ);

for a parametrization of elliptic curves with full level-two-structure in an arithmetic context, however, we need to modify our approach.

One possibility is to classify elliptic curves with full level-two-structure and a non-zero relative invariant differential. This moduli problem is indeed representable; the most straightforward representation of the resulting universal elliptic curve over ℤ[1/2] is the “modified Legendre family”

F(2, ω) : ty^{2} = x(x − 1)(x − λ),

for t ≠ 0, λ(λ − 1) ≠ 0. We pay the price that that the resulting moduli space Y (2, ω) is now an (affine) surface over ℤ[1/2].

One might ask whether in this case there is an analogue to compactifying the affine modular curves Y (n) in the higher level case by adding the cusps. It is remarkable that if we compactify the universal elliptic curve F(2, ω) in the right way, we obtain an elliptic threefold E(2, ω) with good arithmetic properties and with a simple geometric description. In particular, it will turn out that the action of SL_{2}(𝔽_{2}) ≅ S_{3} on the universal level-two-structure of F(2, ω) and the translations by two-torsion sections extend to everywhere regular automorphisms of E(2, ω) (generating an action of S_{4}), and we will exhibit a certain blowdown of E(2, ω) as a ℙ^{1}-bundle over a surface which is ℙ^{2} blown up at four points. It follows that twists of E(2, ω) ⊗ K (for K any field) by the S_{4}-action have many K-rational points, which allows us to apply our construction to questions regarding visibility.

1.2 Visibility

1.2.1 Background

The following definition is taken from [Cr-M]:

Definition 1 Let K be an algebraic number field, and denote the absolute Galois group of K by G_{K}; let E be an elliptic curve over K, and let σ ∈ H^{1}(G_{K}, E) = W C(E/K) be an element of the Weil-Châtelet group of E. Then σ is visible in the abelian variety A if there is an embedding ι : E ↪ A over K such that under the induced map on cohomology

ι_{∗} : H^{1}(G_{K}, E) → H^{1}(G_{K}, A)

σ is sent to 0.

We give a geometric and an arithmetic interpretation of visibility:

Geometric interpretation of visibility.

Recall that to each element σ ∈ H^{1}(G_{K}, E) we can associate a principal homogeneous space C_{σ} of E. C_{σ} is a curve of genus one over K and is acted upon by E, which is its Jacobian. It follows from a simple argument in Galois cohomology that if σ is visible in an abelian variety A in the sense of Definition 1, then the principal homogeneous space C_{σ} corresponding to σ is contained as a subvariety in A, in fact as a translate of E ⊂ A. The converse is also true under some mild technical hypotheses (cf. [Cr-M]).

Arithmetic interpretation of visibility.

For simplicity, suppose that we can visualize σ ∈ H^{1}(G_{K}, E) in an abelian surface A (which will not always be possible – the following idea can easily be modified to apply to higher dimensional scenarios). Note that since E injects into A, A is isogenous to a product E ×F, where F is another elliptic curve over K. This suggests the following construction:

Given a cohomology class σ ∈ H^{1}(G_{K}, E), we lift it to an element η ∈ H^{1}(G_{K}, Φ), where Φ is a finite sub-group scheme of E over K; again for simplicity, let us assume that Φ = E[m] for some number m (e. g. if σ ∈ H^{1}(G_{K}, E)[m]). Then we look for another elliptic curve F over K such that

1. E[m] ≅ F [m] as group schemes over K (equivalently, as G_{K}-modules) and

2. η ∈ H^{1}(G_{K}, E[m]) ≅ H^{1}(G_{K}, F [m]) ↦ 0 ∈ H^{1}(G_{K}, F) under the obvious map on cohomology.

Then, σ is visible in the abelian surface A = (E ×F)/Φ, where Φ is E[m] embedded diagonally into E × F; again, this is an easy exercise in Galois cohomology (cf.[M]).

Pushing this perspective further, consider the following two exact sequences coming from the long exact Galois cohomology sequences associated to multiplication by m on E and F, respectively:

0 → E(K)/mE(K) → H^{1}(G_{K}, E[m]) → H^{1}(G_{K}, E)[m] → 0  (1.1)
0 → F (K)/mF (K) → H^{1}(G_{K}, F [m]) → H^{1}(G_{K}, F)[m] → 0.  (1.2)

We started out with an element σ ∈ H^{1}(G_{K}, E)[m] and lifted it to an element η ∈ H^{1}(G_{K}, E[m]), which we can also see as an element of H^{1}(G_{K}, F [m]). Since η ↦ 0 ∈ H^{1}(G_{K}, F), we see from the exact sequence (1.2) that η actually comes from an element of F (K)/mF (K)

– i. e. from a K-rational point P on F (in the terminology of [M], P explains σ).

To connect this to the geometric picture, recall that we have the exact sequence of abelian varieties over K

0 → E → A → F → 0;

it turns out that the inverse image of the point P which we just found under the surjection A → F is the homogeneous space C_{σ} of E corresponding to our original cohomology class σ ∈ H^{1}(G_{K}, E).

What is known.

We give a brief summary of the known facts regarding how to find abelian varieties in which a given element σ ∈ H^{1}(G_{K}, E) is visible:

• Johan de Jong proved that given any elliptic curve E over a number field K and any σ ∈ H^{1}(G_{K}, E), σ is visible in some abelian variety A (for a summary of the argument, see [Cr-M]). We note that the proof is non-constructive.

• Recently, William Stein exhibited an explicit construction which for any E, K and σ as above produces an abelian variety A in which σ is visible; in particular, he obtains a bound for the dimension of A depending on σ. We give a brief summary of the argument:

Given the data E, K, σ, there is a finite extension L of K such that σ splits over L – i. e. σ is mapped to zero under the restriction map H^{1}(G_{K}, E) → H^{1}(G_{L}, E). Let E_{L} be the base change of E to L, and denote the Weil restriction of scalars of E_{L} from L to K by A. A is an abelian variety over K of dimension [L : K], and it turns out that σ is visible in A.

• Recall that the Shafarevich-Tate group � (E/K) of an elliptic curve E over a number field K is the subgroup of H^{1}(G_{K}, E) given by

� (E/K) = ker(H^{1}(G_{K}, E(K̄)) → ∏_{v} H^{1}(G_{K_{v}}, E(K̄_{v}))),

where the direct product runs over all completions K_{v} of K, and G_{K_{v}} is the absolute Galois group of K_{v}. Geometrically, the principal homogeneous space C_{σ} corresponding to an element σ in the Shafarevich-Tate group has a point over any completion K_{v} of K – i. e. C_{σ} is locally trivial everywhere. � (E/K) is one of the fundamental arithmetic invariants of the elliptic curve E; a famous conjecture states that � (E/K) is always finite.

In [Cr-M], J. Cremona and B. Mazur consider the following scenario: Let E be an elliptic curve over ℚ with conductor N, and suppose that E is a strong Weil curve, i. e. we have an embedding E ↪ J_{0}(N). Experimental data then shows that a surprisingly large part of the Shafarevich-Tate group � (E/ℚ) of E is visible in J_{0}(N), at least for relatively small N. W. Stein has investigated this question experimentally for a larger range of conductors, even though a satisfactory explanation for this phenomenon remains to be found.

1.2.2 Visibility and Universal Elliptic Curves

It turns out that in many cases we can do even “better” than W. Stein’s construction outlined above. One instance is the following theorem, proved by B. Mazur in [M]:

Theorem 1 Let E be an elliptic curve over a number field K, and let σ ∈ � (E/K)[3] be a three-torsion element in the Shafarevich-Tate group of E. Then there are infinitely many abelian surfaces A in which σ is visible.

Note that W. Stein’s result says that every three-torsion element of � (E/K) is visible in an abelian threefold. Theorem 1 does not only give us better information on the dimension of A, but also establishes that there are infinitely many abelian surfaces in which a given σ ∈ � (E/K)[3] is visible.

Recall that in order to construct abelian surfaces in which σ as in Theorem 1 is visible, it is enough to find elliptic curves F such that F [3] ≅ E[3] as G_{K}-modules and such that a lift η ∈ H^{1}(G_{K}, E[3]) ≅ H^{1}(G_{K}, F [3]) of σ is mapped to 0 under H^{1}(G_{K}, F [3]) → H^{1}(G_{K}, F)[3]. It turns out that the existence of many such curves F can be guaranteed by the existence of many K-rational points on certain twists of (the base change to K of) the universal elliptic curve E(3). From the representation of E(3) as the Hesse cubic described at the beginning of this chapter, however, it is not difficult to deduce that the twists in question satisfy the Hasse principle, and that in fact the K-rational points are Zariski-dense in them. We will give an argument for visualizing two-torsion elements of � (E/K) in Section 3.4 below which is entirely analogous to Mazur’s argument for proving Theorem 1.

We note that Mazur’s proof is very much dependent on the particularities of the arithmetic of E(3) and its twists. When trying to generalize his argument to m-torsion elements of � (E/K) (or even W C(E/K)), for m > 3, one faces the problem that the universal (generalized) elliptic curve E(m) with full level-m-structure, as well as its twists, are much more complicated surfaces on which it will be much harder to establish the existence of many K-rational points. It also turns out that for heuristic reasons, for m > 5 one would not expect elements of � (E/K)[m] to be visible in abelian surfaces in general (since the modular curve X(m) then has positive genus).

We will concentrate in this dissertation on visualizing two-torsion elements of W C(E/K). When trying to apply Mazur’s method to this case, one faces the immediate problem that there is no universal elliptic curve with full level-two-structure, for reasons pointed out at the beginning of this chapter. It will turn out, however, that the compactification of the universal elliptic curve with full level-two-structure and an invariant regular differential which we construct in this dissertation has all the necessary properties for running a variant of Mazur’s argument, to visualize elements of W C(E/K)[2] in abelian surfaces (and we can even drop the hypothesis of local triviality of the cohomology classes we want to visualize). Moreover, we can make use of the arithmetic of E(4), the universal generalized elliptic curve with full level-four-structure, to prove that some elements of W C(E/K)[2] are visible in a special class of abelian surfaces.

1.3 Outline and Results

The first two parts of this dissertation will concentrate on constructing a compactification E(2, ω) of the universal elliptic curve F(2, ω) introduced above with good geometric and arithmetic properties. In Chapter 2, we will construct E(2, ω) locally by blowing up a “naive” compactification of F(2, ω) appropriately; the description we obtain is explicit, if complicated (in terms of a number of local coordinate charts), and it will allow us to establish that E(2, ω) has the properties we aim for.

In Chapter 3 we will carry out an alternative construction of E(2, ω), which will yield a simple geometric description. Our main result is Theorem 4 in Section 3.3:

Theorem. There is a compactification E(2, ω) over X(2, ω) of the universal elliptic curve F(2, ω) over the moduli scheme Y (2, ω) over ℤ[1/2], classifying elliptic curves with full level-two-structure and an invariant regular non-vanishing relative differential, such that

• X(2, ω) and E(2, ω) are both smooth, projective and integral over Spec(ℤ[1/2]), E(2, ω) is flat over X(2, ω), and the fiber of E(2, ω) over any point of Spec(ℤ[1/2]) is a regular and rational variety. In any characteristic (other than two), the singular fibers of E(2, ω) over X(2, ω) are of type I_{2}, I_{0}^{∗} or I_{1}^{∗}, referring to the Kodaira-Néron classification. Moreover, E(2, ω) is minimal over X(2, ω) in the appropriate sense.

• The two-torsion sections of the generic fiber extend to regular sections of E(2, ω) over X(2, ω) which are contained in the smooth locus of E(2, ω) over X(2, ω), and the translations by two-torsion sections extend to regular automorphisms of E(2, ω).

• The actions of SL_{2}(𝔽_{2}) ≅ S_{3} and of 𝔾_{m} on F(2, ω) and Y (2, ω) induced from their actions on the universal level structure and the universal differential extend to actions on E(2, ω) over X(2, ω).

Moreover, there is a sequence of morphisms

E(2, ω) → B → V → ℙ^{2}  (1.3)

over ℤ[1/2], where V is the blow-up of ℙ^{2} at four ℤ[1/2]-points in general position, and B is the pullback to V of the projective line bundle ℙ(O⊕O(1)) over ℙ^{2}. The first and the third morphism in (1.3) are birational morphisms, and the middle morphism has two regular sections over ℤ[1/2]. The group S_{4} (as well as 𝔾_{m}) acts on all schemes in (1.3), compatibly with the morphisms and the sections.

We can then use our construction of E(2, ω) to prove that two-torsion of the Weil-Châtelet group of elliptic curves is visible in abelian surfaces (see Theorem 5 in Section 3.4):

Theorem. Let E be an elliptic curve over a number field K, and σ ∈ H^{1}(G_{K}, E)[2] any element in the Weil-Châtelet group of E of order two. Then there are infinitely many abelian surfaces in which σ is visible.

Finally, in Chapter 4 we will investigate the arithmetic of E(4), the universal elliptic curve with full level-four-structure. Our analysis is based on a series of papers by T. Shioda, where he proved that E(4) is the desingularized Kummer surface of a certain abelian surface. It will come out of this analysis that it is not possible (at least in a straightforward way) to apply this representation of E(4) to questions of visualizing elements of W C(E/K) of exact order four; we can, however, use it to show that certain two-torsion elements of W C(E/K)[2] are visible in abelian surfaces half of whose four-torsion Galois-module we can fix, as expressed in Theorem 6 in Section 4.3:

Theorem. Let E be an elliptic curve over a number field K with a nontrivial K-rational two-torsion point P. Suppose that σ ∈ H^{1}(G_{K}, E(K̄))[2] can be lifted to a cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩) under the map on cohomology induced by the injection ⟨P ⟩ ↪ E. We note that H^{1}(G_{K}, ⟨P ⟩) ≅ Hom(G_{K}, ⟨P ⟩) ≅ K^{∗}/(K^{∗})^{2}, canonically; let η correspond to the class of a ∈ K^{∗} in K^{∗}/(K^{∗})^{2} under this isomorphism.

Then, if for some b ∈ K^{∗} the elliptic curves

E_{ab^{2}} : by^{2} = x^{3} + 4ax and E_{a^{3}b^{2}} : by^{2} = x^{3} + 4a^{3}x

both have positive rank over K, the following holds:

There are infinitely many elliptic curves F over K with distinct j-invariants such that

F (K̄)[4] ≅ ℤ/4ℤ × μ_{4}

as G_{K}-modules, and if we denote (0, −1) ∈ ℤ/4ℤ × μ_{4} by P′, then under the map on cohomology

H^{1}(G_{K}, ⟨P ⟩) ˜→H^{1}(G_{K}, ⟨P′⟩) → H^{1}(G_{K}, F (K̄))[2]

induced by the map ⟨P ⟩ ˜→⟨P′⟩ ↪ F, the cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩) is mapped to 0 ∈ H^{1}(G_{K}, F (K̄))[2]. In particular, σ is visible in the abelian surface (E × F)/(P, P′) for any of these curves F.

The Universal Family F(2, ω) and a Compactification

2.1 Introduction

As we mentioned in the previous chapter, the problem of classifying elliptic curves with full level-two-structure is not rigid and hence not representable in the category of schemes. We can, however, modify our moduli problem to classifying elliptic curves with full level-two-structure and a regular invariant relative differential; by an argument along the lines of [KM], we will show that this classification problem is indeed representable. We denote the resulting moduli space by Y (2, ω), and the corresponding universal elliptic curve by F(2, ω). Y (2, ω) is a scheme of relative dimension two over ℤ[1/2]; the fiber in characteristic zero is a two-dimensional variety.

The main purpose of this chapter is to construct a compactification of F(2, ω)/Y (2, ω), by which we mean a scheme X(2, ω) which is proper over ℤ[1/2] and which contains Y (2, ω) as a dense open subset, and a scheme E(2, ω) proper over X(2, ω), containing F(2, ω) as a dense open subset, such that

F(2, ω) ↪ E(2, ω)
↓  ↓
Y (2, ω) ↪ X(2, ω)

is a Cartesian diagram.

Apart from being interesting in its own right, we could hope to use the (characteristic zero fiber of) such a compactification to prove results on the visibility of two-torsion elements of the Shafarevich-Tate group (or, more generally, the Weil-Châtelet group) of elliptic curves in abelian surfaces, along the ideas which Mazur used in proving Theorem 1 as stated in the previous chapter. This will indeed turn out to be the case for the compactification we will construct. However, we will need to appeal to an alternative construction of E(2, ω) in order to ensure that for a number field K, certain twists of E(2, ω) ⊗ K have enough K-rational points, which will be accomplished in the next chapter.

Naturally we are only interested in compactifications with “nice” properties (where, to some degree, what one means by “nice” will depend on personal taste). These include both functoriality properties which the noncompactified family F(2, ω)/Y (2, ω) has by virtue of its universal property – e. g. extension of translation by two-torsion sections and of the actions of 𝔾_{m} and S_{3} ≅ SL_{2}(𝔽_{2}) – as well as properties controlling the singular fibers of E(2, ω) over X(2, ω), such as their shape, and regularity of E(2, ω).

The following is the list of properties which we will aim for E(2, ω)/X(2, ω) to have, and which we will establish in the course of this chapter. We preceed this list be stating two facts, one of which is an obvious consequence of what we have said already: The generic fiber of E(2, ω)/X(2, ω) is an elliptic curve with full level-two-structure; moreover, it will turn out that none of the fibers in E(2, ω) over X(2, ω)\Y (2, ω) is smooth.

1. Extension of two-torsion sections: The two-torsion points of the generic fiber give rise to sections of E(2, ω)/X(2, ω), extending to all nonsmooth fibers, and they are contained in the smooth locus of E(2, ω) over X(2, ω).

2. Control of fiber types: In any characteristic (other than two), the singular fibers of E(2, ω)/X(2, ω) are of type I_{2}, I_{0}^{∗} or I_{1}^{∗}, referring to the Kodaira-Néron classification.

3. The scheme E(2, ω) is smooth over Spec(ℤ[1/2]), and the fiber of E(2, ω) over any point of Spec(ℤ[1/2]) is a regular and rational variety.

4. E(2, ω) is integral as well as flat and projective over X(2, ω), and X(2, ω) is integral as well as smooth and projective over ℤ[1/2].

5. Extension of translations by two-torsion sections: Translations by the two-torsion sections extend to automorphisms of all of E(2, ω) over X(2, ω), including the singular fibers.

6. Extension of S_{3}-action and 𝔾_{m}-action: From their universal properties, F(2, ω)/Y (2, ω) come endowed with actions of SL_{2}(𝔽_{2}) ≅ S_{3} and of 𝔾_{m}; these actions extend to actions on E(2, ω)/X(2, ω).

7. Minimality: In characteristic zero, the structure map E(2, ω)/ℚ → X(2, ω)/ℚ is minimal, in the sense that E(2, ω)/ℚ contains no generically contractible surface whose contractible fibers lie in the fibers of the structure map (cf. the Introduction of [Mir]).

We will also comment on the following fact, which is a property of F(2, ω)/Y (2, ω) and is automatic from their universal property: “Twisting property in characteristic zero”: Given an elliptic curve F over a number field K, we get (up to conjugation) a homomorphism τ : G_{K} → GL_{2}(𝔽_{2}) ≅ S_{3} from the action of G_{K} on the two-torsion points of F. Viewing S_{3} as a subgroup of the automorphism group of the fibers of E(2, ω) and X(2, ω) over K, we can twist these two varieties compatibly by (the cohomology classes of) the cocycles in H^{1}(G_{K}, Aut(E(2, ω)/K)) and H^{1}(G_{K}, Aut(X(2, ω)/K)) induced by τ, respectively, to obtain varieties Ẽ/X̃ over K, say.

Then every elliptic curve F′ over K such that F [2] ≅ F′[2] as Galoismodules appears as the fiber of Ẽ/X̃ over a K-rational point of X̃.

We note that many of the properties listed here have their obvious analogues in the theory of universal generalized elliptic curves with full level-N-structure, for N ≥ 3. We note, however, that in contrast to the higher level-structures, at this point we do not know of a moduli interpretation of the compactification E(2, ω)/X(2, ω) which we are going to construct.

2.2 The Setup

2.2.1 The Universal Family F(2, ω)

We will start by exhibiting the moduli space (over ℤ[1/2]) classifying elliptic curves with full level-two-structure and a nowhere vanishing relative differential. The construction closely follows similar constructions in [KM, Chapter 2].

Let E be an elliptic curve over a base S, such that two is invertible everywhere on S, with full level-two-structure (i. e. an isomorphism α : ℤ/2ℤ × ℤ/2ℤ ˜→E[2] of group schemes over S) and a nowhere vanishing relative differential ω on E over S. Since two is invertible everywhere, Zariski-locally on S we can write E in (affine) Weierstrass form

y^{2} = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

and this fixes y up to a scalar. Translating x such that α(1, 0) has x-coordinate 0, we obtain the equation

y^{2} = x(x − A)(x − B),

where A is the x-coordinate of α(0, 1), say, and B the x-coordinate of α(1, 1), so A, B ≠ 0 and A ≠ B; this fixes x up to a scalar. Finally, substituting x by u^{2}x and y by u^{3}y for some suitable u ∈ 𝔾_{m}(S), we can arrange that ω is the differential dx/2y, and this fixes the equation completely.

We conclude that over ℤ[1/2], the moduli space Y (2, ω) classifying elliptic curves with full level-two-structure and a nowhere vanishing relative differential is given by Spec(ℤ[1/2][A, B]_{AB(A−B)}), and the corresponding universal elliptic curve F(2, ω) over Y (2, ω) is given by the equation

y^{2} = x(x − A)(x − B),

as above.

From their universal property we deduce that F(2, ω)/Y (2, ω) is endowed with canonical actions by S_{3} ≅ GL_{2}(ℤ/2ℤ) (acting on the level-two-structure) and by 𝔾_{m} (acting on the differential). It is this moduli space and universal elliptic curve for which we want to find a compactification.

It will turn out that for our purposes it will be more convenient to write Y (2, ω) and F(2, ω) in a slightly different form. Substituting x by Ax and y by Ay in the equation for F(2, ω) we gave above, and then writing r′ for A and s′ for B for reasons of more tasteful notation, we get that Y (2, ω) is given as Spec(ℤ[1/2][r′, s′]_{r′s′(r′−s′)}) and that F(2, ω) is given as a closed subscheme of ℙ_{ℤ[1/2][r′,s′]_{r′s′(r′−s′)}}^{2} by the homogeneous Weierstrass equation

Y^{2}Z = X(X − Z)(r′X − s′Z),

with the obvious level-two-structure, where the universal differential is still given by dx/2y in affine coordinates.

2.2.2 Initial Setup

The starting point for our construction of E(2, ω) and X(2, ω) is the scheme E over Spec(ℤ[1/2]) which is suggestively given by the equation

E : x_{3}^{2}(rx_{1} − sx_{2}) = tx_{1}x_{2}(x_{1} − x_{2})

in ℙ^{2} × ℙ^{2} in every characteristic not equal to two, where (x_{1} : x_{2} : x_{3}) are homogeneous coordinates of the first factor and (r : s : t) are coordinates of the second factor. (The exact definition of E would have to be given in terms of gluing the obvious affine coordinate charts over ℤ[1/2]; we leave the formulation to the reader, as our notation here should not cause any confusion.) We view the second factor as the base 𝔅 ≅ ℙ_{ℤ[1/2]}^{2}, with the structure morphism from E to 𝔅 being the projection onto the second factor of the ambient space.

It is easy to see that the dense open subset of E which lies over the dense open subset of 𝔅 given by t ≠ 0 ≠ rs(r − s) is isomorphic (over ℤ[1/2]) to the universal elliptic curve F(2, ω) over the moduli space Y (2, ω), the isomorphism being given by

(r : s : t) = (r′ : s′ : −(r′)^{2})
(x_{1} : x_{2} : x_{3}) = (s′/r′(X − Z) : X − s′/r′Z : Y);

E/𝔅 is hence a compactification of F(2, ω)/Y (2, ω), albeit without many of the properties we wish a compactification to have. We will modify E/𝔅 in the coming sections, by performing certain blow-ups, to obtain a compactification with suitable properties.

Before we finish this section with some remarks regarding our “plan of action”, we mention two basic lemmata:

Lemma 1 The four two-torsion sections of E over 𝔅 can be given (again suggestively) by

P_{1} : (x_{1} : x_{2} : x_{3}) = (1 : 0 : 0)
P_{2} : (x_{1} : x_{2} : x_{3}) = (1 : 1 : 0)
P_{3} : (x_{1} : x_{2} : x_{3}) = (0 : 1 : 0)
P_{4} : (x_{1} : x_{2} : x_{3}) = (0 : 0 : 1) (zero-section);

in particular, they extend the two-torsion sections of F(2, ω) over Y (2, ω) under the open immersion given above. The canonical action of SL_{2}(ℤ/2ℤ) ≅ S_{3} on F(2, ω) over Y (2, ω) extends to an action on E/𝔅, fixing the zerosection and permuting the three two-torsion sections, induced by

s_{(12)} : ((x_{1} : x_{2} : x_{3}), (r : s : t)) ⟼ ((x_{1} − x_{2} : −x_{2} : x_{3}), (−r : s − r : t))
s_{(13)} : ((x_{1} : x_{2} : x_{3}), (r : s : t)) ⟼ ((x_{2} : x_{1} : x_{3}), (s : r : t))

on the two generators (12), (13) of S_{3}, where (ij) interchanges the sections P_{i} and P_{j}. Finally, the action of 𝔾_{m} on F(2, ω) over Y (2, ω) also extends to an action of E over 𝔅, induced by

d_{μ} : ((x_{1} : x_{2} : x_{3}), (r : s : t)) ⟼ ((x_{1} : x_{2} : μ^{−1}x_{3}), (r : s : μ^{−2}t))

for any invertible global section μ on the base.

PROOF. The P_{i} can be seen to be two-torsion sections from the isomorphism given above; the rest follows by straightforward calculations.

Lemma 2 In any characteristic other than two (i. e. over every point of Spec(ℤ[1/2])), the fibers of E over the points (r : s : t) of 𝔅 with t ≠ 0 ≠ rs(r − s) are smooth elliptic curves; the fibers of E over points (r : s : t) with t ≠ 0 and rs(r − s) = 0, but not both r, s = 0, are of type I_{2} in the Kodaira-Néron classification of singular fibers of minimal proper regular models of elliptic curves over Dedekind domains, as described, for example, in [Sil2, Section IV. 8.]. Moreover, E is regular in, and translations by the two-torsion sections extend to, the open set lying over the locus in 𝔅 given by t ≠ 0, and not both r, s = 0.

PROOF. The first assertion easily follows from the remarks preceeding these two lemmata; the second follows from an easy computation. Regularity of the fibers of E on the Zariski-open sets in question will follow from the calculations in Section 2.4, and the extension of translation by two-torsion sections is again a straightforward calculation, which we will indicate in Section 2.5.

We also want to remark on some combinatorial data of the I_{2}-fibers of E: In the fibers over the points of 𝔅 where r = 0, s, t ≠ 0, the component intersecting the zero-section also intersects the section P_{1}, whereas the other component intersects the sections P_{2} and P_{3}; in the fibers over points with s = 0, r, t ≠ 0, the component intersecting the zero-section also intersects the section P_{3}, the other component intersects the sections P_{1} and P_{2}; finally, in the fibers over points in the base with r = s ≠ 0, t ≠ 0, the identity component intersects the section P_{2}, whereas the non-identity component intersects the sections P_{1} and P_{3}.

2.2.3 Preliminary Remarks

Our strategy for this chapter in constructing a compactification E(2, ω) over X(2, ω) of the family of elliptic curves F(2, ω) over Y (2, ω) with the properties we desire is as follows: We start out with the compactification we have – the scheme E over 𝔅 – and desingularize it by performing certain blow-ups on E as well as on 𝔅. We then prove that the blow-up we construct has the desired properties. This strategy is, of course, strongly inspired by the procedure of constructing Néron models for elliptic curves over Dedekind domains via the construction of minimal regular proper models.

Higher-dimensional elliptic schemes vs. elliptic surfaces.

That it is possible to desingularize at least the fiber of E over the generic point of ℤ[1/2] by blowing it up along certain subvarieties is clear from previous work. Most generally, we can refer to [Hir], of course; references more specific to our situation (and more explicit) are given by [Mir] and [Szy]. The latter two treat the process of desingularizing schemes given locally in Weierstrass form over the base; [Mir] concentrates on elliptic threefolds in characteristic zero, whereas [Szy] treats a rather more general class of scenarios, including high-dimensional bases with mixed characteristics, as we do here. They both prove the existence and the shape as well as certain properties of the desingularizations they construct.

Both [Mir] and [Szy] use generalizations of Tate’s algorithm for constructing minimal proper regular models of elliptic curves over Dedekind domains (as, for example, described in [Sil2, Section IV. 9.]) in order to desingularize the schemes at hand. However, a priori, the models they obtain lack many properties which minimal proper regular models over Dedekind domains have:

When working with elliptic surfaces over Dedekind domains (or over non-singular curves), one has the notion of a canonical minimal proper regular model, as described in [Sil2, Theorems III. 7. 7., III. 8. 4.] and [Sil2, Theorem IV. 4. 5., Proposition IV. 4. 6.]. These theorems, as well as other facts in the foundations of the theory which make crucial use of the fact we are working with elliptic surfaces, ensure that the models which Tate’s algorithm produces have certain “functorial” properties – e. g. an automorphism of the generic fiber automatically extends to an automorphism of the model, as for example translation by a generic section.

Even when working with elliptic three-folds over surfaces in characteristic zero, we do not have any more the notion of a “universal” minimal regular model in the above sense from which we can deduce functorial properties “for free”. Instead, we have to find a different line of argument to establish, say, properties (5) and (6) we listed in Section 2.1 for the model we are going to construct.

The scheme E vs. Weierstrass schemes.

For this reason, we choose a more explicit approach than [Mir], by starting out with the scheme E (which is not given in Weierstrass form over its base 𝔅) and performing blow-ups explicitly which we then show to desingularize the fibers over all points of ℤ[1/2].

This strategy has the advantage that we start out with a scheme E which already has the desired actions of S_{3} and 𝔾_{m}, as described in Lemma 1, and we will pick our blow-ups in a way which makes it easy enough to see that these actions will lift compatibly to the resulting schemes E(2, ω) and X(2, ω); the same goes for the extension of two-torsion sections.

Moreover, we will use the explicit description of the final blow-up to deduce that translations by two-torsion sections extend to automorphisms of the whole scheme E(2, ω). Starting out with E instead of the Weierstrass presentation provides a definite simplification here: The singular fibers over the lines in 𝔅 given by r = 0, s = 0, and r = s (outside their intersection and the points where t = 0) have already been taken care of, as is the content of Lemma 2; in order to desingularize E in the fibers over t = 0 and over r = s = 0, we only need two blow-ups, whereas Tate’s algorithm starting from a Weierstrass representation would need three.

Plan of the Chapter.

We take our approach at the expense of having to do quite a bit of “formulacrunching”. After going through some of the preliminaries in Section 2.3, we perform the necessary blow-ups of E and 𝔅 to obtain the schemes E(2, ω) and X(2, ω) explicitly in Section 2.4, which certainly has a very computational flavor. We refer the reader to the summary of results at the end of every subsection, where we also comment on some immediate consequences of our construction. In Section 2.5 we will establish the extension of translations by sections of order two to all of E(2, ω), before in Section 2.6 we turn to some geometric and arithmetic properties of the fiber in characteristic zero which are relevant to our study of visibility.

2.3 A Note on Blowing Up, on Multiplicities, and on Regularity

2.3.1 Blowing Up

Our references for blowing up schemes along subschemes are [EH, Section IV. 2.] and [Har, Section II. 7.].

In what follows, we will explicitly calculate certain blow-ups of the scheme E; the basis of these calculations is the following proposition, which seems to be well known (cf. [Sil2, Section IV. 7.]), but which we include for lack of a suitable reference. All the material needed here is in [EH, Section IV. 2.].

Proposition 1 Let X be a scheme over a field k given as X = Spec(A), A = k[x_{1}, x_{2}, …, x_{n}]/f, where f is an irreducible polynomial in x_{1}, x_{2}, ..., x_{n} – i. e. X is a (reduced) hypersurface in 𝔸^{n}(k). Further suppose that the subscheme Y of 𝔸^{n}(k) given by x_{1} = x_{2} = … = x_{m} = 0, n ≥ m ≥ 2, is also a closed subscheme of X. Then Bl_{Y}X, the blow-up of X along Y, can be described as follows:

• Bl_{Y}X is covered by open affines U^{(1)}, U^{(2)}, …, U^{(m)}, where for 1 ≤ i ≤ m, U^{(i)} = Spec(A^{(i)}), A^{(i)} = k[x_{1}^{(i)}, x_{2}^{(i)}, …, x_{n}^{(i)}]/f^{(i)}, with f^{(i)} a polynomial in x_{1}^{(i)}, x_{2}^{(i)}, …, x_{n}^{(i)};

• f^{(i)} can be calculated as follows: Into f, substitute x_{j} = x_{j}^{(i)} for j = i

or for j > m; also, substitute x_{j} = x_{i}^{(i)}x_{j}^{(i)} for j ≠ i, 1 ≤ j ≤ m,

to obtain f_{∗}^{(i)}, a polynomial in x_{1}^{(i)}, x_{2}^{(i)}, …, x_{n}^{(i)}. Then factor f_{∗}^{(i)} = (x_{i}^{(i)})^{e_{i}}f^{(i)} such that f^{(i)} is not divisible by x_{i}^{(i)}; this gives the desired polynomial f^{(i)}.

• The transition maps between the U^{(i)} are as follows: For i ≠ j, there is an isomorphism (A^{(i)})_{x_{j}^{(i)}} ˜→(A^{(j)})_{x_{i}^{(j)}}, inducing an isomorphism be-

tween the corresponding opens of U^{(i)} and U^{(j)}, which is given by

\begin{matrix}x_{l}^{(i)} & ↦ & x_{l}^{(j)}/x_{i}^{(j)} & & for l ≠ i, j; 1 ≤ l ≤ m; \\ x_{i}^{(i)} & ↦ & x_{i}^{(j)}x_{j}^{(j)} & ; \\ x_{j}^{(i)} & ↦ & 1/x_{i}^{(j)} & ; \\ x_{l}^{(i)} & ↦ & x_{l}^{(j)} & & for l > m. \end{matrix}

• The map Bl_{Y}X → X can be described locally on the U^{(i)} as follows: the map U^{(i)} → X is induced by the ring homomorphism A → A^{(i)} given by

x_{l} ↦ x_{l}^{(i)}  for l = i or l > m;
x_{l} ↦ x_{l}^{(i)}x_{i}^{(i)} for l ≠ i, 1 ≤ l ≤ m.

PROOF. By setting A = k in Example IV-17 of [EH], the proposition holds for m = n and f = 0. More generally, setting A = k[x_{1}, …, x_{m−n}] in the same example (and renumbering the indices in the proposition) yields the proposition for f = 0, any m and n.

In order to deduce the proposition for irreducible f, we use Proposition IV-21 of [EH], which says the following:

Proposition 2 Let X be any scheme, Y ⊂ X a subscheme and ϕ : Bl_{Y}X → X the blow-up of X along Y. Let ν : X′ → X be any morphism and set Y′ = ν^{−1}(Y) ⊂ X′. If W is the closure, in the fiber product X′ ×_{X} Bl_{Y}X, of the inverse image π_{1}^{−1}(X′\Y′), then π_{1} : W → X′ is the blow-up of X′ along Y′. (Here, π_{1} denotes the projection of the fiber product to its first factor; π_{1}^{−1}(X′\Y′) denotes (X′\Y′) ×_{X} Bl_{Y}X.)

We apply this proposition with

X = Spec(k[x_{1}, …, x_{n}])
Y = Spec(k[x_{1}, …, x_{n}]/(x_{1}, …, x_{m}))
X′ = Spec(k[x_{1}, …, x_{n}]/f),

ν being the closed immersion of X′ into X. Y′ is, under the hypotheses of the proposition, the same as Y, but seen as a closed subscheme of X′. We now compute the closure of π_{1}^{−1}(X′\Y′) in X′ ×_{X} Bl_{Y} X; note that by the above, we know a description of Bl_{Y} X in terms of open affines V^{(i)} = Spec(B^{(i)}), for 1 ≤ i ≤ m, where B^{(i)} = k[x_{1}^{(i)}, x_{2}^{(i)}, …, x_{n}^{(i)}], with the gluing maps as described.

Computing the scheme-theoretic closure (which exists a priori, since all the schemes involved are noetherian and hence all maps involved are quasicompact and quasi-separated) can be done locally on the V^{(i)}. By rightexactness of tensor products, it follows that the inverse image of X′\Y′ under π_{1} in X′ ×_{X} V^{(i)} is given by Spec((B^{(i)})_{x_{i}^{(i)}}/f_{∗}^{(i)}), with its obvi-

ous map to V^{(i)}; and it is now clear that its closure in V^{(i)} is given by Spec(k[x_{1}^{(i)}, x_{2}^{(i)}, …, x_{n}^{(i)}]/f^{(i)}), with f^{(i)} as stated above, and the proposition follows.

Remark. We note that the same argument certainly works verbatim if we take k to be any UFD (e. g. ℤ[1/2]) instead of a field.

2.3.2 Multiplicity

We want to recall the definition of multiplicity of an irreducible component of a scheme of finite type over a field; the definition we give is taken from [BLR], section 9. 1., Definition 3:

Definition 2 Let X be a scheme of finite type over a field k; denote by X_{1}, …, X_{r} the (reduced) irreducible components of X, and for i = 1, …, r, let η_{i} ∈ X be the generic point corresponding to X_{i}. The multiplicity m_{i} of X_{i} in X is then the length of the artinian local ring O_{X,η_{i}}:

m_{i} = l(O_{X,η_{i}}).

We will refer to this definition when computing multiplicities of components of singular fibers of our (generalized) elliptic curve E and its blow-ups. The “base case” for us is the multiplicity of the affine line in 𝔸^{2}(k) = Spec(k[x, y]) given by x^{n} = 0, i. e. calculating the length of the artinian ring R = (k[x, y]/x^{n})_{(x)}. Note that R is isomorphic to the ring k(y)[x]/x^{n} with length n, so the line has multiplicity n, as it should.

2.3.3 Regularity and Smoothness

Regularity.

We will want to check whether the schemes we obtain are regular or not. It turns out that we will be reduced to checking regularity on schemes of finite type over a perfect field k; by Proposition 6. 6. 7. of [EGA IV], it is hence equivalent to check whether their base changes to k̄ are regular. These base changes will, in fact, always be varieties in the sense of [Har], i. e. integral separated schemes of finite type over an algebraically closed field; this will follow from Proposition 7. 16. in [Har, Section II. 7.] and from our constructions.

Since on varieties it is enough to check regularity at closed points (cf. [Mats, Theorem 19. 3.]), we can use the following criterion, taken from the definitions and Theorem 5. 1. of [Har, Section I. 5.], to check regularity locally on open affines:

Lemma 3 Let Y ⊆ 𝔸^{n} be an affine variety over an algebraically closed field k; let f_{1}, …, f_{t} ∈ A = k[x_{1}, …, x_{n}] be a set of generators for the ideal of Y. Let P be a closed point of Y; then the local ring O_{P,Y} is regular if and only if the rank of the matrix |(∂f_{i}/∂x_{j})(P)| is n − r, where r is the dimension of Y.

We will, in fact, apply this lemma only to affine hypersurfaces, i. e. for t = 1, r = n − 1, in which case we only have to check whether at least one of the partial derivatives of f is non-zero at P.

Note: In what follows, we will drop the requirement of the base field being algebraically closed when speaking about varieties, so we can, e. g., talk about varieties over ℚ.

Smoothness.

We just mention the following fact:

Suppose that X is an S-scheme, where S is a noetherian scheme with all residue fields perfect, and suppose that X is projective over S; also, suppose that for any point s ∈ S, the fiber X_{s} is a regular variety over the residue field k(s) of s. Then it follows from the “Jacobian criterion” for regularity of varieties which we just gave and from Proposition 7 of [BLR, Section 2. 2.] that X is, in fact, smooth over S.

2.4 Performing the Blow-ups

This section’s aim is the following: Starting with the schemes E over 𝔅 introduced previously, which constitute a compactification of the universal family F(2, ω) over the moduli scheme Y (2, ω), we will perform certain blow-ups (both on E and on 𝔅) in order to desingularize E. The result will be a scheme which we will call E(2, ω) over a base X(2, ω), all over Spec(ℤ[1/2]); E(2, ω)/X(2, ω) will be another compactification of F(2, ω)/Y (2, ω) by virtue of the fact that all the subschemes along which we are going to blow up E and 𝔅 are contained in E\F(2, ω) and 𝔅\Y (2, ω), respectively. Our blow-ups will be chosen such that E(2, ω) will be smooth over Spec(ℤ[1/2]) – in the hope that, in analogy to the construction of minimal proper regular models of elliptic curves over Dedekind domains, such a blow-up has the properties which we desire (and this hope will be justified).

The blow-ups which we are going to perform are very much inspired by Tate’s algorithm for determining the singular fibers of the Néron model of an elliptic curve over a Dedekind domain, as in the exposition in [Sil2, Section IV. 9.], and by its generalization to other bases as in [Mir] and [Szy]. Note that we only have to “improve” the portion of E over the subschemes of 𝔅 given by t = 0 and by r = s = 0; since those two subschemes do not meet, we can analyze the corresponding blow-ups independently.

In the following, we will first restrict our attention to the fibers of E and 𝔅 over the prime fields ℚ and 𝔽_{p}, for p ≠ 2. (We note, however, that everything we do also works in characteristic two – it just doesn’t have to do anything with elliptic curves!) We will note later that everything we have done works, in fact, over Spec(ℤ[1/2]).

Recall that E is given by the equation

x_{3}^{2}(rx_{1} − sx_{2}) = tx_{1}x_{2}(x_{1} − x_{2})

inside ℙ^{2} × ℙ^{2}, where the first factor has homogeneous coordinates (x_{1} : x_{2} : x_{3}), and the second factor has homogeneous coordinates (r : s : t) and is seen as the base 𝔅.

In order to explicitly compute the blow-ups we want to perform, we will make use of Proposition 1 above (if with a slightly less formal (and more suggestive) notation along the lines of [Sil2, Section IV.7.]). Along the way we will also compute the shape of the emerging singular fibers over closed points of the base, and determine which components intersect the lifts of the two-torsion sections P_{i}; we will also show that the actions of S_{3} and of 𝔾_{m}, as introduced above, lift to our blow-ups.

Finally, we will observe that the resulting scheme is integral as well as smooth and projective over Spec(ℤ[1/2]), and that E(2, ω) is flat over X(2, ω). Summarizing our construction in this section, we will obtain the following

Proposition 3 There is a compactification E(2, ω) over X(2, ω) of F(2, ω) over Y (2, ω), all over Spec(ℤ[1/2]), which has properties (1), (2), (3), (4) and (6) as listed in Section 2.1.

In addition, our calculations will give us an explicit (if involved) description of the scheme E(2, ω), which we can use in our further analysis in subsequent sections.

We will first perform the blow-ups which desingularize our scheme E in the fibers over the line t = 0 in 𝔅. When desingularizing the fiber over the point r = s = 0 in 𝔅, it turns out that we have to blow up 𝔅 along that point first to obtain the variety 𝔅′, pull back our family of elliptic curves to 𝔅′ and then desingularize the pull-back, if we want our final family of curves to be flat over the base; this will be carried out in the second half of this section. All our calculations will be done on affine coordinate charts.

2.4.1 The Fibers over the Line t = 0

Program.

Recall that for the time being, we work over a perfect field.

The fibers of E over the line t = 0 in 𝔅 each consist of a double line (V : x_{3}^{2} = 0) intersecting a single line (L_{4} : rx_{1} = sx_{2}) transversely (the reasons for the numbering of fiber components will become clear in the next chapter). It turns out that we can desingularize E on these fibers by performing two successive blow-ups:

• First we blow up E along the irreducible surface W_{1} ≅ ℙ^{1} × ℙ^{1}, given by t = x_{3} = 0, to obtain E_{1}; W_{1} consists of the (reduced) lines V in the fibers over t = 0. We then see that there is an irreducible surface W_{2} inside E_{1} such that the blow-down map E_{1} → E induces an isomorphism from W_{2} to W_{1}.

• Secondly, we blow up E_{1} along W_{2} to obtain E_{2}.

After performing these blow-ups we will indicate the shape of the resulting fibers of E_{2} over the line (t = 0) ⊂ 𝔅, and discuss how to lift the two torsion sections and the actions by S_{3} and G_{m} to E_{2}.

We now proceed to compute and analyze the blow-ups we have described, on those coordinate charts of E which are affected by them.

The coordinate chart r ≠ 0, x_{1} ≠ 0.

We introduce affine coordinates x_{2,1} = x_{2}/x_{1}, x_{3,1} = x_{3}/x_{1}, s_{r} = s/r, t_{r} = t/r. The equation for E in this coordinate chart becomes

x_{3,1}^{2}(1 − s_{r}x_{2,1}) = t_{r}x_{2,1}(1 − x_{2,1}).  (2.1)

Note that the affine scheme given by this equation has singular points, for example at (x_{2,1}, x_{3,1}, s_{r}, t_{r}) = (0, 0, s_{r}, 0) for any value of s_{r}, as follows easily from the criterion given in Lemma 3.

We hence perform a blow-up along the subscheme W_{1}, given on this coordinate chart by the ideal (t_{r}, x_{3,1}). Applying Proposition 1, we obtain two coordinate charts for the resulting blow-up E_{1}:

• Chart ℭ_{1}: Set x_{3,1} = B′_{2,3}t_{r} in equation (2.1) – the reason for the naming of variables will become clear in the next chapter – and divide both sides by t_{r} to obtain the equation

B_{2,3}^{′2}t_{r}(1 − s_{r}x_{2,1}) = x_{2,1}(1 − x_{2,1})

inside 𝔸^{4}[x_{2,1}, B′_{2,3}, s_{r}, t_{r}]. Using Lemma 3, one easily checks that the affine variety given by this equation is regular. The fiber over t_{r} = 0 (for any value for s_{r}) in this coordinate chart consists of two disjoint single lines: L_{1} : x_{2,1} = 0, L_{2} : x_{2,1} = 1.

• Chart ℭ_{2}: Set t_{r} = B′_{3,2}x_{3,1} in equation (2.1) and divide both sides by x_{3,1} to obtain the equation

x_{3,1}(1 − s_{r}x_{2,1}) = B′_{3,2}x_{2,1}(1 − x_{2,1})  (2.2)

inside 𝔸^{4}[x_{2,1}, x_{3,1}, s_{r}, B′_{3,2}]. This affine variety is singular at the point (x_{2,1}, x_{3,1}, s_{r}, B′_{3,2}) = (1, 0, 1, 0).

It turns out that each fiber of E_{1} over a point of the line t = 0 in 𝔅 contains precisely one line of multiplicity two on the chart ℭ_{2}. We denote by W_{2} the subscheme of E_{1} consisting of these (reduced) double lines.

On the chart ℭ_{2}, W_{2} is given by the ideal (x_{3,1}, B′_{3,2}). It can be easily checked that W_{2} maps isomorphically to W_{1} under the blow-down map E_{1} → E. We continue by blowing up E_{1} along W_{2}, to obtain the following two coordinate charts over ℭ_{2} of the resulting variety E_{2}:

– Chart ℭ_{21}: Set x_{3,1} = Γ′_{0,01}B′_{3,2} in equation (2.2) and divide both sides by B′_{3,2} to obtain the equation

Γ′_{0,01}(1 − s_{r}x_{2,1}) = x_{2,1}(1 − x_{2,1})

inside 𝔸^{4}[x_{2,1}, Γ′_{0,01}, s_{r}, B′_{3,2}] (no condition on B′_{3,2}). Looking at partial derivatives, this affine variety is easily seen to be regular.

– Chart ℭ_{22}: Set B′_{3,2} = Γ′_{01,0}x_{3,1} in equation (2.2) and divide both sides by x_{3,1} to obtain the equation

(1 − s_{r}x_{2,1}) = Γ′_{01,0}x_{2,1}(1 − x_{2,1})

inside 𝔸^{4}[x_{2,1}, x_{3,1}, s_{r}, Γ′_{01,0}] (no condition on x_{3,1}) – again, this gives a regular affine variety.

Note that W_{2} ⊂ E_{1} does not intersect the coordinate chart ℭ_{1}, so we have to perform the second blow-up only on the chart ℭ_{2}.

The resulting fibers over t_{r} = 0. Let us compute the fibers over the line t_{r} = 0 in the charts ℭ_{21} and ℭ_{22}. By using the gluing maps between different coordinate charts which are induced by the gluing maps between the standard open affines of projective space and by the gluing maps mentioned in Proposition 1, we can easily detect when the same component turns up on several charts.

Tracing back the maps to the original coordinate chart, the components of the fiber over a closed point (s_{r}, t_{r}) = (s_{r}, 0) in ℭ_{21} correspond to minimal primes of the ring

R = k[x_{2,1}, Γ′_{0,01}, s_{r}, B′_{3,2}]/(Γ′_{0,01}(1 − s_{r}x_{2,1}) − x_{2,1}(1 − x_{2,1}), (B′_{3,2})^{2}Γ′_{0,01}),

where we substitute some number for s_{r} corresponding to the particular fiber we are looking at, and k denotes the field over which we are working. Depending on the value of s_{r}, there are three or four components, corresponding to

• the (images in R of the) prime ideals (Γ′_{0,01}, x_{2,1}), (Γ′_{0,01}, x_{2,1} − 1): these correspond to single lines – in fact, they correspond to the lines L_{1} and L_{2}, respectively, which we have already seen on the chart ℭ_{1};

• the prime ideal 𝔓 = (B′_{3,2}, Γ′_{0,01}(1−s_{r}x_{2,1})−x_{2,1}(1−x_{2,1})) if s_{r} ≠ 1, or the two prime ideals 𝔓′ = (B′_{3,2}, Γ′_{0,01} − x_{2,1}) and 𝔓′′ = (B′_{3,2}, 1 − x_{2,1}) if s_{r} = 1. These ideals all correspond to lines with multiplicity two. 𝔓 and 𝔓′ correspond to lines which under the blow-down map E_{2} → E inject into the surface W_{1} ⊂ E we described above, so we denote these lines by V; the line corresponding to 𝔓′′ in the fiber over s_{r} = 1 we call V_{(1:1)}.

Aside: We will indicate one multiplicity computation here for the chart ℭ_{21}; all subsequent such computations can be done in similar ways and will not be spelt out. By Definition 2, in order to calculate the multiplicity of the line corresponding to 𝔓 in the case s_{r} ≠ 1, we have to compute the length of the artinian ring R_{𝔓}, where the ring R is as described above. To this end, note that the images of neither 1 − s_{r}x_{2,1} nor 1 − x_{2,1} in R are in 𝔓, and that they hence become units in the ring R_{𝔓}; it follows easily that R_{𝔓} ≅ (k[B′_{3,2}, x_{2,1}]/(B′_{3,2})^{2}x_{2,1})_{(B′_{3,2})} ≅ (k[B′_{3,2}, x_{2,1}]/(B′_{3,2})^{2})_{(B′_{3,2})}, so the corresponding line has multiplicity two by the “base case” we discussed at the end of Section 2.3.2.

In the chart ℭ_{22}, the fibers over t_{r} = 0 can be computed in an analogous way; in an “informal” notation, we obtain the following:

• the proper transform of the line L_{4}, namely the single line given by Γ′_{01,0} = 0, s_{r}x_{2,1} = 1 – if s_{r} ≠ 0;

• the double line x_{3,1} = 0, 1 − s_{r}x_{2,1} = Γ′_{01,0}x_{2,1}(1 − x_{2,1}), which is the proper transform of V, if s_{r} ≠ 1; or the two double lines V : x_{3,1} = 0, Γ′_{01,0}x_{2,1} = 1 and V_{(1:1)} : x_{3,1} = 0, x_{2,1} = 1 if s_{r} = 1.

The coordinate chart r ≠ 0, x_{2} ≠ 0.

Much of what follows is analogous to the analysis of the coordinate chart (2.1). We introduce affine coordinates x_{1,2} = x_{1}/x_{2}, x_{3,2} = x_{3}/x_{2}, s_{r} = s/r, t_{r} = t/r as before, and the equation for E in this coordinate chart becomes

x_{3,2}^{2}(x_{1,2} − s_{r}) = t_{r}x_{1,2}(x_{1,2} − 1).  (2.3)

Note that the affine variety given by this equation again has singular points, for example at (x_{1,2}, x_{3,2}, s_{r}, t_{r}) = (0, 0, s_{r}, 0) for any value of s_{r}.

In analogy to the first coordinate chart we looked at, we perform a blowup along the subscheme W_{1}, given by the ideal (t_{r}, x_{3,2}) on this chart, to obtain two new coordinate charts for E_{1}:

• Chart ℭ_{3}: Set x_{3,2} = B_{0,3}t_{r} in equation (2.3) and divide both sides by t_{r} to obtain the equation

B_{0,3}^{2}t_{r}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, B_{0,3}, s_{r}, t_{r}]. The affine variety given by this equation can be checked to be regular using Lemma 3; the fiber over t_{r} = 0 (any s_{r}) consists of two disjoint single lines: L_{3} : x_{1,2} = 0, and the line x_{1,2} = 1, which is actually L_{2}.

• Chart ℭ_{4}: Set t_{r} = B_{3,0}x_{3,2} in equation (2.3) and divide both sides

by x_{3,2}:

x_{3,2}(x_{1,2} − s_{r}) = B_{3,0}x_{1,2}(x_{1,2} − 1)  (2.4)

inside 𝔸^{4}[x_{1,2}, x_{3,2}, s_{r}, B_{3,0}]. The affine variety given by this equation has two singular points: (x_{1,2}, x_{3,2}, s_{r}, B_{3,0}) = (0, 0, 0, 0), (1, 0, 1, 0).

As above, we perform one more blow up along the subscheme W_{2}, corresponding on this coordinate chart to the ideal (x_{3,2}, B_{3,0}); we obtain the following two new coordinate charts for E_{2}:

– Chart ℭ_{41}: Set x_{3,2} = Γ′′_{0,01}B_{3,0} in equation (2.4) and divide both sides by B_{3,0} to obtain

Γ′′_{0,01}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, Γ′′_{0,01}, s_{r}, B_{3,0}] (no condition on B_{3,0}); this gives a regular affine variety.

– Chart ℭ_{42}: Set B_{3,0} = Γ′′_{01,0}x_{3,2} in equation (2.4) and divide both sides by x_{3,2} to obtain

(x_{1,2} − s_{r}) = Γ′′_{01,0}x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, x_{3,2}, s_{r}, Γ′′_{01,0}] (no condition on x_{3,2}); again, this affine variety is regular.

The resulting fibers over t_{r} = 0. The computation of the fibers over t_{r} = 0 in the charts ℭ_{41} and ℭ_{42} is very similar to the computations for the charts ℭ_{21} and ℭ_{22} above. The fibers in ℭ_{41} consist of

• the single lines L_{2} : Γ′′_{0,01} = 0, x_{1,2} = 1; L_{3} : Γ′′_{0,01} = x_{1,2} = 0;

• the double line V : B_{3,0} = 0, Γ′′_{0,01}(x_{1,2}−s_{r}) = x_{1,2}(x_{1,2}−1) if s_{r} ≠ 0, 1, or the two double lines V : B_{3,0} = 0, Γ′′_{0,01} = x_{1,2} − 1 (or Γ′′_{0,01} = x_{1,2}) and V_{(1:0)} (or V_{(1:1)}) : B_{3,0} = 0, x_{1,2} = s_{r} if s_{r} = 0 (or s_{r} = 1, respectively).

Likewise, in ℭ_{42}, the fibers over t_{r} = 0 consist of

• the single line L_{4} : Γ′′_{01,0} = 0, x_{1,2} = s_{r};

• the double line V : x_{3,2} = 0, x_{1,2} − s_{r} = Γ′′_{01,0}x_{1,2}(x_{1,2} − 1) if s_{r} ≠ 0, 1, or the two double lines V_{(1:0)} (or V_{(1:1)}) : x_{3,2} = 0, x_{1,2} = s_{r} and V : x_{3,2} = 0, 1 = Γ′′_{01,0}(x_{1,2} − 1) (or 1 = Γ′′_{01,0}x_{1,2}) if s_{r} = 0 (or s_{r} = 1).

The remaining coordinate charts.

In addition to the coordinate charts (2.1) and (2.3) of E, we have to deal with

• the affine opens given by s ≠ 0, x_{1} ≠ 0 and s ≠ 0, x_{2} ≠ 0: the analysis of the blow-ups on these charts are, in fact, entirely analogous to the analysis of the charts (2.3) and (2.1), respectively, if we simply switch the roles of r and s and of x_{1} and x_{2}. Alternatively, we will give an argument below that (13) ∈ Aut(E) can be lifted to an automorphism of E_{2}, and this maps the open set of E_{2} over the charts s ≠ 0, x_{1} ≠ 0 and s ≠ 0, x_{2} ≠ 0 of E isomorphically to the open set of E_{2} over the charts r ≠ 0, x_{2} ≠ 0 and r ≠ 0, x_{1} ≠ 0 of E which we just dealt with.

• the affine opens where x_{3} ≠ 0; since those opens do not intersect the subscheme t = x_{3} = 0 along which we are blowing up, they remain unchanged. Note: For future reference, we denote the coordinate chart given by r ≠ 0, x_{3} ≠ 0 by ℭ_{5}.

Two-torsion sections, fiber types and the actions of S_{3} and 𝔾_{m}.

We are now ready to put together our analyses of the coordinate charts of E_{2} which we just computed: Two-torsion sections. We want to lift the two-torsion sections P_{1}, P_{2}, P_{3}, P_{4}, as defined earlier, to sections of E_{2} over 𝔅, and determine which components of the singular fibers of E_{2} over t = 0 their images meet. By abuse of language, we will sometimes identify the section P_{i} with its scheme-theoretic image.

We note that by Proposition 2 above, the proper transform of the schemetheoretic image of the two-torsion section P_{i} to E_{2} is the blow-up of P_{i} along its intersection with W_{1}, blown up in turn along its intersection with W_{2}. The section P_{4} does not meet W_{1}, hence it remains unchanged. For i = 1, 2, 3, P_{i} ∩ W_{1} is, in fact, a Cartier subscheme of P_{i} in the sense of [EH, Definition IV-15], so it follows easily from [EH, Definition IV-16 and Lemma IV-19] that the blow-up of P_{i} along P_{i} ∩ W_{1} is isomorphic to P_{i}; likewise for the second blow-up. We hence deduce that the two-torsion sections lift to sections of E_{2} over 𝔅. In particular, we obtain:

• The zero-section P_{4} still meets the single line L_{4} in all fibers of E_{2} over t = 0.

• In ℭ_{1}, the lift of P_{1} can be seen to be given by the equations x_{2,1} = B′_{2,3} = 0, and the lift of P_{2} is given by x_{2,1} = 1, B′_{2,3} = 0; hence, in the fibers over t = 0, P_{1} meets the component L_{1}, P_{2} meets the component L_{2}.

• In ℭ_{3}, the lift of P_{3} is given by x_{1,2} = B_{0,3} = 0; hence, P_{3} meets the fibers of E_{2} over t = 0 in the component L_{3}.

Note that in what follows, we will drop the linguistic distinction between the sections P_{i} and their lifts to E_{2}; this should not cause any confusion. Fiber types. Summarizing our analysis of E_{2}, and looking at how the components of the fibers of E_{2} over the line (t = 0) ⊂ 𝔅 intersect, we obtain the following picture:

• The fibers of E_{2} over the points of 𝔅 with t = 0, rs(r−s) ≠ 0 are of the type called I_{0}^{∗} in the Kodaira-Néron classification; V is the double line, L_{1}, L_{2}, L_{3}, L_{4} are the single lines, intersecting one two-torsion section each as described above.

• The fibers over the points t = 0 = rs(r − s) of 𝔅 are each of type I_{1}^{∗}; the two double lines over t = 0, (r : s) = (0 : 1)/(1 : 1)/(1 : 0) are V and V_{(0:1)}/V_{(1:1)}/V_{(1:0)}. The four single lines are again L_{1}, L_{2}, L_{3}, L_{4}. Over r = 0, L_{1} and L_{4} intersect V_{(0:1)}, and L_{2} and L_{3} meet V; over r = s, L_{2} and L_{4} intersect V_{(1:1)}, whereas L_{1} and L_{3} intersect V; and over s = 0, L_{3} and L_{4} intersect V_{(1:0)}, whereas L_{1} and L_{2} meet V. Again, each of the single lines meets one two-torsion section, as described above.

Extending the actions of S_{3} and 𝔾_{m}. Recall the universal property of blow-ups given in [EH, Definition IV-16]:

Let X be a scheme, Y ⊂ X a subscheme, and ϕ : Bl_{Y} X → X the blowdown map. Then ϕ^{−1}(Y) is a Cartier subscheme of Bl_{Y} X, and ϕ is universal with respect to the following property: If f : W → X is any morphism such that f^{−1}(Y) is a Cartier subscheme in W, there is a unique morphism g : W → Bl_{Y} X such that f = ϕ ∘ g.

Applying this universal property to X = E, Y = W_{1}, W = E_{1} = Bl_{W_{1}}E and f = s∘ϕ for any s ∈ S_{3} ⊂ Aut(E), and noting that W_{1} is invariant under the action of S_{3} on E, we deduce that we can lift s to an automorphism of E_{1} over E. Applying the same argument to the second blow-up, noting that W_{2} is invariant under the induced action of S_{3} on E_{1} (since it is mapped isomorphically onto W_{1} under the blow-down map E_{1} → E), we conclude that the action of S_{3} lifts to E_{2}. Finally, a similar argument applied to (the scheme-theoretic images of) the two-torsion sections P_{i} gives that the action of S_{3} on E_{2} still permutes the P_{i} as before.

An analogous argument shows that the 𝔾_{m}-action lifts to an action on E_{2} over 𝔅, too. It is, in fact, an easy task to make this action explicit on the coordinate charts by simply going through the definitions of the new coordinates; an easier description will be given in the next chapter.

2.4.2 The Fibers over the Point r = s = 0

Program.

In the variety E_{2} which we constructed in the previous subsection, the fiber over the point (r : s : t) = (0 : 0 : 1) ∈ 𝔅 consists of three single lines, L′_{1} : x_{2} = 0, L′_{2} : x_{1} = x_{2}, and L′_{3} : x_{1} = 0, all meeting in the point (x_{1} : x_{2} : x_{3}) = (0 : 0 : 1), which is a singular point of E_{2}. (Note that the fiber over (x_{1} : x_{2} : x_{3}) = (0 : 0 : 1) remains unchanged under the blow-ups of E we performed earlier.)

In order to desingularize this fiber to obtain a flat family of curves, we have to first blow up 𝔅 along the point (r : s : t) = (0 : 0 : 1) and pull back E_{2} to the new base 𝔅′ to obtain a family of curves E_{3}, which we then desingularize by blowing it up along appropriate subschemes. This process corresponds to ensuring that the discriminant locus on the base (i. e. the divisor on the base over which the fibers of our family are singular) has only normal crossings, as explained in [Mir] and [Szy]. It turns out that we can desingularize E_{3} by performing two more successive blow-ups. Finally, we again discuss the resulting fiber types and the lifting of the two-torsion sections, as well as lifting the actions of S_{3} and 𝔾_{m} to the resulting variety.

Blowing up the base.

We can specialize to the coordinate chart of 𝔅 where t ≠ 0; we introduce affine coordinates r_{t} = r/t, s_{t} = s/t. E_{2} over this chart has the same equation as E, namely

x_{3}^{2}(r_{t}x_{1} − s_{t}x_{2}) = x_{1}x_{2}(x_{1} − x_{2}).

By Proposition 1, the blow-up of the affine open t ≠ 0 of 𝔅 at the point r_{t} = s_{t} = 0 is covered by the following two coordinate charts, along with the maps to Spec(k[r_{t}, s_{t}]) as indicated (here k again denotes the field over which we are working):

𝔅_{1} = Spec(k[ρ_{σ}, s_{t}]), r_{t} = ρ_{σ}s_{t};
𝔅_{2} = Spec(k[σ_{ρ}, r_{t}]), s_{t} = σ_{ρ}r_{t}.

We will exhibit and analyze a sequence of two blow-ups which desingularize the pullback of E_{2} to 𝔅′. We will carry out a detailed analysis only over 𝔅_{1}; the analysis over 𝔅_{2} is entirely analogous, switching the roles of x_{1} and x_{2} and of r and s. (Alternatively, to obtain the picture over the chart 𝔅_{2}, we can use the lift of the action of S_{3} to the blow-up we construct.)

We will denote the exceptional divisor in 𝔅′ by D. Pulling back E_{2} to 𝔅′ gives rise to the variety E_{3}, which over 𝔅_{1} can easily be seen to be given by the equation

s_{t}x_{3}^{2}(ρ_{σ}x_{1} − x_{2}) = x_{1}x_{2}(x_{1} − x_{2}).  (2.5)

On 𝔅_{1}, D is the line given by s_{t} = 0. The singular fibers of E_{3} over D all consist of the three single lines L′_{1} : x_{2} = 0, L′_{2} : x_{1} = x_{2}, L′_{3} : x_{1} = 0, all meeting in the point x_{1} = x_{2} = 0, which is singular on E_{3}.

Blowing up E_{3}.

The first blow-up. Let W′_{1} be the one-dimensional subscheme of E_{3} given by x_{1} = x_{2} = 0 over D; we blow up E_{3} along W′_{1}, obtaining the variety E_{4}. Note that the two affine coordinate charts for (2.5) given by x_{1} ≠ 0 and x_{2} ≠ 0 do not meet W′_{1}, so they remain unaffected. On the coordinate chart x_{3} ≠ 0, we introduce affine coordinates x_{1,3} = x_{1}/x_{3}, x_{2,3} = x_{2}/x_{3}; the equation for E_{3} on this chart is

s_{t}(ρ_{σ}x_{1,3} − x_{2,3}) = x_{1,3}x_{2,3}(x_{1,3} − x_{2,3}),  (2.6)

and on this open affine, the subscheme W′_{1} corresponds to the ideal (s_{t}, x_{1,3}, x_{2,3}). Applying Proposition 1, we obtain three coordinate charts for E_{4} over the chart given by (2.6):

• Chart ℭ′_{1}: Set x_{1,3} = A_{3,2}s_{t}, x_{2,3} = A′_{3,2}s_{t} in equation (2.6) and divide both sides by s_{t}^{2} (the reason for the naming of variables will become clear in the next chapter) ; this gives rise to the affine open of E_{4} given by

ρ_{σ}A_{3,2} − A′_{3,2} = s_{t}A_{3,2}A′_{3,2}(A_{3,2} − A′_{3,2})

inside 𝔸^{4}[A_{3,2}, A′_{3,2}, ρ_{σ}, s_{t}]. Note that E_{4} is regular on this coordinate chart; the fibers over D (given by (ρ_{σ}, s_{t}) = (ρ_{σ}, 0)) consist of the single line ρ_{σ}A_{3,2} = A′_{3,2} for each value of ρ_{σ}, which we call L′_{4}.

• Chart ℭ′_{2}: Set s_{t} = A_{2,3}x_{1,3}, x_{2,3} = Δ_{12,01}x_{1,3} in equation (2.6) and divide both sides by x_{1,3}^{2}; this gives rise to the chart

A_{2,3}(ρ_{σ} − Δ_{12,01}) = x_{1,3}Δ_{12,01}(1 − Δ_{12,01})  (2.7)

inside 𝔸^{4}[x_{1,3}, Δ_{12,01}, ρ_{σ}, A_{2,3}]. Note that there are two singular points on this chart, given by (x_{1,3}, Δ_{12,01}, ρ_{σ}, A_{2,3}) = (0, 0, 0, 0) and (0, 1, 1, 0).

• Chart ℭ′_{3}: Set s_{t} = A′_{2,3}x_{2,3}, x_{1,3} = Δ_{01,12}x_{2,3} in equation (2.6) and divide both sides by x_{2,3}^{2} to obtain the chart

A′_{2,3}(ρ_{σ}Δ_{01,12} − 1) = x_{2,3}Δ_{01,12}(Δ_{01,12} − 1)  (2.8)

inside 𝔸^{4}[Δ_{01,12}, x_{2,3}, ρ_{σ}, A′_{2,3}]. There is one singular point on this chart, (Δ_{01,12}, x_{2,3}, ρ_{σ}, A′_{2,3}) = (1, 0, 1, 0).

The second blow-up. We note that on the charts ℭ′_{2} and ℭ′_{3}, the fibers over D (= the line s_{t} = 0 in 𝔅_{1}) contain exactly one line of multiplicity two (which we call V′) given by A_{2,3} = x_{1,3} = 0 on ℭ′_{2} and by A′_{2,3} = x_{2,3} = 0 on ℭ′_{3}. The fibers of E_{4} over D also contain four single lines each: L′_{4}, which we have seen in the chart ℭ′_{1} already, as well as the proper transforms of the lines L′_{1}, L′_{2} and L′_{3}, which are disjoint and each intersect V′ transversely.

We perform a final blow-up, along the subscheme W′_{2} ⊂ E_{4}, consisting of the (reduced) lines V′ in the fibers over D, in order to obtain the variety E_{5}. Since W′_{2} does not intersect the chart ℭ′_{1}, we only have to perform the blow-up on the charts ℭ′_{2} and ℭ′_{3}:

• We blow up ℭ′_{2} along the subscheme given by the ideal (x_{1,3}, A_{2,3}), obtaining the following two charts:

– Chart ℭ′_{21}: In equation (2.7), set x_{1,3} = A′_{12,2}A_{2,3} and divide both sides by A_{2,3} to obtain the affine chart in 𝔸^{4}[A′_{12,2}, Δ_{12,01}, ρ_{σ}, A_{2,3}] given by

ρ_{σ} − Δ_{12,01} = A′_{12,2}Δ_{12,01}(1 − Δ_{12,01}).

– Chart ℭ′_{22}: In equation (2.7), set A_{2,3} = A′_{2,12}x_{1,3} and divide both sides by x_{1,3} to obtain the affine chart in 𝔸^{4}[x_{1,3}, Δ_{12,01}, ρ_{σ}, A′_{2,12}] given by

A′_{2,12}(ρ_{σ} − Δ_{12,01}) = Δ_{12,01}(1 − Δ_{12,01}).

• We blow up ℭ′_{3} along the subscheme given by the ideal (x_{2,3}, A′_{2,3}), obtaining the following two charts:

– Chart ℭ′_{31}: In equation (2.8), set x_{2,3} = A′′_{12,2}A′_{2,3} and divide both sides by A′_{2,3} to obtain the affine chart in 𝔸^{4}[Δ_{01,12}, A′′_{12,2}, ρ_{σ}, A′_{2,3}] given by

ρ_{σ}Δ_{01,12} − 1 = A′′_{12,2}Δ_{01,12}(Δ_{01,12} − 1).

– Chart ℭ′_{32}: In equation (2.8), set A′_{2,3} = A′′_{2,12}x_{2,3} and divide both sides by x_{2,3} to obtain the affine chart in 𝔸^{4}[Δ_{01,12}, x_{2,3}, ρ_{σ}, A′′_{2,12}] given by

A′′_{2,12}(ρ_{σ}Δ_{01,12} − 1) = Δ_{01,12}(Δ_{01,12} − 1).

We can easily check by Lemma 3 that E_{5} is regular on all of these coordinate charts. The resulting fibers. We compute the fibers of E_{5} over D on the coordinate charts ℭ′_{21}, ℭ′_{22}, ℭ′_{31}, ℭ′_{32}. Calculations analogous to the previous subsection give the following result:

• On ℭ′_{21} we have

– the single line given by A′_{12,2} = 0, Δ_{12,01} = ρ_{σ}, which is the proper transform of L′_{4};

– and the double line

V′ : A_{2,3} = 0, ρ_{σ} − Δ_{12,01} = A′_{12,2}Δ_{12,01}(1 − Δ_{12,01}) if ρ_{σ} ≠ 0, 1;

or the two double lines V′ : A_{2,3} = 0, 1 = A′_{12,2}(Δ_{12,01} − 1) (or 1 = A′_{12,2}Δ_{12,01}) and V′_{(0:1)} (or V′_{(1:1)}): A_{2,3} = 0, Δ_{12,01} = ρ_{σ}, if ρ_{σ} = 0 (or 1). We note that under the blow-down map E_{5} → E_{4}, the lines V′ inject onto the subscheme W′_{2} of E_{4}.

• On ℭ′_{22} we have

– the single lines given by A′_{2,21} = 0, Δ_{12,01} = 0 (proper transform of L′_{1}) and by A′_{2,21} = 0, Δ_{12,01} = 1 (proper transform of L′_{2});

– and the double line

V′ : x_{1,3} = 0, A′_{2,21}(ρ_{σ} − Δ_{12,01}) = Δ_{12,01}(1 − Δ_{12,01}) if ρ_{σ} ≠ 0, 1;

or the two double lines V′ : x_{1,3} = 0, A′_{2,21} = Δ_{12,01} − 1 (or A′_{2,21} = Δ_{12,01}) and V′_{(0:1)} (or V′_{(1:1)}): x_{1,3} = 0, Δ_{12,01} = ρ_{σ}, if ρ_{σ} = 0 (or ρ_{σ} = 1, respectively).

• On ℭ′_{31} we have

– the single line given by A′′_{12,2} = 0, 1 = ρ_{σ}Δ_{01,12}, if ρ_{σ} ≠ 0 – this is the proper transform of L′_{4};

– and the double line

V′ : A′_{2,3} = 0, ρ_{σ}Δ_{01,12} − 1 = A′′_{12,2}Δ_{01,12}(Δ_{01,12} − 1) if ρ_{σ} ≠ 1;

or the two double lines V′ : A′_{2,3} = 0, 1 = A′′_{12,2}Δ_{01,12} and V′_{(1:1)} : A′_{2,3} = 0, Δ_{01,12} = 1, if ρ_{σ} = 1.

• On ℭ′_{32} we have

– the single lines given by A′′_{2,12} = 0, Δ_{01,12} = 0 (proper transform of L′_{3}) and A′′_{2,12} = 0, Δ_{01,12} = 1 (proper transform of L′_{2});

– and the double line

V′ : x_{2,3} = 0, A′′_{2,12}(ρ_{σ}Δ_{01,12} − 1) = Δ_{01,12}(Δ_{01,12} − 1) if ρ_{σ} ≠ 1;

or the two double lines V′ : x_{2,3} = 0, A′′_{2,12} = Δ_{01,12} and V′_{(1:1)} : x_{2,3} = 0, Δ_{01,12} = 1, if ρ_{σ} = 1.

Two-torsion sections, fiber types, and the action of S_{3} and 𝔾_{m}.

We remark once more that the analysis of the blow-ups of E_{3} over 𝔅_{2} ⊂ 𝔅′ is entirely analogous to the calculations for the blow-ups of E_{3} over 𝔅_{1} which we just performed. Putting everything together, we obtain the following picture: Two-torsion sections. We first note that all four two-torsion sections clearly extend to E_{3}, by the same formulae as in Lemma 1 above. Moreover, (the scheme-theoretic images of) P_{1}, P_{2}, P_{3} do not meet the subschemes along which we have performed the last two blow-ups; hence they automatically lift to sections of E_{4} and E_{5} over 𝔅′, and meet the singular fibers over D in the lines L′_{1}, L′_{2}, L′_{3}, respectively.

The subscheme W′_{1} along which we blew up E_{3} is contained in the image of the section P_{4} as a Cartier subscheme; by an argument as in the previous subsection, P_{4} hence lifts to E_{4}. On the coordinate chart ℭ′_{1} above, it is moreover easy to see that P_{4} is given by A_{3,2} = A′_{3,2} = 0; we deduce that the image of P_{4} in E_{4} does not meet the scheme W′_{2} along which we blew up E_{4} to obtain E_{5}, so P_{4} automatically lifts to a section of E_{5} over 𝔅′. By looking at the chart ℭ′_{1} we also deduce that P_{4} meets the fibers over D in the line L′_{4}. Fiber types. Summarizing our construction of E_{5}, we obtain the following fiber types over D, the exceptional divisor of 𝔅′:

• The fibers of E_{5} over the closed points (ρ_{σ}, s_{t}) of 𝔅_{1} with s_{t} = 0, ρ_{σ} ≠ 0, 1 are of type I_{0}^{∗} in the Kodaira-Néron classification; V′ is the double line, L′1, L′_{2}, L′_{3}, L′_{4} are the single lines, intersecting one twotorsion section each as described above.

• The fibers over the three remaining closed points of D are each of type I_{1}^{∗}; the two double lines over r = s = 0, (ρ : σ) = (0 : 1)/ (1 : 1)/(1 : 0) are V′ and V′_{(0:1)}/V′_{(1:1)}/V′_{(1:0)}, the four single lines are again L′_{1}, L′_{2}, L′_{3}, L′_{4}. Over r = s = 0 = ρ, L′_{1} and L′_{4} intersect V′_{(0:1)}, and L′_{2} and L′_{3} meet V′; over r = s = 0 = ρ − σ, L′_{2} and L′_{4} intersect V′_{(1:1)}, whereas L′_{1} and L′_{3} intersect V′; and over r = s = 0 = σ, L′_{3} and L′_{4} intersect V′_{(1:0)}, whereas L′_{1} and L′_{2} meet V′. Again, each of the single lines meets one two-torsion section, as described above.

Extending the actions of S_{3} and 𝔾_{m}. Using the universal property of blow-ups as in the previous subsection, we can lift the action of S_{3} to E_{5}. We only need to show that at each stage, the subscheme along which we blow up is invariant under the action.

• Applying Proposition 2 to the schemes X = ℙ^{2} = 𝔅, Y the point in X given by (r : s : t) = (0 : 0 : 1), and X′ = E_{2}, it is not difficult to see (by an argument as in the proof of Proposition 1) that E_{3} is the blow-up of E_{2} along its fiber over the point (r : s : t) = (0 : 0 : 1). Since this fiber is certainly preserved under the action of S_{3}, the action of S_{3} can be lifted to an action on E_{3} over 𝔅′. (It is also easy to give the action of S_{3} on E_{3}/𝔅′ explicitly and hence verify directly that it lifts.)

• Since the action of S_{3} on E_{3} permutes the lines L′_{1}, L′_{2}, L′_{3} over the exceptional divisor D ⊂ 𝔅′, their intersection is preserved; since this is the subscheme W′_{1} along which we blow up to obtain E_{4}, the action of S_{3} lifts to E_{4}.

• Looking at the action of S_{3} on the fibers of E_{4} over D, it is clear that the subscheme W′_{2}, consisting of the only double line in those fibers, is invariant; hence, the action of S_{3} lifts to an action on E_{5} over 𝔅′.

It follows immediately from the above that S_{3} permutes the two-torsion sections, as lifted to E_{5}, as before.

Again, an analogous argument shows that we can also lift the action of 𝔾_{m} to an action on E_{5} over 𝔅′; as before, it is an easy exercise to give this action explicitly on the coordinate charts.

2.4.3 Conclusion

We now observe that we can “glue” everything we have done in the previous two subsections to a construction over Spec(ℤ[1/2]): We can perform all our blow-ups over ℤ[1/2], which, by the remark after Proposition 1, yields coordinate charts which – in terms of equations – look exactly like the ones we obtained above, so the argument looks like above word by word. We denote the resulting final blow-ups E_{5} of E and 𝔅′ of 𝔅 over Spec(ℤ[1/2]) by E(2, ω) and X(2, ω).

A diagram illustrating the discriminant locus in X(2, ω) along with the singular fibers of E(2, ω), as well as the combinatorial information which we have established on the intersection of the fiber components with each other and with the two-torsion sections, is given in Figure 1 at the end of this chapter.

We can easily see that E(2, ω) and X(2, ω) have many of the properties we desire:

E(2, ω)/X(2, ω) is a compactification of F(2, ω)/Y (2, ω), since we only performed blow-ups along subschemes in E\F(2, ω) and 𝔅\Y (2, ω). We can easily deduce that the two-torsion sections of F(2, ω) over Y (2, ω) extend to sections of E(2, ω) over X(2, ω) (e. g. from the explicit formulae we gave); moreover, our calculations show that in every characteristic (other than two) the singular fibers of E(2, ω) over X(2, ω) are of the types I_{2}, I_{0}^{∗} or I_{1}^{∗} in the Kodaira-Néron classification. In the course of our calculations we noted that the fiber of E(2, ω) over any point in Spec(ℤ[1/2]) is a regular variety (so by the remark we made at the end of section 2.3.3, E(2, ω) is smooth over Spec(ℤ[1/2])) which is clearly rational (since the fiber of E over that point was rational to start with). Finally, the extension of the S_{3}- and the 𝔾_{m}-action to E(2, ω)/X(2, ω) over Spec(ℤ[1/2]) follows by the same argument which we gave for the fiber in each individual characteristic.

Last but not least, X(2, ω) is ℙ_{Spec(ℤ[1/2])}^{2} blown up along a Spec(ℤ[1/2])- valued point, so it certainly is integral, projective and smooth over Spec(ℤ[1/2]). From [Har, Proposition II. 7. 10.] it follows that E(2, ω) is projective over X(2, ω), and from [Har, Theorem III. 9. 9.] it follows that E(2, ω) is flat over X(2, ω); the fact that E(2, ω) is integral follows from our explicit description. We have hence proved Proposition 3; it remains to check whether our candidate E(2, ω) satisfies the properties (5) and (7) as laid out in Section 2.1.

Remark. It deserves mentioning that the fiber types of the singular fibers E(2, ω) over X(2, ω) are, of course, exactly the ones predicted in [Mir] and [Szy]. We also want to refer to Section 14 of [Mir], on the fibers over the intersection of two components of the discriminant locus in the base in characteristic zero. There, Miranda points out that our I_{1}^{∗}-fibers are actually I_{2}^{∗}-fibers “in disguise”: If we take, for example, a line L in X(2, ω) which goes through a point P, the fiber over which in E(2, ω) has type I_{1}^{∗}, but such that the generic fiber over L is a smooth elliptic curve, then the pullback of E(2, ω) to L is singular over P, and the fiber over P of the minimal proper regular model of the pullback has type I_{2}^{∗}. Looking at the group of components of fibers of Kodaira-Néron type I_{n}^{∗}, as described, for example, in [Sil2, Section IV. 9.], this gives us hope that the translations by two-torsion sections extend to automorphisms of E(2, ω) – a hope which, in the following section, will turn out to be justified.

2.5 Translation by Two-torsion Sections

2.5.1 Extension of Translations by Sections

The aim of this section is to prove the following:

Proposition 4 The translations by the two-torsion sections in F(2, ω) extend (uniquely) to automorphisms of the scheme E(2, ω) which we constructed in the previous section.

We want to comment briefly on the approach we will take in the proof below:

As we already mentioned, when working with proper minimal regular models of elliptic curves over Dedekind domains, a proposition analogous to Proposition 4 would follow immediately from the functorial properties of those models (any automorphism of the generic fiber extends to an automorphism of the model).

In the case of higher-dimensional bases, we do not have the notion of a canonical minimal model in this sense, and we cannot prove a statement like Proposition 4 in this way. (In fact, even when working with regular elliptic schemes over surfaces in characteristic zero which are “minimal” in the sense we described in Section 2.1 (taken from [Mir, Section 0]), we do not expect propositions of the above kind to be true in general: Looking at the normal bundles of the components of the fibers over singular points in the discriminant locus, as described in [Mir, Section 11], we do not expect translations by two-torsion sections to extend to those fibers in certain scenarios.) We will prove the above proposition by explicit computations instead, using the description of E(2, ω) in terms of coordinate charts we obtained in the previous sections.

2.5.2 Proof of the Proposition

Preliminary Remarks:

• Since X(2, ω) and E(2, ω) are separated schemes, Y (2, ω) and F(2, ω) are Zariski-dense in X(2, ω) and E(2, ω), respectively, and the translations by two-torsion sections are automorphisms of F(2, ω) which are compatible with the structure morphism to Y (2, ω), it follows easily that the extensions of those translations (once we have established their existence) are automorphisms of E(2, ω) which are compatible with the structure morphism to X(2, ω).

• On the fibers of F(2, ω) over Y (2, ω), we can explicitly calculate the translations by the two-torsion sections by the chord-tangent method. In terms of the homogeneous coordinates which we used to describe E over 𝔅 in Section 2.2.2, we can describe t_{P_{1}}, translation by the section P_{1}, by the formulae

(x_{1} : x_{2} : x_{3}) ↦ (t(x_{2}^{2} − x_{1}x_{2}) + rx_{3}^{2} : tx_{2}^{2} : −tx_{2}x_{3})
= (sx_{2}(x_{1} − x_{2}) : x_{2}(rx_{1} − sx_{2}) : −x_{3}(rx_{1} − sx_{2}))
= (sx_{3}(x_{1} − x_{2}) : x_{3}(rx_{1} − sx_{2}) : −tx_{1}(x_{1} − x_{2})).

From these equations we can easily deduce that t_{P_{1}} extends to the fibers of E over the locus in 𝔅 where r = 0, s = 0 or r = s, but not r = s = 0 or t = 0 – in other words, t_{P_{1}} extends to the fibers of E(2, ω) with multiplicative reduction. Since t_{P_{i}} = s_{(1i)} ∘ t_{P_{1}} ∘ s_{(1i)} for i = 2, 3, the same follows for t_{P_{2}} and t_{P_{3}}. It hence suffices to extend the t_{P_{i}} to the fibers of E(2, ω) over X(2, ω) with additive reduction.

• By an argument similar to the one outlined above, since F(2, ω) is Zariski-dense in E(2, ω) and the latter is a separated scheme, it suffices to show that we can cover E(2, ω) by Zariski-opens to each of which we can extend the translations t_{P_{i}}.

By the last of the above remarks, in order to extend the translations by two-torsion sections to the fibers of E(2, ω) with additive reduction, it suffices to find open sets in the coordinate charts for E(2, ω) to which we can extend the t_{P_{i}} and whose union covers those fibers. This is the strategy which we are going to carry out now.

Computations on the Coordinate Charts:

On the basis of the representation of t_{P_{1}} in terms of homogeneous coordinates on E, it is an easy task to give t_{P_{1}} on the coordinate charts of E(2, ω) we calculated in the previous section in terms of their local coordinates. We will use the resulting formulae to deduce that t_{P_{1}} extends to a large part of the fibers in E(2, ω) with additive reduction. Finally, using the relationships t_{P_{i}} = s_{(ij)} ∘ t_{P_{j}} ∘ s_{(ij)} for i ≠ j (where s_{(ij)} refers to the action of S_{3}), it will follow that all the t_{P_{i}} will extend to automorphisms of all of E(2, ω).

We will first analyze the fibers of E(2, ω) over the line in X(2, ω) corresponding to the line t = 0 in 𝔅 (we will identify those two lines in the sequel by abuse of language), and afterwards the fibers over the exceptional divisor of X(2, ω). The fibers over the line t = 0 ⊂ 𝔅. Recall that the coordinate chart ℭ_{41} is given (over ℤ[1/2]) by the equation

ℭ_{41} : Γ′′_{0,01}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside Spec(ℤ[1/2][x_{1,2}, Γ′′_{0,01}, s_{r}, B_{3,0}]). A straightforward computation gives that t_{P_{1}} on ℭ_{41} is given by the ring homomorphism induced by

t_{P_{1}} : (x_{1,2}, Γ′′_{0,01}, s_{r}, B_{3,0}) ↦ (1 + Γ′′_{0,01} − x_{1,2}, Γ′′_{0,01}, s_{r}, −B_{3,0}),

so t_{P_{1}} extends to the coordinate chart ℭ_{41}.

Likewise, we get that on the coordinate chart ℭ_{3} given by the equation

ℭ_{3} : B_{0,3}^{2}t_{r}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside Spec(ℤ[1/2][x_{1,2}, B_{0,3}, s_{r}, t_{r}]), t_{P_{1}} is given by the ring homomorphism induced by

t_{P_{1}} : (x_{1,2}, B_{0,3}, s_{r}, t_{r}) ↦ (B_{0,3}^{2}t_{r} − x_{1,2} + 1, −B_{0,3}, s_{r}, t_{r}),

so t_{P_{1}} also extends to the coordinate chart ℭ_{3}.

Using the transition maps between different coordinate charts, we can deduce after a quick survey that given the two previous paragraphs, we only need to find Zariski-open neighborhoods of the fiber components which we called L_{4} in the coordinate charts ℭ_{42} and ℭ_{5}, as well as Zariski-open neighborhoods of the fiber components L_{1} in ℭ_{1} and ℭ_{21}, to which we can extend t_{P_{1}}, in order to deduce that t_{P_{1}} extends to the fibers over the line t = 0 in 𝔅 where r ≠ 0. These are easy calculations; we will write out one of them, and only indicate the others.

Recall that the coordinate chart ℭ_{1} of E(2, ω) is given by the equation

ℭ_{1} : (B′_{2,3})^{2}t_{r}(1 − s_{r}x_{2,1}) = x_{2,1}(1 − x_{2,1})

inside Spec(ℤ[1/2][x_{2,1}, B′_{2,3}, s_{r}, t_{r}]), whereas the coordinate chart ℭ_{5}, given by r ≠ 0, x_{3} ≠ 0 as defined in Section 2.4, can be described by the equation

ℭ_{5} : x_{1,3} − s_{r}x_{2,3} = t_{r}x_{1,3}x_{2,3}(x_{1,3} − x_{2,3})

inside Spec(ℤ[1/2][x_{1,3}, x_{2,3}, s_{r}, t_{r}]). Again by a straightforward calculation, we dedude that that t_{P_{1}}, mapping a certain Zariski-open set of ℭ_{1} into ℭ_{5}, is given by the ring homomorphism induced by

t_{P_{1}} : (x_{1,3}, x_{2,3}, s_{r}, t_{r}) ↦ (−s_{r}B′_{2,3}, \frac{B′_{2,3}(s_{r}x_{2,1} − 1)}{1 − x_{2,1}}, s_{r}, t_{r}),

and it follows easily that t_{P_{1}} extends to a Zariski open neighborhood of the lines L_{1} in ℭ_{1}, given by x_{2,1} = 0 = t_{r}.

We can dispose of the other cases similarly, by looking at how t_{P_{1}} maps a certain open subset of ℭ_{5} into ℭ_{1}, an open subset of ℭ_{21} into ℭ_{42}, and an open subset of ℭ_{42} back into ℭ_{21}. t_{P_{1}} extends thus to the fibers over the line t = 0 in 𝔅 where r ≠ 0.

Finally, since t_{P_{3}} = s_{(13)} ∘ t_{P_{1}} ∘ s_{(13)}, we deduce that t_{P_{3}} extends to the fibers over the line t = 0 in 𝔅 where s ≠ 0. In particular, both t_{P_{1}} and t_{P_{3}} extend to the fiber over (r : s : t) = (1 : 1 : 0) ∈ 𝔅, and hence so does t_{P_{2}} = t_{P_{1}} ∘t_{P_{3}}. By once again using that t_{P_{i}} = s_{(ij)} ∘t_{P_{j}} ∘s_{(ij)} for all i ≠ j, it follows immediately that all t_{P_{1}}, t_{P_{2}} and t_{P_{3}} extend to the fibers of E(2, ω) over the line t = 0 in 𝔅. The fibers over the exceptional divisor of X(2, ω). The analysis here is entirely similar to the previous one. Recall that the coordinate chart ℭ′_{32} is given (over ℤ[1/2]) by the equation

ℭ′_{32} : A′′_{2,12}(ρ_{σ}Δ_{01,12} − 1) = Δ_{01,12}(Δ_{01,12} − 1)

inside Spec(ℤ[1/2][Δ_{01,12}, x_{2,3}, ρ_{σ}, A′′_{2,12}]). A straightforward computation gives that t_{P_{1}} on ℭ′_{32} is given by the ring homomorphism induced by

t_{P_{1}} : (Δ_{01,12}, x_{2,3}, ρ_{σ}, A′′_{2,12}) ↦ (−\frac{A′′_{2,12}}{Δ_{01,12}} , −x_{2,3}, ρ_{σ}, A′′_{2,12})
= (\frac{1 − Δ_{01,12}}{ρ_{σ}Δ_{01,12} − 1}, −x_{2,3}, ρ_{σ}, A′′_{2,12}),

and it follows that t_{P_{1}} extends to the coordinate chart ℭ′_{32}.

We also have to write out two coordinate charts which we did not calculate explicitly earlier: the coordinate charts given by x_{1} ≠ 0 and x_{2} ≠ 0 respectively in the Zariski open set of E_{3} given by equation 2.5 in the previous section (on which also t ≠ 0 ≠ σ); these charts remained unaffected in the subsequent blow-ups. With the usual notation for affine coordinates, we have the chart ℭ′_{01} (on which x_{1} ≠ 0), given by the equation

ℭ′_{01} : s_{t}x_{3,1}^{2}(ρ_{σ} − x_{2,1}) = x_{2,1}(1 − x_{2,1})

inside Spec(ℤ[1/2][x_{2,1}, x_{3,1}, ρ_{σ}, s_{t}]), and the chart ℭ′_{02} (x_{2} ≠ 0) given by the equation

ℭ′_{02} : s_{t}x_{3,2}^{2}(ρ_{σ}x_{1,2} − 1) = x_{1,2}(x_{1,2} − 1)

inside Spec(ℤ[1/2][x_{1,2}, x_{3,2}, ρ_{σ}, s_{t}]). We can calculate t_{P_{1}} to be given on ℭ′_{02} by the ring homomorphism induced by

t_{P_{1}} : (x_{1,2}, x_{3,2}, ρ_{σ}, s_{t}) ↦ (\frac{x_{1,2} − 1}{x_{1,2}ρ_{σ} − 1}, −x_{3,2}, ρ_{σ}, s_{t})
= (\frac{s_{t}x_{3,2}^{2}}{x_{1,2}} , −x_{3,2}, ρ_{σ}, s_{t}),

so t_{P_{1}} also extends to the coordinate chart ℭ′_{02}.

Using the transition maps between coordinate charts, we are reduced at this point – in analogy to the analysis above – to finding Zariski open neighborhoods of the fiber components L′_{4} (as defined in the previous section) in the coordinate charts ℭ′_{1}, ℭ′_{21} and ℭ′_{31}, neighborhoods of L′_{1} in the charts ℭ′_{01} and ℭ′_{22}, and of L′_{2} in ℭ′_{22}, to which t_{P_{1}} extends, in order to show that t_{P_{1}} extends to all fibers over D ∩ 𝔅_{1}. (Recall that D is the exceptional divisor of X(2, ω), and 𝔅_{1} is the coordinate chart of X(2, ω) as given in the previous section.)

We can find those neighborhoods in the charts by looking at the rational maps induced by t_{P_{1}} from ℭ′_{01} to ℭ′_{1} and vice versa, and inbetween the charts ℭ′_{21}, ℭ′_{22}, ℭ′_{31} and ℭ′_{32}; the calculations are in precise analogy to the calculations we spelt out above.

Finally, by an argument analogous to the one given earlier, making use of the relations t_{P_{i}} = s_{(ij)} ∘ t_{P_{j}} ∘ s_{(ij)} for i ≠ j, we can deduce that all t_{P_{1}}, t_{P_{2}} and t_{P_{3}} extend to the fibers of E(2, ω) over the exceptional divisor of X(2, ω). This completes the proof of Proposition 4.

2.6 Minimality and Twists of E(2, ω)

In this section we are going to analyze some geometric and arithmetic properties of the characteristic zero fiber of E(2, ω). We will first establish minimality of E(2, ω) over X(2, ω), as defined in Section 2.1 (which is essentially taken from [Mir] and spelt out), and thereby finish our program laid out at the beginning of this chapter. To conclude, we will comment on the “twisting property” (also introduced in Section 2.1), which is a property of F(2, ω) rather than E(2, ω) and follows automatically from its universal properties; we mention it here for a lack of a suitable reference and because we will use it towards the end of the next chapter, wenn we apply our construction of E(2, ω) to questions of visibility.

For the remainder of this section, we will write E_{ℚ} for E(2, ω) ⊗_{ℤ[1/2]} ℚ, and F_{ℚ} for F(2, ω) ⊗_{ℤ[1/2]} ℚ; similarly, we write X_{ℚ} for X(2, ω) ⊗_{ℤ[1/2]} ℚ and Y_{ℚ} for Y (2, ω) ⊗_{ℤ[1/2]} ℚ.

2.6.1 Minimality

We want to prove the following

Proposition 5 The structure map E_{ℚ} → X_{ℚ} induced by the structure map p : E(2, ω) → X(2, ω) is minimal: There is no surface Σ ⊂ E_{ℚ} such that E_{ℚ} can be blown down along Σ to a non-singular variety 𝔙 over X_{ℚ} such that the blow-down map is compatible with the structure maps to X_{ℚ}.

PROOF. As we mentioned above, we elaborate on an argument (for our special case) which Miranda indicates in his article [Mir] to prove minimality for the elliptic threefolds he constructs there.

We may without loss of generality work over the complex numbers; the pullbacks of the varieties E_{ℚ} and X_{ℚ} to ℂ will be denoted by E_{ℂ} and X_{ℂ}, respectively.

Suppose that, contrary to the proposition, there is a surface Σ along which we can blow down E_{ℂ} to a variety 𝔙 as described in the statement of the proposition. The fibers of the blow-down map must then be contained in the fibers of the structure morphism E_{ℂ} → X_{ℂ}. We may assume that those fibers are generically isomorphic to ℙ^{1}, and hence they must be components of the singular fibers of E_{ℂ}/X_{ℂ}.

Let 𝔉 be such a fiber of the blow-down map, and let N_{Σ/E_{ℂ}} denote the normal bundle of Σ in E_{ℂ}. From [GH, Section 4. 6., Blowing Up Submanifolds] it follows that (generically) we have N_{Σ/E_{ℂ}}|_{𝔉} ≅ O_{𝔉}(−1).

Now suppose that T is a regular surface inside E_{ℂ} such that Σ and T intersect transversely along 𝔉 – that is, if we denote the (holomorphic) tangent bundle of a variety X by T_{X}, then T_{E_{ℂ}}|_{𝔉} is spanned by T_{Σ}|_{𝔉} and T_{T} |_{𝔉}. Using the fact that every vector bundle of rank two on ℙ^{1} is decomposable (cf. [Har, Chapter V, Corollary 2. 16.]), it is not hard to deduce that

N_{𝔉/T} ≅ N_{Σ/E_{ℂ}}|_{𝔉},

and hence

N_{𝔉/T} ≅ O_{𝔉}(−1).  (2.10)

On the other hand, let the discriminant locus Δ be the divisor in X_{ℂ} over which the fibers in E_{ℂ} are singular, and let 𝔉 be a component of the fiber of E_{ℂ} over a smooth point P of Δ; hence, Σ is (locally) the surface in E_{ℂ} consisting of the corresponding component in the fibers over the component of Δ containing P.

We can then construct a surface T as above in the following way: Take a line L in X_{ℂ} such that Δ∩L is contained in the smooth part of Δ, contains P, and such that Δ and L intersect transversely everywhere (this can certainly be done). Then the pullback of E_{ℂ} to L, seen as a closed subvariety of E_{ℂ}, is a regular elliptic surface, as can be checked by routine calculations using the explicit description of E_{ℂ} which we have in terms of coordinate charts; moreover, Σ and T do intersect transversely along 𝔉.

But now c_{1}(N_{𝔉/T}) is the self-intersection of 𝔉 seen as a divisor on T. From the shape of the fibers we see that it is impossible that (𝔉.𝔉)_{T} = −1, and we have arrived at a contradiction to (2.10). Hence our original assumption that p : E_{ℂ} → X_{ℂ} is not minimal must have been false.

2.6.2 Summary of Results

Summarizing the construction we have carried out, we obtain the following Theorem 2 The compactification E(2, ω) over X(2, ω) of the universal elliptic curve F(2, ω) over Y (2, ω) constructed in Section 2.4 satisfies properties (1), (2), (3), (4), (5), (6), and (7) from Section 2.1.

PROOF. Put together Proposition 3, Proposition 4 and Proposition 5.

2.6.3 The Twisting Property of F(2, ω)

Before we analyze the arithmetic of certain twists of the varieties X_{ℚ}, Y_{ℚ}, E_{ℚ} and F_{ℚ} and their base changes to general number fields, let us recall the corresponding situations for higher level modular curves and universal elliptic curves.

For n > 2, let E(n) be the generalized universal elliptic curve with full level-n-structure over the corresponding modular curve X(n). Following the exposition in [M], X(n) is the moduli space (over ℤ[1/n]) for the data (F, α), where F is a generalized elliptic curve in the sense of [DR], and α is an isomorphism of group schemes

α : ℤ/nℤ × μ_{n} ˜→F [n]

inducing the identity map μ_{n} ≅ Λ^{2}(ℤ/nℤ × μ_{n}) → Λ^{2}F [n] ≅ μ_{n}. Here the first isomorphism is the natural one, and the second isomorphism is induced by the Weil-pairing on F [n]. Also, the corresponding (generalized) universal elliptic curve E(n) can be viewed as the moduli space classifying triples (F, α, P), where F and α are as above, and P is a point (or section) on F.

Given an elliptic curve E over a number field K, the action of G_{K}, the absolute Galois group of K, on the n-torsion of E gives us a (continuous) representation

ρ_{E} : G_{K} → GL_{2}(ℤ/nℤ),

up to conjugacy (depending on the choice of a basis of E[n] as a free ℤ/nℤ-module of rank two). Let χ : G_{K} → (ℤ/nℤ)^{∗} be the cyclotomic character induced by the action of G_{K} on μ_{n}, and define ρ_{0} : G_{K} → GL_{2}(ℤ/nℤ) by

ρ_{0} : g ↦ (\begin{matrix}1 & 0 \\ 0 & χ(g) \end{matrix}) .

The definition c(g) := ρ_{E}(g)ρ_{0}^{−1}(g) then gives rise to cocycles

c_{X(n),E} : G_{K} → SL_{2}(ℤ/nℤ) ⊂ AutK̄(X(n)K̄)

and

c_{E(n),E} : G_{K} → SL_{2}(ℤ/nℤ) ⊂ AutK̄(E(n)K̄);

here, X(n)K̄ is the base change of X(n) to K̄, E(n)K̄ is the base change of E(n) to K̄, and the actions of SL_{2}(ℤ/nℤ) are induced by the actions on the level structure of the underlying classification problems: In terms of the moduli interpretation of their points, for λ ∈ SL_{2}(ℤ/nℤ), let λ(F, α) = (F, α ∘ λ), and λ(F, α, P) = (F, α ∘ λ, P). (Here, we need to implicitly choose an isomorphism μ_{n} ≅ ℤ/nℤ in order to make sense of the composite α ∘ λ.)

Twisting X(n)_{K} by the cocycle c_{X(n),E} and E(n)_{K} by c_{E(n),E}, we obtain varieties { \atop X(n)} and Ẽ(n) over K. (For a reference on twisting of algebraic varieties, see [Ser2, Chapter 3].) It can be shown that { \atop X(n)} is the moduli space for pairs (F, α), where F is a generalized elliptic curve over a K-scheme S and α is an isomorphism of group schemes α : (E ×_{K} S)[n] ˜→F [n] over S respecting the Weil-pairing; likewise, Ẽ(n) is the moduli space for triples (F, α, P), with F, α as above, and P an S-section of F.

We will prove a similar result for the analogous twists of F_{K} = F_{ℚ}⊗_{ℚ}K and Y_{K} = Y_{ℚ} ⊗_{ℚ} K by certain cocycles, coming from the representation of G_{K} induced by its action on the two-torsion of a given elliptic curve E. The argument we give is entirely analogous to the argument one would give in the case of twisting higher level affine modular curves and the corresponding universal elliptic curves; we sketch it here for lack of a suitable reference (to our knowledge). We note that working with the varieties F_{K} and Y_{K} is clearly sufficient for our purposes, since their twists are open subvarieties of the corresponding twists of the compactifications we have constructed – which is just as well, as we do not have a moduli interpretation for E(2, ω) and X(2, ω). We also note that we do not have to worry about compatibility of determinants arising from the Weil-pairing, since this is automatic in the level two case.

We now fix a number field K for the rest of this subsection. Recall that the K-variety Y_{K} represents the moduli problem of classifying triples (F, α, ω), where F is an elliptic curve over a K-scheme S, α is an isomorphism of group schemes α : (ℤ/2ℤ × ℤ/2ℤ)_{S} ˜→F [2] over S, and ω is a nowhere vanishing relative differential on F over S; likewise, F_{K} represents the moduli problem of classifying quadruples (F, α, ω, P), where F, α, ω are as above, and P is an S-valued section of F.

Proposition 6 Let E be an elliptic curve over K; denote the absolute Galois group of K by G_{K}. From the action of G_{K} on E(K̄)[2], we get (up to conjugation) a homomorphism

ρ_{E} : G_{K} → GL_{2}(𝔽_{2}) ≅ S_{3}.

Using the S_{3}-action on F_{K}, ρ_{E} gives rise to a cohomology class c_{E} ∈ H^{1}(G_{K}, AutK̄(FK̄)); likewise, we obtain a cohomology class c′_{E} ∈ H^{1}(G_{K}, AutK̄(YK̄)). We twist F_{K} by c_{E} to obtain the variety F̃_{K} over K, likewise we twist Y_{K} by c′_{E} to obtain the variety Ỹ_{K} over K. Note that the two twists are compatible, so from the structure map p : F_{K} → Y_{K} we automatically obtain a morphism p̃: F̃_{K} → Ỹ_{K} over K.

Then Ỹ_{K} represents the moduli problem of classifying triples (F, α, ω), where F is an elliptic curve over a K-scheme S, α is an isomorphism of group schemes α : E[2]_{S} ˜→F [2] over S, and ω is a nowhere vanishing relative differential on F over S; and F̃_{K} represents the moduli problem of classifying quadruples (F, α, ω, P), where F, α, ω are as above, and P is an S-valued section of F.

An easy consequence is then the following

Corollary. With the notation of the proposition, F̃_{K} is an elliptic curve over Ỹ_{K}; E is isomorphic over K to a fiber of F̃_{K} over a K-rational point of Ỹ_{K}; and E[2]Ỹ_{K} ≅ F̃_{K}[2] as group schemes over Ỹ_{K}.

It is the corollary which we will use in our discussion of visibility questions in the next chapter. After outlining a proof of Proposition 6, we will briefly indicate how one can deduce the Corollary directly, in a much more concrete, if less insight-providing way.

PROOF OF PROPOSITION. Given an elliptic curve E over K, consider the moduli problems of classifying triples (F, α, ω) and of classifying quadruples (F, α, ω, P), where F is an elliptic curve over a K-scheme S, ω is a nowhere vanishing regular relative differential on F over S, α : E[2] ×_{K} S ˜→F [2] is an isomorphism of group schemes over S, and P is an S-valued point of F. It follows from [KM, Chapter 4] that both moduli problems are representable by K-schemes Ỹ and F̃, respectively: In the terminology of [KM], these problems are clearly rigid, and they can easily be seen to be relatively representable and affine, by explicitly exhibiting the relatively representing schemes in analogy to the standard case of higher level modular curves. Clearly, F̃ is the universal elliptic curve corresponding to the moduli problem represented by Ỹ, and we have a canonical structure map p̃: F̃ → Ỹ, induced by the forgetful functor on the moduli data.

After some finite extension of number fields L/K, the moduli problems represented by Y_{K} and Ỹ become equivalent (and likewise the moduli problems represented by F_{K} and by F̃), by choosing an isomorphism β : E[2] ×_{K} L ˜ ⟶(ℤ/2ℤ × ℤ/2ℤ)_{L}. From the universal properties we hence obtain isomorphisms (depending on our choice of β)

ϕ : Y_{K} ×_{K} L ˜ ⟶ Ỹ ×_{K} L,
ψ : F_{K} ×_{K} L ˜ ⟶ F̃ ×_{K} L,

defined over L and compatible with the structure maps p and p̃. All that remains to show is that ϕ and ψ exhibit Ỹ and F̃ as the twists of Y_{K} and F_{K} induced by the cocycles c′_{E} and c_{E}, respectively – i. e. to show that for every σ ∈ Gal(L/K),

ϕ^{−1} ∘ σ ∘ ϕ ∘ σ^{−1} = ρ′_{E}(σ),
ψ^{−1} ∘ σ ∘ ψ ∘ σ^{−1} = ρ_{E}(σ),

up to conjugation by elements of S_{3} ⊂ Aut_{K}(Y_{K}) or Aut_{K}(F_{K}) (this ambiguity comes from the fact that ϕ and ψ depended on our choice of β, and that two cocyles in H^{1}(G_{K}, S_{3}) = Hom(G_{K}, S_{3}) are cohomologous if and only if they differ from each other by conjugation).

We will establish the second equation; the first follows by an exactly analogous argument. We will use the moduli interpretation of our setup.

We have to show that the following diagram of varieties and maps over L

F_{K} ×_{K} L →^{σ} F_{K} ×_{K} L(∗) →^{ρ_{E}(σ)} F_{K} ×_{K} L
ψ ↓  ↓ ψ
F̃ ×_{K} L  ⟶^{σ}  F̃ ×_{K} L

commutes, up to conjugation by an element of S_{3} ⊂ Aut(F_{K}) at (*), for every σ ∈ G_{K}.

Let us see how the maps act on the moduli data over L: It is easy to see that both σ ∘ ψ and ψ ∘ ρ_{E}(σ) ∘ σ send an elliptic curve F to F^{σ}, that they send a point P on F to the point P^{σ} on F^{σ}, and a nowhere vanishing regular differential ω on F to the nowhere vanishing regular differential ω^{σ} on F^{σ}. Finally, by carefully looking at how σ acts on the two-torsion on each moduli problem, we deduce that σ ∘ ψ and ψ ∘ ρ_{E}(σ) ∘ σ also have the same effect on the level-two-structure, up to conjugation by an element of S_{3} at (*), which finishes the proof of the proposition.

Explicit proof of the Twisting Property. As we mentioned above, the Corollary can be proved directly and explicitly, by means of an interesting exercise on explicit twists of elliptic curves. The only non-obvious assertion is the statement that the original elliptic curve E is isomorphic over K to a fiber of the twist F̃_{K} over a K-rational point of Ỹ_{K}. We will outline an argument and indicate the entailed calculations.

The idea is the following: Given an elliptic curve E over K, we will exhibit a set of fibers F = ⋃_{i=1}^{n} F_{i} of F_{K} over Y_{K} such that F_{i} is the fiber over the point P_{i} ∈ Y_{K}(K̄), and such that M = ⋃_{i=1}^{n} P_{i} is a subset of Y_{K}(K̄) which is invariant both under the action of G_{K} and under the action of S_{3} (hence, the same holds for F).

Viewing S_{3} as a subset of Aut(F/M), we can then twist F/M by the cocycle induced by the homomorphism ρ_{E} as defined in the statement of Proposition 6 to obtain F̃/M̃. But it is not hard to see that M̃ is then a K-rational set of points in Ỹ_{K}, and F̃ is the set of fibers of F̃_{K} over M̃ under p̃.

All which remains to be done is to choose the set M ⊂ Y_{K}(K̄) in such a way that there is a K-rational point in M̃, the fiber over which in F̃ is isomorphic to E over K. We can do this in a very explicit manner:

Recall that in Section 2.2.1, we introduced the representation of F_{K} over Y_{K} given by the “Legendre-like” family

Y^{2}Z = X(X − Z)(r′X − s′Z),

where r′s′(r′−s′) ≠ 0. From Section 2.2.2, it follows by an easy computation that the action of S_{3} on F_{K}/Y_{K} in this representation is given by

s_{(12)} : ((r′, s′), (X : Y : Z)) ↦ ((−r′, s′ − r′), (Z − X : −Y : Z)),
s_{(13)} : ((r′, s′), (X : Y : Z)) ↦ ((s′, r′), (r′X : r′Y : s′Z)).

Given an elliptic curve E over K, we can take E to be given by some Weierstrass equation of the form

E : y^{2}z = (x − dz)(x − ez)(x − fz),

where d, e, f ∈ K̄. We note that the action of G_{K} on E(K̄)[2] and the resulting homomorphism ρ_{E} : G_{K} → S_{3} ≅ GL_{2}(𝔽_{2}) correspond to the action of G_{K} on the triple (d, e, f); we fix the homomorphism ρ_{E}, and hence the resulting cocycle (as opposed to cohomology class) with values in S_{3}, by decreeing the following:

Let Q_{1} = (d : 0 : 1), Q_{2} = (e : 0 : 1), Q_{3} = (f : 0 : 1) be the three non-trivial two-torsion points of E; then for σ ∈ G_{K}, ρ_{E}(σ) = (ij) if σ, acting on E, interchanges Q_{i} and Q_{j} and leaves the third two-torsion point fixed, and likewise ρ_{E}(σ) = (ijk) if, under the action of σ on E, σ : Q_{i} ↦ Q_{j} ↦ Q_{k} ↦ Q_{i}.

We are now ready to describe our choice of fibers of F_{K}/Y_{K} in terms of their “Legendre-like” representation, satisfying the conditions as stated above. The fibers are as follows:

F_{1} : y_{1}^{2}z_{1} = x_{1}(x_{1} − z_{1})((d − e)x_{1} − (f − e)z_{1}) over (d − e, f − e)
F_{2} : y_{2}^{2}z_{2} = x_{2}(x_{2} − z_{2})((f − e)x_{2} − (d − e)z_{2}) over (f − e, d − e)
F_{3} : y_{3}^{2}z_{3} = x_{3}(x_{3} − z_{3})((f − d)x_{3} − (e − d)z_{3}) over (f − d, e − d)
F_{4} : y_{4}^{2}z_{4} = x_{4}(x_{4} − z_{4})((e − d)x_{4} − (f − d)z_{4}) over (e − d, f − d)
F_{5} : y_{5}^{2}z_{5} = x_{5}(x_{5} − z_{5})((e − f)x_{5} − (d − f)z_{5}) over (e − f, d − f)
F_{6} : y_{6}^{2}z_{6} = x_{6}(x_{6} − z_{6})((d − f)x_{6} − (e − f)z_{6}) over (d − f, e − f).

⋃_{i=1}^{6} F_{i} is a closed subscheme of F_{K} which is clearly invariant under the action of G_{K} and under the action of S_{3}.

Also consider the following set of elliptic curves ⋃_{i=1}^{6} F′_{i} over K̄:

F′_{1} : (y′_{1})^{2}z′_{1} = (x′_{1} − dz′_{1})(x′_{1} − ez′_{1})(x′_{1} − fz′_{1})
over P (1)
F′_{2} : (y′_{2})^{2}z′_{2} = x′_{2}(x′_{2} − (d − e)z′_{2})(x′_{2} − (f − e)z′_{2})
over P ((d − e)(f − e))
F′_{3} : (y′_{3})^{2}z′_{3} = (x′_{3} − ez′_{3})(x′_{3} − fz′_{3})(x′_{3} − dz′_{3})
over P ((e − f)(d − f)(e − d))
F′_{4} : (y′_{4})^{2}z′_{4} = x′_{4}(x′_{4} − (e − f)z′_{4})(x′_{4} − (d − f)z′_{4})
over P ((e − f)(d − f))
F′_{5} : (y′_{5})^{2}z′_{5} = (x′_{5} − fz′_{5})(x′_{5} − dz′_{5})(x′_{5} − ez′_{5})
over P ((f − e)(f − d)(d − e))
F′_{6} : (y′_{6})^{2}z′_{6} = x′_{6}(x′_{6} − (f − d)z′_{6})(x′_{6} − (e − d)z′_{6})
over P ((f − d)(e − d))

where the labelling of the base points indicates the way in which we decree G_{K} to permute the F′_{i}: The fiber over P (θ), for some term θ, is mapped by σ ∈ G_{K} to the fiber over P (σ(θ)), letting σ act on the projective coordinates as usual. It can easily be checked that this gives, indeed, a well defined action of G_{K} on ⋃_{i=1}^{6} F′_{i}.

In particular, note that F′_{1} over P (1) is stable under the action of G_{K}, and is isomorphic to E over K. So in order to prove the Corollary, it is sufficient to show that ⋃_{i=1}^{6} F′_{i} is the twist of ⋃_{i=1}^{6} F_{i} corresponding to the cohomology class in H^{1}(G_{K}, Aut(⋃_{i=1}^{6} F_{i})) induced by ρ_{E}.

To this end, consider the following isomorphism ϕ′ : ⋃_{i=1}^{6} F_{i} ˜→ ⋃_{i=1}^{6} F′_{i} over K̄:

ϕ′ : ((x_{1} : y_{1} : z_{1}), (x_{2} : y_{2} : z_{2}), (x_{3} : y_{3} : z_{3}), …, (x_{6} : y_{6} : z_{6})) ⟼
((x′_{1} : y′_{1} : z′_{1}), (x′_{2} : y′_{2} : z′_{2}), (x′_{3} : y′_{3} : z′_{3}), …, (x′_{6} : y′_{6} : z′_{6}))
= ((x_{1} + \frac{ez_{1}}{d−e} : y_{1} : \frac{z_{1}}{d−e}), (x_{2} : y_{2} : \frac{z_{2}}{f−e}),
(x_{3} + \frac{dz_{3}}{f−d} : y_{3} : \frac{z_{3}}{f−d}), (x_{4} − \frac{f−d}{e−d}z_{4} : y_{4} : \frac{z_{4}}{e−d}),
(x_{5} + \frac{fz_{5}}{e−f} : y_{5} : \frac{z_{5}}{e−f}), (x_{6} − z_{6} : y_{6} : \frac{z_{6}}{d−f})).

By [Ser2, Chapter 3], all we need to check at this point is that, for every σ ∈ G_{K}, the automorphism (ϕ′)^{−1} ∘ σ ∘ ϕ′ ∘ σ^{−1} : ⋃_{i=1}^{6} F_{i} → ⋃_{i=1}^{6} F_{i} is the same as the automorphism corresponding to the image of ρ_{E}(σ) under the map S_{3} ↪ Aut(⋃_{i=1}^{6} F_{i}) induced by our choice of S_{3} ↪ Aut(F_{K}). This is a straightforward calculation; note that it is enough to perform the check for at most two elements in G_{K} whose images under ρ_{E} generate im(ρ_{E}).

Figure 1: Singular fibers of E(2, ω) over the discriminant locus of X(2, ω) 50

[figure: t = 0; “-1”; (r = s = 0); r = 0; r = s; s = 0; ρ = 0; ρ = σ; σ = 0; V_{(0:1)}; V; L_{1}; L_{4}; L_{2}; L_{3}; P_{1}; P_{4}; P_{2}; P_{3}; I_{1}^{∗}; V_{(1:1)}; V; L_{2}; L_{4}; L_{1}; L_{3}; P_{2}; P_{4}; P_{1}; P_{3}; I_{1}^{∗}; V_{(1:0)}; V; L_{3}; L_{4}; L_{1}; L_{2}; P_{3}; P_{4}; P_{1}; P_{2}; I_{1}^{∗}; V′_{(0:1)}; V′; L′_{1}; L′_{4}; L′_{2}; L′_{3}; P_{1}; P_{4}; P_{2}; P_{3}; I_{1}^{∗}; V′_{(1:1)}; V′; L′_{2}; L′_{4}; L′_{1}; L′_{3}; P_{2}; P_{4}; P_{1}; P_{3}; I_{1}^{∗}; V′_{(1:0)}; V′; L′_{3}; L′_{4}; L′_{1}; L′_{2}; P_{3}; P_{4}; P_{1}; P_{2}; I_{1}^{∗}; P_{4}; P_{1}; P_{2}; P_{3}; I_{2}; P_{4}; P_{2}; P_{1}; P_{3}; I_{2}; P_{4}; P_{3}; P_{1}; P_{2}; I_{2}; V; L_{1}; L_{2}; L_{3}; L_{4}; P_{1}; P_{2}; P_{3}; P_{4}; I_{0}^{∗}; V′; L′_{1}; L′_{2}; L′_{3}; L′_{4}; P_{1}; P_{2}; P_{3}; P_{4}; I_{0}^{∗}]

51

Alternative Construction of E(2, ω)

Having constructed the scheme E(2, ω) as a desingularization of a certain elliptic threefold, we will in this chapter exhibit an alternative description of E(2, ω) as a blow-up of a certain projective line bundle over projective two-space. We will derive two benefits from this: First, we will be able to describe E(2, ω) explicitly as a subscheme of a product of projective spaces; secondly, we can use our alternative construction to show that certain twists of (the characteristic zero fiber of) E(2, ω) have many rational points, which will allow us then to deduce a visibility type result.

We will start by showing how to explicitly construct certain elementary blow-ups of projective spaces and line bundles; we will then give an explicit description of E(2, ω) and finally show how to apply our construction to answer questions on visibility.

3.1 Some Elementary Blow-ups

In this section we perform some “explicit” blow-ups (in the sense that we describe them as subvarieties of products of projective spaces instead of patching together coordinate charts) which will help us towards our aim to give an explicit alternative description of the scheme E(2, ω), to be achieved in the next section. We assume that the reader is familiar with the description of blow-ups of affine space at a point or along linear subspaces, along the lines of Excercise IV-26 of [EH]; until further notice, we will work over some (arbitrary) fixed field k.

We start with two-dimensional projective space ℙ^{2} with homogeneous coordinates (X_{0} : X_{1} : X_{2}). Let P_{1}, P_{2}, P_{3}, P_{4} be the four closed points in general position with homogeneous coordinates

P_{1} = (1 : 0 : 0)
P_{2} = (0 : 1 : 0)
P_{3} = (0 : 0 : 1)
P_{4} = (1 : −1 : 1).

Since any set of four points in ℙ^{2} in general position is mapped to any other such set by a unique linear homogeneous transformation, the action of S_{4} on the set {P_{1}, P_{2}, P_{3}, P_{4}} by permutation induces an action of S_{4} on ℙ^{2}.

Lemma 4 Let V be the blow-up of ℙ^{2} along the four points P_{1}, P_{2}, P_{3}, P_{4} as given above. Then V can be given as a closed subvariety

V ⊂ ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1}

by the equations

β_{0}(γ_{0} + γ_{1})(δ_{01} − δ_{12}) = (β_{0} − β_{2})γ_{0}δ_{01}  (3.1)
α_{1}(γ_{0} + γ_{1})δ_{12} = (α_{1} + α_{2})γ_{1}δ_{01}  (3.2)
(α_{1} + α_{2})β_{2}(δ_{01} − δ_{12}) = α_{2}(β_{0} − β_{2})δ_{12}  (3.3)
α_{1}β_{2}γ_{0} = α_{2}β_{0}γ_{1}.  (3.4)

The blow-down V → ℙ^{2} can be given on homogeneous coordinates by

ν : (X_{0} : X_{1} : X_{2}) = (α_{2}β_{0} : α_{1}β_{2} : α_{2}β_{2})
= (α_{1}γ_{0} : α_{1}γ_{1} : α_{2}γ_{1})
= (β_{0}γ_{0} : β_{0}γ_{1} : β_{2}γ_{0}),

and the action of S_{4} can be lifted to an action on V, given on generators by

(12) : ((α_{1} : α_{2}), (β_{0} : β_{2}), (γ_{0} : γ_{1}), (δ_{01} : δ_{12}))
↦ ((−β_{0} : β_{2}), (−α_{1} : α_{2}), (γ_{1} : γ_{0}), (δ_{01} : δ_{01} − δ_{12}))
(13) : ((α_{1} : α_{2}), (β_{0} : β_{2}), (γ_{0} : γ_{1}), (δ_{01} : δ_{12}))
↦ ((γ_{1} : γ_{0}), (β_{2} : β_{0}), (α_{2} : α_{1}), (δ_{12} : δ_{01}))
(14) : ((α_{1} : α_{2}), (β_{0} : β_{2}), (γ_{0} : γ_{1}), (δ_{01} : δ_{12}))

↦ ((δ_{01} : δ_{12} − δ_{01}), (β_{0} : β_{0} − β_{2}), (−γ_{0} : γ_{0} + γ_{1}), (α_{1} : α_{1} + α_{2})).

PROOF. It is easy to see that V can be given inside

ℙ_{(X_{0}:X_{1}:X_{2})}^{2} × ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1}

by the equations

X_{1}α_{2} = X_{2}α_{1}
X_{0}β_{2} = X_{2}β_{0}
X_{0}γ_{1} = X_{1}γ_{0}
(X_{0} + X_{1})δ_{12} = (X_{1} + X_{2})δ_{12},

which provides the intuition for the adopted notation and the starting point for the representation of V as stated in the lemma. S_{4} clearly acts on this variety as given. For the other assertions, one first shows that the morphism ν as given in the lemma is indeed an isomorphism away from the points P_{i} and their inverse images in the variety given by equations (3.1) to (3.4). An easy check on local coordinate charts around the points P_{i} and their inverse images under ν then finishes the proof.

Lemma 5 The variety V can also be given as a subvariety inside

V ⊂ ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1} × ℙ_{(ρ:σ)}^{1}

by the equations

α_{1}(β_{0} − β_{2})ρ = (ρ − σ)(α_{1} + α_{2})β_{0}  (3.5)
γ_{1}(δ_{01} − δ_{12})ρ = (ρ − σ)γ_{0}δ_{12}  (3.6)
α_{2}(γ_{0} + γ_{1})ρ = σ(α_{1} + α_{2})γ_{0}  (3.7)
β_{2}δ_{01}ρ = σβ_{0}δ_{12}  (3.8)
α_{1}(δ_{01} − δ_{12})σ = (ρ − σ)α_{2}δ_{01}  (3.9)
γ_{1}(β_{0} − β_{2})σ = (ρ − σ)β_{2}(γ_{0} + γ_{1});  (3.10)

the action of S_{4} here is given as in Lemma 4, plus the action on ℙ_{(ρ:σ)}^{1} given on generators by

\begin{matrix}(12) : & (ρ : σ) & ↦ & (σ − ρ : σ) \\ (13) : & (ρ : σ) & ↦ & (σ : ρ) \\ (14) : & (ρ : σ) & ↦ & (ρ : ρ − σ). \end{matrix}

PROOF. Denote the variety given by equations (3.5) to (3.10) by V′. Then there is a morphism from V to V′ given by

(ρ : σ) = ((α_{1} + α_{2})β_{0} : (α_{1} + α_{2})β_{0} − α_{1}(β_{0} − β_{2}))
= (γ_{0}δ_{12} : γ_{0}δ_{12} − γ_{1}(δ_{01} − δ_{12}))
= ((α_{1} + α_{2})γ_{0} : α_{2}(γ_{0} + γ_{1})) etc.,

following equations (3.5) to (3.10); using equations (3.1) to (3.4), it is an easy task to verify that all these formulae coincide where they are defined, and that the domains of definition cover all of V. Vice versa, one can deduce equations (3.1) to (3.4) from equations (3.5) to (3.10).

We give one example: Multiplying equation (3.9) by β_{2} and substituting for β_{2}δ_{01}ρ using equation (3.8), one obtains equation (3.3) multiplied by σ; likewise, mutliplying equation (3.5) by δ_{12} and substituting for β_{0}δ_{12}σ again using equation (3.8), one obtains equation (3.3) multiplied by ρ, from which we deduce that equation (3.3) holds. Equations (3.1), (3.2) and (3.4) can be deduced from equations (3.5) to (3.10) by analogous arguments.

This easily shows that V and V′ are isomorphic; the assertion about the action of S_{4} is clear.

Having established an explicit description for the blow-up of ℙ^{2} along four points, we also need equations for the projective line bundle ℙ(O(1)⊕O) over ℙ_{(X_{0}:X_{1}:X_{2})}^{2}. It is an easy task to verify that ℙ(O(1) ⊕ O) can in fact be given as a closed subvariety of the space ℙ_{(X_{0}:X_{1}:X_{2})}^{2} × ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} by the equations

X_{1}Y_{2} = X_{2}Y_{1}
X_{0}Y_{2} = X_{2}Y_{0}
X_{0}Y_{1} = X_{1}Y_{0},

and that ℙ(O(1) ⊕ O) is isomorphic to the blow-up of ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} at the point (0 : 0 : 0 : 1). We also remark that the fibration ℙ(O(1) ⊕ O) → ℙ^{2} has two disjoint sections, one given by Y_{3} = 0, the other one given by Y_{0} = Y_{1} = Y_{2} = 0, and that the action of S_{4} on ℙ_{(X_{0}:X_{1}:X_{2})}^{2} as discussed above can be lifted to an action on ℙ(O(1) ⊕ O), compatibly with the fibration, given on generators by

(12) : ((X_{0} : X_{1} : X_{2}), (Y_{0} : Y_{1} : Y_{2} : Y_{3}))
↦ ((X_{1} : X_{0} : −X_{2}), (Y_{1} : Y_{0} : −Y_{2} : −Y_{3}))
(13) : ((X_{0} : X_{1} : X_{2}), (Y_{0} : Y_{1} : Y_{2} : Y_{3}))
↦ ((X_{2} : X_{1} : X_{0}), (Y_{2} : Y_{1} : Y_{0} : Y_{3}))
(14) : ((X_{0} : X_{1} : X_{2}), (Y_{0} : Y_{1} : Y_{2} : Y_{3}))

↦ ((−X_{0} : X_{1} + X_{0} : X_{2} − X_{0}), (−Y_{0} : Y_{1} + Y_{0} : Y_{2} − Y_{0} : Y_{3})).

(We remark that the minus-sign in the Y_{3}-component of the automorphism (12) is necessary to make this an action of S_{4}; also, this way the automorphisms come out as what we want them to be in the end, when we compare our construction here with the variety E(2, ω) constructed in the previous chapter.)

Putting all of this together with Lemma 4, we easily obtain

Proposition 7 Let B be the pullback of the projective line bundle ℙ(O(1) ⊕ O) over ℙ^{2} to the variety V introduced above. Then B can be given as a closed subvariety of

ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1} × ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3}

by equations (3.1) to (3.4) from Lemma 4 above plus the equations

α_{1}Y_{2} = α_{2}Y_{1}
β_{0}Y_{2} = β_{2}Y_{0}
γ_{0}Y_{1} = γ_{1}Y_{0}
δ_{01}(Y_{1} + Y_{2}) = δ_{12}(Y_{0} + Y_{1}).

The fibration B → V has two disjoint sections, one given by Y_{3} = 0 (which we will call V), the other given by Y_{0} = Y_{1} = Y_{2} = 0 (which we will call V′). Moreover, the action of S_{4} on ℙ(O(1) ⊕ O) given above pulls back to an action on B, compatibly with the fibration and the sections, and is given by the formula on the five factors of the ambient space as in the previous lemmas.

We note that we can also construct B as the blow-up of the projective line bundle ℙ(O(1) ⊕ O) over ℙ^{2} along the fibers over the points P_{1}, P_{2}, P_{3}, P_{4} in ℙ_{(X_{0}:X_{1}:X_{2})}^{2}.

Now denote by Q_{1}, Q_{2}, Q_{3}, Q_{4} the four points in ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} given by

Q_{1} : (Y_{0} : Y_{1} : Y_{2} : Y_{3}) = (1 : 0 : 0 : 0)
Q_{2} : (Y_{0} : Y_{1} : Y_{2} : Y_{3}) = (0 : 1 : 0 : 0)
Q_{3} : (Y_{0} : Y_{1} : Y_{2} : Y_{3}) = (0 : 0 : 1 : 0)
Q_{4} : (Y_{0} : Y_{1} : Y_{2} : Y_{3}) = (1 : −1 : 1 : 0).

For our next step, we want to compute the blow-up of B along the lines q_{1}, q_{2}, q_{3}, q_{4} which map to the points Q_{1}, Q_{2}, Q_{3}, Q_{4} ∈ ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3}, respectively, under the obvious projection. We need the following preliminary

Lemma 6 Let W be the blow-up of ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} at the four points Q_{1}, Q_{2}, Q_{3}, Q_{4} given above. Then W is the subvariety of

ℙ_{(a_{1}:a_{2}:a_{3})}^{2} × ℙ_{(b_{0}:b_{2}:b_{3})}^{2} × ℙ_{(c_{0}:c_{1}:c_{3})}^{2} × ℙ_{(d_{01}:d_{12}:d_{3})}^{2}

given by the ten equations

a_{2}b_{3} = a_{3}b_{2}  (3.11)
(c_{0} + c_{1})d_{3} = c_{3}d_{01}  (3.12)
a_{1}c_{3} = a_{3}c_{1}  (3.13)
(b_{0} − b_{2})d_{3} = b_{3}(d_{01} − d_{12})  (3.14)
(a_{1} + a_{2})d_{3} = a_{3}d_{12}  (3.15)
b_{0}c_{3} = b_{3}c_{0}  (3.16)
b_{0}(c_{0} + c_{1})(d_{01} − d_{12}) = (b_{0} − b_{2})c_{0}d_{01}  (3.17)
a_{1}(c_{0} + c_{1})d_{12} = (a_{1} + a_{2})c_{1}d_{01}  (3.18)
(a_{1} + a_{2})b_{2}(d_{01} − d_{12}) = a_{2}(b_{0} − b_{2})d_{12}  (3.19)
a_{1}b_{2}c_{0} = a_{2}b_{0}c_{1}.  (3.20)

Moreover, the action of S_{4} on ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} induced by the above action on ℙ(O(1) ⊕ O) can be lifted to an action on W and is given on generators by

(12) : ((a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3}))
↦ ((−b_{0} : b_{2} : b_{3}), (−a_{1} : a_{2} : a_{3}),
(c_{1} : c_{0} : −c_{3}), (d_{01} : d_{01} − d_{12} : −d_{3}))
(13) : ((a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3}))
↦ ((c_{1} : c_{0} : c_{3}), (b_{2} : b_{0} : b_{3}), (a_{2} : a_{1} : a_{3}), (d_{12} : d_{01} : d_{3}))
(14) : ((a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3}))
↦ ((d_{01} : d_{12} − d_{01} : d_{3}), (−b_{0} : b_{2} − b_{0} : b_{3}),
(−c_{0} : c_{1} + c_{0} : c_{3}), (a_{1} : a_{2} + a_{1} : a_{3})).

PROOF. As in Lemma 4, the starting point for this representation of W, which also provides the intuition for the notation, is its representation inside

ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} × ℙ_{(a_{1}:a_{2}:a_{3})}^{2} × ℙ_{(b_{0}:b_{2}:b_{3})}^{2} × ℙ_{(c_{0}:c_{1}:c_{3})}^{2} × ℙ_{(d_{01}:d_{12}:d_{3})}^{2}

by the equations

\begin{aligned}a_{2}Y_{3} & = & a_{3}Y_{2} & b_{2}Y_{3} & = & b_{3}Y_{2} \\ a_{1}Y_{3} & = & a_{3}Y_{1} & b_{0}Y_{3} & = & b_{3}Y_{0} \\ a_{1}Y_{2} & = & a_{2}Y_{1} & b_{0}Y_{2} & = & b_{2}Y_{0} \\ c_{1}Y_{3} & = & c_{3}Y_{1} & d_{12}Y_{3} & = & d_{3}(Y_{1} + Y_{2}) \\ c_{0}Y_{3} & = & c_{3}Y_{0} & d_{01}Y_{3} & = & d_{3}(Y_{0} + Y_{1}) \\ c_{0}Y_{1} & = & c_{1}Y_{0} & d_{01}(Y_{1} + Y_{2}) & = & d_{12}(Y_{0} + Y_{1}). \end{aligned}

Let W′ denote the variety given by equations (3.11) to (3.20). Then we find as in Lemma 4 that there is a morphism μ from W′ to ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} (which, of course, turns out to be the blow-down map) given by the formulae

μ : (Y_{0} : Y_{1} : Y_{2} : Y_{3})
= (a_{2}b_{0} : a_{1}b_{2} : a_{2}b_{2} : a_{2}b_{3} = a_{3}b_{2})
= (c_{0}d_{01} : c_{1}d_{01} : c_{0}d_{12} − c_{1}d_{01} + c_{1}d_{12} : c_{3}d_{01} = (c_{0} + c_{1})d_{3})
= (a_{1}c_{0} : a_{1}c_{1} : a_{2}c_{1} : a_{3}c_{1} = a_{1}c_{3})
= (b_{0}(d_{01} − d_{12}) : b_{0}d_{12} − b_{2}d_{01} : b_{2}(d_{01} − d_{12}) : b_{3}(d_{01} − d_{12})
= (b_{0} − b_{2})d_{3})
= (a_{1}d_{01} − a_{1}d_{12} + a_{2}d_{01} : a_{1}d_{12} : a_{2}d_{12} : a_{3}d_{12} = (a_{1} + a_{2})d_{3})
= (b_{0}c_{0} : b_{0}c_{1} : b_{2}c_{0} : b_{3}c_{0} = b_{0}c_{3})
= (a_{3}b_{0} : a_{1}b_{3} : a_{2}b_{3} = a_{3}b_{2} : a_{3}b_{3})
= (c_{0}d_{3} : c_{1}d_{3} : c_{3}(d_{12} − d_{01}) + c_{0}d_{3} : c_{3}d_{3})
= (a_{3}c_{0} : a_{1}c_{3} = a_{3}c_{1} : a_{2}c_{3} : a_{3}c_{3})
= (b_{0}d_{3} : b_{3}d_{12} − b_{2}d_{3} : b_{2}d_{3} : b_{3}d_{3})
= (a_{3}d_{01} − a_{1}d_{3} : a_{1}d_{3} : a_{2}d_{3} : a_{3}d_{3})
= (b_{3}c_{0} = b_{0}c_{3} : b_{3}c_{1} : b_{2}c_{3} : b_{3}c_{3});

it is a straightforward check that the domains of definition cover all of W′ and (using the equations for W′ given in the statement of the lemma) that each pair of formulae agree wherever they both make sense. It is moreover clear that the rational map from ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} to W′ given by

\begin{aligned}& ((a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3})) \\ = & ((Y_{1} : Y_{2} : Y_{3}), (Y_{0} : Y_{2} : Y_{3}), (Y_{0} : Y_{1} : Y_{3}) : (Y_{0} + Y_{1} : Y_{1} + Y_{2} : Y_{3})) \end{aligned}

is defined outside the points Q_{1}, Q_{2}, Q_{3}, Q_{4}, and it is easy to verify that it is in fact an inverse to μ : W′ → ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} outside the points Q_{i} and their inverse images (hence μ is an isomorphism away from those closed sets). Finally one checks on the appropriate local coordinate charts that μ does exhibit W′ as the blow-up of ℙ_{(Y_{0}:Y_{1}:Y_{2}:Y_{3})}^{3} at the points Q_{i}. The assertion about the action of S_{4} is clear.

Proposition 8 Let C denote the blow-up of the variety B described in Proposition 7 along the four lines q_{1}, q_{2}, q_{3}, q_{4} given by

\begin{aligned}q_{1} : & Y_{1} & = & Y_{2} & = & Y_{3} & = & 0 \\ q_{2} : & Y_{0} & = & Y_{2} & = & Y_{3} & = & 0 \\ q_{3} : & Y_{0} & = & Y_{1} & = & Y_{3} & = & 0 \\ q_{4} : & Y_{0} + Y_{1} & = & Y_{1} + Y_{2} & = & Y_{3} & = & 0. \end{aligned}

Then C can be given as a subvariety of

\begin{matrix}ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1} \\ ×ℙ_{(a_{1}:a_{2}:a_{3})}^{2} × ℙ_{(b_{0}:b_{2}:b_{3})}^{2} × ℙ_{(c_{0}:c_{1}:c_{3})}^{2} × ℙ_{(d_{01}:d_{12}:d_{3})}^{2} \end{matrix}

by equations (3.1) to (3.4) given in Lemma 4, equations (3.11) to (3.16) given in Lemma 6, and the four equations

α_{1}a_{2} = α_{2}a_{1}  (3.21)
β_{0}b_{2} = β_{2}b_{0}  (3.22)
γ_{0}c_{1} = γ_{1}c_{0}  (3.23)
δ_{01}d_{12} = δ_{12}d_{01}.  (3.24)

The blow-down to B can be described by the formulae given in the proof of Lemma 6. Moreover, S_{4} acts on C compatibly with the blow-down and the action on B; the action can be described by the combination of the formulae given in Lemma 4 and the formulae given in Lemma 6.

PROOF. We begin by observing that equations (3.17) to (3.20) in Lemma 6 can be easily deduced from equations (3.1) to (3.4) given in Lemma 4 and from equations (3.21) to (3.24). Let C denote the variety given by the equations listed in the proposition; it follows from Lemma 6 that the morphism C → B which is described in the proposition is an isomorphism outside the lines q_{i} and their inverse images; the assertions about the S_{4}-action are again clear.

It remains to check over suitably chosen open sets of B that C is locally the desired blow-up. Consider, e. g., the open set U of B given by Y_{0} ≠ 0 ≠ Y_{0} + Y_{1} (so q_{1} ⊂ U, but q_{2}, q_{3}, q_{4} do not intersect U). It follows easily that on U we have β_{0} ≠ 0, γ_{0} ≠ 0, δ_{01} ≠ 0, and that on the inverse image of U in C we also have b_{0} ≠ 0, c_{0} ≠ 0, d_{01} ≠ 0. We thus define affine coordinates Y_{1,0} := Y_{1}/Y_{0}, Y_{2,0} := Y_{2}/Y_{0}, Y_{3,0} := Y_{3}/Y_{0} on U, and c_{1,0} := c_{1}/c_{0}, b_{2,0} := b_{2}/b_{0}, b_{3,0} := b_{3}/b_{0} = c_{3}/c_{0} =: c_{3,0} on the inverse image of U in C.

Chasing through the equations defining B and C on U and its inverse image, and eliminating affine variables to the extent possible, it follows that the open set U is just the closed variety inside 𝔸_{(Y_{1,0},Y_{2,0},Y_{3,0})}^{3} ×ℙ_{(α_{1}:α_{2})}^{1} given by

Y_{1,0} ≠ −1; α_{1}Y_{2,0} = α_{2}Y_{1,0},

with the line q_{1} ⊂ U given by Y_{1,0} = Y_{2,0} = Y_{3,0} = 0. The inverse image of U in C can be described as the variety in 𝔸_{(c_{1,0},b_{2,0},b_{3,0}=c_{3,0})}^{3}×ℙ_{(α_{1}:α_{2})}^{1}×ℙ_{(a_{1}:a_{2}:a_{3})}^{2} given by

\begin{matrix}c_{1,0} ≠ −1; & & α_{1}b_{2,0} = α_{2}c_{1,0}; \\ b_{2,0}a_{3} = b_{3,0}a_{2}; & c_{1,0}a_{3} = c_{3,0}a_{1}; & α_{1}a_{2} = α_{2}a_{1}, \end{matrix}

and the morphism C → B is just

Y_{1,0} = c_{1,0}; Y_{2,0} = b_{2,0}; Y_{3,0} = b_{3,0}.

From this description it can be seen immediately that locally over U ⊂ B, C is the blow-up of B along the line q_{1}; similar considerations over other open subsets of B then give the desired result.

3.2 Constructing E(2, ω)

In order to finish our second construction of E(2, ω), it will turn out that all which is left to do is to blow up the variety C introduced above in Proposition 8 along four lines. We will first perform this blow-up before showing that the scheme we have constructed is, indeed, the same as the one we constructed in the previous chapter. For the time being, we continue to work over some arbitrary field k.

3.2.1 Performing The Final Blow-up

Before we perform the final blow-up of our construction, we make a few comments on the special fibers of the blow-ups we have performed so far:

As we remarked after Proposition 7, the projective line bundle B over V can be constructed as the blow-up of the projective line bundle ℙ(O(1) ⊕ O) over ℙ_{(X_{0}:X_{1}:X_{2})}^{2} along the fibers over the points P_{1}, P_{2}, P_{3}, P_{4}. Let L′_{1}, L′_{2}, L′_{3}, L′_{4} denote the corresponding collections of special fibers in B, as well as their proper transforms in C. An easy calculation shows that the surfaces L′_{i} ⊂ C can be described by the following equations:

L′_{1}  L′_{2}  L′_{3}  L′_{4}
a_{1} = a_{2} = 0 a_{2} = 0 = α_{2} a_{1} = 0 = α_{1} a_{1} + a_{2} = 0 = α_{1} + α_{2}
b_{2} = 0 = β_{2} b_{0} = b_{2} = 0 b_{0} = 0 = β_{0} b_{0} − b_{2} = 0 = β_{0} − β_{2}
c_{1} = 0 = γ_{1} c_{0} = 0 = γ_{0} c_{0} = c_{1} = 0 c_{0} + c_{1} = 0 = γ_{0} + γ_{1}
d_{12} = 0 = δ_{12} d_{01} − d_{12} = 0 d_{01} = 0 = δ_{01}  d_{01} = d_{12} = 0
= δ_{01} − δ_{12}
c_{0}d_{3} = c_{3}d_{01} c_{1}d_{3} = c_{3}d_{01} b_{2}d_{3} = b_{3}d_{12}  b_{0}c_{3} = b_{3}c_{0}
b_{0}d_{3} = b_{3}d_{01} a_{1}d_{3} = a_{3}d_{12} a_{2}d_{3} = a_{3}d_{12}  a_{1}c_{3} = a_{3}c_{1}
b_{0}c_{3} = b_{3}c_{0} a_{1}c_{3} = a_{3}c_{1} a_{2}b_{3} = a_{3}b_{2}  a_{2}b_{3} = a_{3}b_{2}

From this description it follows that each of the L′_{i} is isomorphic to ℙ^{1} × ℙ^{1}.

Also recall that in Proposition 8 we constructed C as the blow-up of B along the four lines q_{1}, q_{2}, q_{3}, q_{4}. Denote the corresponding collections of special fibers in C by L_{1}, L_{2}, L_{3}, L_{4}, respectively. Then the surfaces L_{i} ⊂ C can be described by the following equations:

L_{1}  L_{2}  L_{3}  L_{4}
α_{1}a_{2} = α_{2}a_{1}  α_{2} = 0  α_{1} = 0  α_{1} + α_{2} = 0
a_{2} = a_{3} = 0 a_{1} = a_{3} = 0 a_{1} + a_{2} = a_{3} = 0
β_{2} = 0  β_{0}b_{2} = β_{2}b_{0}  β_{0} = 0  β_{0} − β_{2} = 0
b_{2} = b_{3} = 0  b_{0} = b_{3} = 0 b_{0} − b_{2} = b_{3} = 0
γ_{1} = 0  γ_{0} = 0  γ_{0}c_{1} = γ_{1}c_{0}  γ_{0} + γ_{1} = 0
c_{1} = c_{3} = 0 c_{0} = c_{3} = 0  c_{0} + c_{1} = c_{3} = 0
δ_{12} = 0  δ_{01} = δ_{12}  δ_{01} = 0  δ_{01}d_{12} = δ_{12}d_{01}
d_{12} = d_{3} = 0 d_{01} − d_{12} d_{01} = d_{3} = 0
= d_{3} = 0

Note that each of the surfaces L_{i} is isomorphic to ℙ^{2} blown up at a point.

We also have inside C the proper transforms of the surfaces V and V′ introduced in Proposition 7 (which we will again also denote by the letters V and V′, respectively). V is then given by the equations

V : a_{3} = b_{3} = c_{3} = d_{3} = 0

plus equations (3.1) to (3.4) and equations (3.21) to (3.24), and V′ can be given by

V′ : a_{1} = a_{2} = b_{0} = b_{2} = c_{0} = c_{1} = d_{01} = d_{12} = 0

and equations (3.1) to (3.4).

We can now define the four lines s_{1}, s_{2}, s_{3}, s_{4} inside C by setting s_{i} := L_{i} ∩ L′_{i}. Chasing through the equations giving C, it is easily seen that inside the ambient space

\begin{matrix}ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1} \\ ×ℙ_{(a_{1}:a_{2}:a_{3})}^{2} × ℙ_{(b_{0}:b_{2}:b_{3})}^{2} × ℙ_{(c_{0}:c_{1}:c_{3})}^{2} × ℙ_{(d_{01}:d_{12}:d_{3})}^{2}, \end{matrix}

s_{i} is the intersection of C with the subspace Σ_{i}, where the Σ_{i} are given by

\begin{matrix}Σ_{1} : & a_{1} = a_{2} = b_{3} = c_{3} = d_{3} = 0 \\ Σ_{2} : & a_{3} = b_{0} = b_{2} = c_{3} = d_{3} = 0 \\ Σ_{3} : & a_{3} = b_{3} = c_{0} = c_{1} = d_{3} = 0 \\ Σ_{4} : & a_{3} = b_{3} = c_{3} = d_{01} = d_{12} = 0. \end{matrix}

We can now describe the final blow-up of our construction:

Proposition 9 Let D be the blow-up of C along the four lines s_{i} given above. Then D can be given as a closed subvariety of the space

\begin{matrix}ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(β_{0}:β_{2})}^{1} × ℙ_{(γ_{0}:γ_{1})}^{1} × ℙ_{(δ_{01}:δ_{12})}^{1} × ℙ_{(ρ:σ)}^{1} \\ ×ℙ_{(a_{1}:a_{2}:a_{3})}^{2} × ℙ_{(b_{0}:b_{2}:b_{3})}^{2} × ℙ_{(c_{0}:c_{1}:c_{3})}^{2} × ℙ_{(d_{01}:d_{12}:d_{3})}^{2} × ℙ_{(r:s:t)}^{2} \end{matrix}

by the six equations

α_{1}(β_{0} − β_{2})ρ = (ρ − σ)(α_{1} + α_{2})β_{0}  (3.25)
γ_{1}(δ_{01} − δ_{12})ρ = (ρ − σ)γ_{0}δ_{12}  (3.26)
α_{2}(γ_{0} + γ_{1})ρ = σ(α_{1} + α_{2})γ_{0}  (3.27)
β_{2}δ_{01}ρ = σβ_{0}δ_{12}  (3.28)
α_{1}(δ_{01} − δ_{12})σ = (ρ − σ)α_{2}δ_{01}  (3.29)
γ_{1}(β_{0} − β_{2})σ = (ρ − σ)β_{2}(γ_{0} + γ_{1})  (3.30)

(these are equations (3.5) to (3.10) from Lemma 5), the five equations

α_{1}a_{2} = α_{2}a_{1}  (3.31)
β_{0}b_{2} = β_{2}b_{0}  (3.32)
γ_{0}c_{1} = γ_{1}c_{0}  (3.33)
δ_{01}d_{12} = δ_{12}d_{01}  (3.34)
ρs = σr  (3.35)

(the first four of which are equations (3.21) to (3.24) from Proposition 8), the six equations

a_{2}b_{3} = a_{3}b_{2}  (3.36)
(c_{0} + c_{1})d_{3} = c_{3}d_{01}  (3.37)
a_{1}c_{3} = a_{3}c_{1}  (3.38)
(b_{0} − b_{2})d_{3} = b_{3}(d_{01} − d_{12})  (3.39)
(a_{1} + a_{2})d_{3} = a_{3}d_{12}  (3.40)
b_{0}c_{3} = b_{3}c_{0}  (3.41)

(which are equations (3.11) to (3.16) from Lemma 6), and the six equations

a_{3}b_{3}(r − s) = ta_{1}(b_{0} − b_{2})  (3.42)
c_{3}d_{3}(r − s) = tc_{1}(d_{01} − d_{12})  (3.43)
a_{3}c_{3}r = t(a_{1} + a_{2})c_{0}  (3.44)
b_{3}d_{3}r = tb_{0}d_{12}  (3.45)
a_{3}d_{3}s = ta_{2}d_{01}  (3.46)
b_{3}c_{3}s = tb_{2}(c_{0} + c_{1}).  (3.47)

The blow-down map from D to C is induced by the projection onto the obvious factors of the ambient spaces. Moreover, the action of S_{4} on C given above lifts to an action on D, compatibly with the blow-down map; the action is given on the factors ℙ_{(ρ:σ)}^{1} × ℙ_{(r:s:t)}^{2} by the formulae

\begin{matrix}(12) : & ((ρ : σ), (r : s : t)) ↦ (σ − ρ : σ), (s − r : s : −t)) \\ (13) : & ((ρ : σ), (r : s : t)) ↦ (σ : ρ), (s : r : t)) \\ (14) : & ((ρ : σ), (r : s : t)) ↦ (ρ : ρ − σ), (r : r − s : −t)) \end{matrix}

and is defined as previously on the remaining factors.

Before we give the proof of Proposition 9, we point out that in addition to equations (3.25) to (3.47), the following six equations also hold on D:

a_{3}b_{3}r = t(a_{1} + a_{2})b_{0}  (3.48)
c_{3}d_{3}r = tc_{0}d_{12}  (3.49)
a_{3}c_{3}s = ta_{2}(c_{0} + c_{1})  (3.50)
b_{3}d_{3}s = tb_{2}d_{01}  (3.51)
a_{3}d_{3}(r − s) = ta_{1}(d_{01} − d_{12})  (3.52)
b_{3}c_{3}(r − s) = t(b_{0} − b_{2})c_{1}.  (3.53)

We indicate briefly how to deduce equation (3.48) from the equations for D given in the proposition:

• Multiplying equation (3.44) by b_{0} and then substituting c_{3}b_{0} by b_{3}c_{0} (using equation (3.41)), we obtain equation (3.48) multiplied by c_{0}.

• Multiplying equation (3.41) by t(a_{1} + a_{2}), we obtain the equation

b_{3}t(a_{1} + a_{2})c_{0} = t(a_{1} + a_{2})b_{0}c_{3};

using equation (3.44) again to substitute t(a_{1} + a_{2})c_{0} by a_{3}c_{3}r, we obtain equation (3.48) multiplied by c_{3}.

• Finally, multiplying equation (3.38) by b_{3}s, we obtain

a_{1}sb_{3}c_{3} = c_{1}a_{3}b_{3}s;

using equation (3.47) to substitute tb_{2}(c_{0} + c_{1}) for the factor sb_{3}c_{3} on the left hand side, and using equation (3.42) to substitute a_{3}b_{3}r + ta_{1}(b_{2} − b_{0}) for the factor a_{3}b_{3}s on the right hand side, we deduce after some cancellation of terms that

c_{1}a_{3}b_{3}r = t(a_{1}b_{2}c_{0} + a_{1}b_{0}c_{1}) (*).

Last but not least, we note that by Lemma 5 in the previous subsection, equation (3.4) (as stated Lemma 4) holds on D; using equations (3.31) to (3.33), one easily obtains the equation

a_{1}b_{2}c_{0} = a_{2}b_{0}c_{1}.

Using this equation to substitute for the term a_{1}b_{2}c_{0} in equation (*), we obtain equation (3.48) mutliplied by c_{1}.

It follows immediately from these calculations that equation (3.48) holds on D. Equations (3.49) to (3.53) can be deduced similarly. (We could also have arrived at this conclusion after any one of the above three calculations by noting that the variety D is irreducible and is not contained in any of the closed sets c_{0} = 0, c_{1} = 0, or c_{3} = 0.)

We note that, having established equations (3.48) to (3.53), the assertions about the action of S_{4} made in Proposition 9 are clear. In particular, as has been the case with all the other varieties in this chapter so far, the action of S_{4} extends to an action on the ambient projective space!

PROOF OF PROPOSITION 9. Let D be the variety given by equations (3.25) to (3.47). We start by observing that outside the closed sets on C and D each given by

a_{3}b_{3} = c_{3}d_{3} = a_{3}c_{3} = b_{3}d_{3} = a_{3}d_{3} = b_{3}c_{3}
= a_{1}(b_{0} − b_{2}) = c_{1}(d_{01} − d_{12}) = (a_{1} + a_{2})c_{0}
= b_{0}d_{12} = a_{2}d_{01} = b_{2}(c_{0} + c_{1}) = 0,

the two varieties are isomorphic. The twelve terms above are the “coefficients” of t and r, s, r − s in equations (3.42) to (3.47); if one of those coefficients is non-zero, it is an easy task to check on coordinate charts that C and D are indeed locally isomorphic there. We mention in particular that the “compatibility” of equations (3.42) to (3.47) with each other can be easily checked using equations (3.25) to (3.41); i. e. for any

((a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3}))

allowed by equations (3.25) to (3.41), there is a triple (r : s : t) satisfying equations (3.42) to (3.47).

It follows from chasing through equations (3.25) to (3.47) that all that is left to check is what happens locally around the lines s_{1}, s_{2}, s_{3}, s_{4} in C and their inverse images in D. Having established that S_{4} acts on C and D compatibly with the map D → C, we only need to do this for s_{1}. An easy chase through the equations for C shows that s_{1} ⊂ C is contained in the Zariski open set U_{C,1} of C given by

a_{3} ≠ 0, b_{0} ≠ 0, c_{0} ≠ 0, c_{0} + c_{1} ≠ 0, d_{01} ≠ 0.  (3.54)

Going through the equations defining C, it is a routine calculation to verify that the open set U_{C,1} is isomorphic to the variety inside

𝔸^{3}[a_{1,3}; a_{2,3}; c_{3,0}] × ℙ_{(α_{1}:α_{2})}^{1}

given by

α_{1}a_{2,3} = α_{2}a_{1,3}, 1 + a_{1,3}c_{3,0} ≠ 0;

here, a_{1,3}, a_{2,3}, c_{3,0} denote the affine coordinates a_{1}/a_{3}, a_{2}/a_{3}, c_{3}/c_{0} respectively. The line s_{1} is given on U_{C,1}by the equations a_{1,3} = a_{2,3} = c_{3,0} = 0.

Similarly, the Zariski open set U_{D,1} of D given by the same equations (3.54) as above is isomorphic to the variety inside

𝔸^{3}[a_{1,3}; a_{2,3}; c_{3,0}] × ℙ_{(α_{1}:α_{2})}^{1} × ℙ_{(r:s:t)}^{2}

given by the equations

r(1 + a_{1,3}c_{3,0})α_{2} = s(α_{1} + α_{2})
α_{1}a_{2,3} = α_{2}a_{1,3}
c_{3,0}r = t(a_{1,3} + a_{2,3})
c_{3,0}s = ta_{2,3}(1 + a_{1,3}c_{3,0})
1 + a_{1,3}c_{3,0} ≠ 0,

and the map U_{D,1} → U_{C,1} is just the projection map on coordinates. But now it is a simple matter (e. g. using Proposition 1 from Section 2.3 on coordinate charts) to verify that U_{D,1} is the blow-up of U_{C,1} along the projective line s_{1}, and the proposition is proved.

Having established Proposition 9, we will finish this subsection with a remark on some subsurfaces inside D. Let S_{i} be the collection of special fibers of the blow-down map D → C over the line s_{i} ⊂ C, for i = 1, 2, 3, 4.

Then the surfaces S_{i} can be described as follows:

S_{1}  S_{2}  S_{3}  S_{4}
a_{1} = a_{2} = 0  a_{2} = 0 = a_{3}  a_{1} = 0 = a_{3} a_{1} + a_{2} = 0 = a_{3}
b_{2} = 0 = b_{3}  b_{0} = b_{2} = 0  b_{0} = 0 = b_{3} b_{0} − b_{2} = 0 = b_{3}
c_{1} = 0 = c_{3}  c_{0} = 0 = c_{3}  c_{0} = c_{1} = 0 c_{0} + c_{1} = 0 = c_{3}
d_{12} = 0 = d_{3} d_{01} − d_{12} = 0 = d_{3} d_{01} = 0 = d_{3} d_{01} = d_{12} = 0
α_{2} = 0  α_{1} = 0  α_{1} + α_{2} = 0
β_{2} = 0  β_{0} = 0  β_{0} − β_{2} = 0
γ_{1} = 0  γ_{0} = 0  γ_{0} + γ_{1} = 0
δ_{12} = 0  δ_{01} − δ_{12} = 0  δ_{01} = 0
(α_{1} + α_{2})σ  β_{0}σ  (γ_{0} + γ_{1})ρ  δ_{01}ρ
= α_{2}ρ  = β_{2}ρ  = γ_{0}σ  = δ_{12}σ
ρs = σr  ρs = σr  ρs = σr  ρs = σr.

In particular, each S_{i} is isomorphic to ℙ^{2} blown up at a point.

It is, moreover, easy to see that for each i = 1, 2, 3, 4 the proper transform of the surface L_{i} (introduced above as a subsurface of C) in D, which we will also call L_{i}, is given by the same equations as listed earlier, plus the condition t = 0 and one extra equation each, which are as follows:

\begin{matrix}L_{1} & L_{2} & L_{3} & L_{4} \\ (α_{1} + α_{2})σ & β_{0}σ & (γ_{0} + γ_{1})ρ & δ_{01}ρ \\ = α_{2}ρ & = β_{2}ρ & = γ_{0}σ & = δ_{12}σ \end{matrix}

(these are the penultimate equations in the description of the surfaces S_{i} in the previous paragraph). Also, the proper transform of each of the surfaces L′_{i} in D is again defined by the same equations as earlier, plus the condition r = s = 0 and one extra equation, which is the same as the extra equation for L_{i} just stated. – We moreover have the proper transforms of the surfaces V and V′ in D, which can be given as follows: The equations for V are

V : a_{3} = b_{3} = c_{3} = d_{3} = t = 0,

plus equations (3.25) to (3.35); the equations for V′ are

V′ : a_{1} = a_{2} = b_{0} = b_{2} = c_{0} = c_{1} = d_{01} = d_{12} = r = s = 0,

plus equations (3.25) to (3.30).

3.2.2 Comparison with the Previous Construction

Having constructed the variety D, we now need to establish that it is the same variety as E(2, ω) we constructed in the previous chapter, and that the actions of S_{4} we have defined on each coincide. We start by making the following observation:

Lemma 7 The equation

d_{3}^{2}(rd_{01} − sd_{12}) = td_{01}d_{12}(d_{01} − d_{12})  (3.55)

holds on D.

PROOF. In order to deduce that equation (3.55) holds, we will show that it holds when multiplied by c_{0}, by c_{1} or by c_{3}, by using the equations we established for D in the previous subsection:

• We start with the equations

c_{0}(r − s)d_{12}
+[−c_{0}(r − s)d_{12} + (c_{0} + c_{1})r(d_{01} − d_{12})] = (c_{0} + c_{1})r(d_{01} − d_{12})
c_{1}(d_{01} − d_{12})r = (r − s)c_{0}d_{12};

the first equation is trivial to verify, and the second equation follows easily from equation (3.26) using equations (3.33) to (3.35). We use the second equation to substitute the terms inside the square brackets in the first equation by c_{0}r(d_{01} − d_{12}) and then mutliply both sides by d_{3}^{2}, to deduce after some elementary arithmetic that

c_{0}d_{3}^{2}(rd_{01} − sd_{12}) = (d_{01} − d_{12})(c_{0} + c_{1})d_{3}^{2}r
(by eqn (3.37)) = (d_{01} − d_{12})d_{01}c_{3}d_{3}r
(by eqn (3.49)) = c_{0}td_{01}d_{12}(d_{01} − d_{12}),

which is equation (3.55) mutliplied by c_{0}.

• If we start with the equations

c_{1}(r − s)d_{12} + c_{0}(r − s)d_{12} = (c_{0} + c_{1})(r − s)d_{12}
c_{1}(d_{01} − d_{12})r = (r − s)c_{0}d_{12},

where the first equation is trivial and the second one is as above, we can again use the second equation to substitute for the term c_{0}(r − s)d_{12} in the first equation and mutliply through by d_{3}^{2}, to deduce that

c_{1}d_{3}^{2}(rd_{01} − sd_{12}) = d_{12}(c_{0} + c_{1})d_{3}^{2}(r − s)
(by eqn (3.37)) = d_{12}d_{01}c_{3}d_{3}(r − s)
(by eqn (3.43)) = c_{1}td_{01}d_{12}(d_{01} − d_{12}),

which is equation (3.55) mutliplied by c_{1}.

• Finally, we can start with the (trivial) equation

tc_{1}(d_{01} − d_{12})d_{12} + tc_{0}(d_{01} − d_{12})d_{12} = td_{12}(d_{01} − d_{12})(c_{0} + c_{1});

using equation (3.43) to substitute in the first summand on the left hand side and equation (3.49) to substitute in the second summand, we deduce that

c_{3}d_{3}(r − s)d_{12} + c_{3}d_{3}r(d_{01} − d_{12}) = td_{12}(d_{01} − d_{12})(c_{0} + c_{1}).

Mutliplying through by d_{3} and using (3.37) to substitute on the right hand side of the resulting equation, we finally obtain equation (3.55) mutliplied by c_{3}.

From these calculations, the lemma follows immediately. (We could again also have deduced our lemma after any one of the above three calculations by noting that D is an irreducible variety and is not contained in any of the closed sets c_{0} = 0, c_{1} = 0 or c_{3} = 0.)

It is clear from Lemma 7 that there is a morphism q : D → E, where E is the variety with which we started our construction of E(2, ω) in the previous chapter. Recall that E was given as a subvariety of ℙ_{(x_{1}:x_{2}:x_{3})}^{2} × ℙ_{(r:s:t)}^{2} by the equation

x_{3}^{2}(rx_{1} − sx_{2}) = tx_{1}x_{2}(x_{1} − x_{2}),

and q is thus given by simply setting (x_{1} : x_{2} : x_{3}) = (d_{01} : d_{12} : d_{3}). It remains to show that q exhibits D as E(2, ω):

Theorem 3 There is an isomorphism i : D ˜→E(2, ω) such that the diagram

D  →^{i}  E(2, ω)
q ↘  ↙  p

E commutes (here, p : E(2, ω) → E is the blowdown map from the last chapter); moreover, i is compatible with the actions of S_{4} on D and E(2, ω), as defined in this chapter and the last.

PROOF. We start by noting that the action of S_{4} on D is generically compatible with the (birational) action of S_{4} on E and the morphism q. To show this, we can simply write out the action of S_{4} on each of the two varieties.

It is an easy task from the previous lemmata and propositions to calculate that the action of S_{3} ⊂ S_{4} on D is given on the factors ℙ_{(d_{01}:d_{12}:d_{3})}^{2} × ℙ_{(r:s:t)}^{2} by

\begin{matrix}(12) :((d_{01} : d_{12} : d_{3}), (r : s : t)) ↦ & ((d_{01} : d_{01} − d_{12} : d_{3}), (s − r : s : −t)) \\ (13) :((d_{01} : d_{12} : d_{3}), (r : s : t)) ↦ & ((d_{12} : d_{01} : d_{3}), (s : r : t)), \end{matrix}

which, looking back to Lemma 1 of the previous chapter, is clearly compatible with the action of S_{3} on E. For the translations by two-torsion sections, note that the automorphism (14)(23) of D is given on the factors ℙ_{(d_{01}:d_{12}:d_{3})}^{2} × ℙ_{(r:s:t)}^{2} by

(14)(23) :((d_{01} : d_{12} : d_{3}), (r : s : t)) ↦ ((a_{2} : a_{2} + a_{1} : −a_{3}), (r : s : t));

on the other hand, it is an easy task to calculate (using the chord tangent method) that generically on E, translation by the two-torsion section P_{1} is given by the rational map

(x_{1} : x_{2} : x_{3}) ↦ (sx_{2}(x_{1} − x_{2}) : x_{2}(rx_{1} − sx_{2}) : x_{3}(sx_{2} − rx_{1}))

(cf. Section 2.5.2). It is now an easy calculation using the equations given in Proposition 9 to verify that on D, we have

(a_{2} : a_{2} + a_{1} : −a_{3}) = (sd_{12}(d_{01} − d_{12}) : d_{12}(rd_{01} − sd_{12}) : d_{3}(sd_{12} − rd_{01}))

wherever the latter is defined; since (14)(23) and S_{3} generate the group S_{4}, this proves our assertion.

We are going to use the compatibility of the actions of S_{4} on D and E in the sequel in order to minimize the number of coordinate charts over which we have to check that D and E(2, ω) are isomorphic: If we can find an open set on D and an open set on E(2, ω) which are isomorphic compatibly with the morphisms q and p to E, and whose images under the S_{4}-action cover D and E(2, ω), we are done, since both p and q are birational morphisms (for q, this will follow from the first point below), and since all varieties involved are separated, so morphisms on dense Zariski-open sets extend uniquely (if they extend at all).

We split the rest of the proof of the theorem into three parts, by constructing the isomorphism i on appropriate open sets. We will carry out our construction by comparing the coordinate charts for E(2, ω) we computed in the previous chapter with suitably chosen coordinate charts of D; straightforward calculations establish that the definitions of the isomorphism i on different Zariski-open sets are compatible. The equations for D on the coordinate charts below follow in a straightforward manner from the equations given in Proposition 9 and Lemma 7; in each case, it remains to check that the affine variables by which we describe D locally determine all the coordinates in such a way that they satisfy all the equations in Proposition 9. These are easy exercises left to the reader.

• The morphism q is an isomorphism between D and E on the Zariskiopen sets given by t ≠ 0 and r ≠ 0 or s ≠ 0 or r ≠ s, and not r = s = 0 (i. e. over the open set of E consisting of the smooth fibers and the fibers with multiplicative reduction over ℙ_{(r:s:t)}^{2}).

In order to prove this assertion, we first note that since the action of S_{3} on D and E is compatible with q, it is enough to consider the Zariski open sets given by t ≠ 0, r ≠ 0, s ≠ 0. Moreover, using the translations by two-torsion sections on E (which is defined on the open set in question) and the action of ⟨(12)(34), (13)(24), (14)(23)⟩ ⊂ S_{4} on D, it is not hard to see that we can assume that d_{3} ≠ 0 ≠ x_{3}, as well. But it is now an easy task to chase through the equations for D and show that on this open set, we also have a_{2} ≠ 0, b_{0} ≠ 0, c_{0} ≠ 0, and that D on this coordinate chart is actually isomorphic to the affine variety inside 𝔸^{4}[r_{t}, s_{t}, d_{01,3}, d_{12,3}] (these are the obvious affine coordinates, i. e. r_{t} = r/t, d_{01,3} = d_{01}/d_{3}, etc.) given by the equation

r_{t}d_{01,3} − s_{t}d_{12,3} = d_{01,3}d_{12,3}(d_{01,3} − d_{12,3})

(cf. Lemma 7), and our assertion follows. Note that p : E(2, ω) → E is an isomorphism over the open set of E under investigation here, so i = p^{−1} ∘ q gives an isomorphism between the corresponding open sets of D and E(2, ω).

• The varieties D and E(2, ω) are isomorphic over a neighborhood of the fibers of E(2, ω) with additive reduction over X(2, ω) over the line t = 0.

Again, using the action of S_{3} on both varieties, we may assume that r ≠ 0 ≠ s; using the translations by two-torsion sections on E(2, ω) and the action of ⟨(12)(34), (13)(24), (14)(23)⟩ ⊂ S_{4} on D, we may also assume that d_{01} ≠ 0 ≠ d_{12}, and consequently δ_{01} ≠ 0 ≠ δ_{12}. We distinguish the following cases:

– b_{3} ≠ 0.

On this open set of D, we may assume that 1 − t_{r}b_{2,3}b_{0,3} ≠ 0 ≠ s_{r} − t_{r}b_{2,3}b_{0,3}, again with the obvious affine coordinates (since we only need to construct i on some Zariski-open set containing the fibers over t = 0). By chasing through the equations giving D in Proposition 9, and in particular from equations (3.28), (3.39) and (3.45), it follows that D is given on the open set in question by the equation

b_{0,3}^{2}t_{r}(d_{01,12} − s_{r}) = d_{01,12}(d_{01,12} − 1)

inside 𝔸^{4}[d_{01,12}, b_{0,3}, s_{r}, t_{r}]. But recall from the previous chapter (Section 2.4) that E(2, ω) is given on the coordinate chart ℭ_{3} by the equation

B_{0,3}^{2}t_{r}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, B_{0,3}, s_{r}, t_{r}]; hence, the morphism given by setting x_{1,2} = d_{01,12}, B_{0,3} = b_{0,3} is an isomorphism between the open set of D in question and a certain open set on the coordinate chart ℭ_{3} of E(2, ω). Note that this isomorphism (as well as the following cases) explains the naming of variables in Section 2.4, as promised earlier!

– b_{0} ≠ 0 (⇔ b_{2} ≠ 0 on the open set of D in question, by equations (3.28) and (3.32) of Proposition 9). Here we need to consider two subcases:

∗ γ_{0} ≠ 0. Using equation (3.26) from Proposition 9, it turns out that D is given on this open set by the equation

d_{01,12} − s_{r} = \frac{1 + γ_{1,0}}{d_{01,12}} d_{01,12}(d_{01,12} − 1)

inside 𝔸^{4}[d_{01,12}, d_{3,12}, s_{r}, γ_{1,0}] (no condition on d_{3,12}). But E(2, ω) is given on the coordinate chart ℭ_{42} by the equation

x_{1,2} − s_{r} = Γ′′_{01,0}x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, x_{3,2}, s_{r}, Γ′′_{01,0}] (no condition on x_{3,2}). Hence the morphism given by setting x_{1,2} = d_{01,12}, x_{3,2} = d_{3,12}, Γ′′_{01,0} = \frac{1+γ_{1,0}}{d_{01,12}} (note that d_{01,12} ≠ 0 on the open set of D on which we are working) gives an isomorphism between our open set of D and a certain open set on the chart ℭ_{42} of E(2, ω).

∗ γ_{01} := γ_{0} + γ_{1} ≠ 0. Again from equation (3.26), it follows that on this open set, D is given by the equation

(γ_{0,01}d_{01,12})(d_{01,12} − s_{r}) = d_{01,12}(d_{01,12} − 1)

inside 𝔸^{4}[d_{01,12}, γ_{0,01}, s_{r}, b_{3,0}] (no condition on b_{3,0}), where γ_{0,01} := γ_{0}/γ_{01}, and the other variables are the obvious affine coordinates. But on the coordinate chart ℭ_{41} from the previous chapter, E(2, ω) is given by the equation

Γ′′_{0,01}(x_{1,2} − s_{r}) = x_{1,2}(x_{1,2} − 1)

inside 𝔸^{4}[x_{1,2}, Γ′′_{0,01}, s_{r}, B_{3,0}] (no condition on B_{3,0}), so the morphism given by setting x_{1,2} = d_{01,12}, Γ′′_{0,01} = d_{01,12}γ_{0,01}, B_{3,0} = b_{3,0} gives an isomorphism from the open set of D in question to a certain open subset of the chart ℭ_{41} of E(2, ω) (recall again that d_{01,12} ≠ 0 on the open set on which we are working).

These cases cover the open set of D under consideration, so the above assertion is proved.

• The varieties D and E(2, ω) are isomorphic over a neighborhood of the fibers of E(2, ω) with additive reduction over X(2, ω) over the exceptional fiber of X(2, ω).

Clearly we may assume here that t ≠ 0; again, by using the S_{3}-actions, we may also assume that ρ ≠ 0 ≠ σ, and by making use of the action of ⟨(12)(34), (13)(24), (14)(23)⟩ ⊂ S_{4} on D, we may also assume that δ_{01} ≠ 0 ≠ δ_{12}.

On the coordinate chart d_{01} ≠ 0, it is an easy task to deduce from Lemma 7 that D is given by the equation

s_{t}d_{3,01}^{2}(ρ_{σ} − d_{12,01}) − d_{12,01}(1 − d_{12,01})

inside 𝔸^{4}[d_{12,01}, d_{3,01}, ρ_{σ}, s_{t}]. But note that on the coordinate chart ℭ′_{01} (introduced in Section 2.5), E(2, ω) is given by the equation

s_{t}x_{3,1}^{2}(ρ_{σ} − x_{2,1}) = x_{2,1}(1 − x_{2,1})

inside 𝔸^{4}[x_{2,1}, x_{3,1}, ρ_{σ}, s_{t}], so that x_{2,1} = d_{12,01}, x_{3,1} = d_{3,01} gives a local isomorphism between D and E(2, ω). Also note that since we work on an open set of D on which δ_{01} ≠ 0 ≠ δ_{12}, the coordinate charts d_{01} ≠ 0 and d_{12} ≠ 0 coincide (cf. equation (3.34)), so we can assume that d_{3} ≠ 0 in the sequel. We have to again distinguish several cases:

– a_{2} ≠ 0.

From equations (3.40) and (3.46), it follows that the equation

ta_{2}δ_{01}(a_{1} + a_{2}) = sa_{3}^{2}δ_{12}

holds on D. Multiplying this equation by σ(δ_{01} − δ_{12}) and using equation (3.29), we deduce that

ρ_{σ}a_{3,2} − δ_{12,01}a_{3,2} = s_{t}a_{3,2}^{3}δ_{12,01}(1 − δ_{12,01})

holds on the open set of D in question, and it turns out that that D is actually given by this equation there. But recall that on the coordinate chart ℭ′_{1}, E(2, ω) is given by the equation

ρ_{σ}A_{3,2} − A′_{3,2} = s_{t}A_{3,2}A′_{3,2}(A_{3,2} − A′_{3,2})

inside 𝔸^{4}[A_{3,2}, A′_{3,2}, ρ_{σ}, s_{t}]. Hence, the map given by A_{3,2} = a_{3,2}, A′_{3,2} = δ_{12,01}a_{3,2} gives an isomorphism from our open set of D to a certain open subset of ℭ′_{1} (note that δ_{12,01} ≠ 0 on the open set of D on which we are working).

– a_{3} ≠ 0. We need to consider two subcases:

∗ α_{2} ≠ 0. On this open set, D is an open subset of affine three-space 𝔸^{3}[α_{1,2}, δ_{12,01}, a_{2,3}]. But looking back to the equation for the coordinate chart ℭ′_{21}, we see that E(2, ω) on that chart is just affine three-space 𝔸^{3}[A′_{12,2}, Δ_{12,01}, A_{2,3}], so that the map given by Δ_{12,01} = δ_{12,01}, A_{2,3} = a_{2,3}, A′_{12,2} = (1 + α_{1,2})/δ_{12,01} (recall again that δ_{12,01} ≠ 0 here) gives an isomorphism between the open set of D in question and a certain open set in the coordinate chart ℭ′_{21} of E(2, ω).

∗ α_{12} := α_{1} + α_{2} ≠ 0. Let α_{2,12} := α_{2}/α_{12}. Then it follows from equation (3.29) that D on this coordinate chart is given by the equation

(δ_{12,01}α_{2,12})(ρ_{σ} − δ_{12,01}) = δ_{12,01}(1 − δ_{12,01})

inside 𝔸^{4}[d_{01,3}, δ_{12,01}, ρ_{σ}, α_{2,12}] (no condition on d_{01,3}). But E(2, ω) on the coordinate chart ℭ′_{22} is given by the equation

A′_{2,12}(ρ_{σ} − Δ_{12,01}) = Δ_{12,01}(1 − Δ_{12,01})

inside 𝔸^{4}[x_{1,3}, Δ_{12,01}, ρ_{σ}, A′_{2,12}] (no condition on x_{1,3}), so that the map given by x_{1,3} = d_{01,3}, Δ_{12,01} = δ_{12,01}, A′_{2,12} = δ_{12,01}α_{2,12} gives an isomorphism between the open set of D here and (an open subset of) the coordinate chart ℭ′_{22} of E(2, ω).

We finally note that the open set of D under consideration does not intersect the closed set given by a_{2} = a_{3} = 0, so it is covered by the above cases, and the assertion is proved.

By our remarks above, we have defined the isomorphism i : D → E(2, ω) on a large enough open set in order to extend it to all of D via the action of S_{4}. We also repeat that it can be established by straightforward calculations that the local definitions of the isomorphism i which we gave do agree on overlaps of their domains. The proof of the theorem is thus complete.

We conclude this section with three remarks:

1. Our construction of the variety D works over fields in any characteristic, and since D has been constructed by a series of successive blowups along projective lines starting with a smooth variety, D is itself smooth. It is moreover clear that we can “globalize” our construction to obtain a scheme D (projective and smooth) over Spec(ℤ), which is isomorphic (over ℤ[1/2], say) to E(2, ω). Note that the constructions of both D and E(2, ω) do actually work in characteristic two, as well; there, however, the threefold we obtain “has nothing to do with elliptic curves”, as the fibers over ℙ_{(r:s:t)}^{2} are all singular.

2. It is easy to give the action of 𝔾_{m} on D which corresponds to the 𝔾_{m}-action on E(2, ω) induced by the action on the universal differential of the moduli problem on the smooth fibers over ℙ_{(r:s:t)}^{2}. If μ is an invertible section on the base over which we are working (i. e. a field, or Spec(ℤ[1/2]), or ...), then the corresponding automorphism ∂_{μ} is given in terms of projective coordinates by the formulae

∂_{μ} :((α_{1} : α_{2}), (β_{0} : β_{2}), (γ_{0} : γ_{1}), (δ_{01} : δ_{12}), (ρ : σ),

(a_{1} : a_{2} : a_{3}), (b_{0} : b_{2} : b_{3}), (c_{0} : c_{1} : c_{3}), (d_{01} : d_{12} : d_{3}), (r : s : t)) ↦ ((α_{1} : α_{2}), (β_{0} : β_{2}), (γ_{0} : γ_{1}), (δ_{01} : δ_{12}), (ρ : σ),

(a_{1} : a_{2} : μ^{−1}a_{3}), (b_{0} : b_{2} : μ^{−1}b_{3}), (c_{0} : c_{1} : μ^{−1}c_{3}),

(d_{01} : d_{12} : μ^{−1}d_{3}), (r : s : μ^{−2}t)).

This action descends through all the blow-ups to the schemes C, B and W which we constructed in the first section of this chapter; in particular, it descends to a trivial action on the scheme V.

3. Comparing the subsurfaces of D which we mentioned at the end of Section 3.2.1 with some of the subsurfaces which we saw inside E(2, ω) in the previous chapter, we obtain the following correspondences: Under the isomorphism between D and E(2, ω) which we established, the surface S_{i} ⊂ D corresponds to the two-torsion section P_{i} of E(2, ω) over X(2, ω), for i = 1, 2, 3, 4; the surface L_{i} ⊂ D corresponds to the surface inside E(2, ω) consisting of the simple component L_{i} in each of the fibers with additive reduction over the line t = 0 in X(2, ω), and the surface V ⊂ D corresponds to the surface inside E(2, ω) consisting of the double component(s) in each of those fibers; and, finally, for i = 1, 2, 3, 4, the surface L′_{i} ⊂ D corresponds to the surface inside E(2, ω) consisting of the single component L′_{i} in each of the fibers with additive reduction over the exceptional divisor of X(2, ω), and the surface V′ ⊂ D corresponds to the surface inside E(2, ω) consisting of the double component(s) in each of those fibers. (This correspondence explains the way in which we numbered the fiber components in the previous chapter.)

3.3 Statement of Main Result

Summarizing our two constructions of E(2, ω) in this chapter and the last, we obtain the following Theorem 4 There is a compactification E(2, ω) over X(2, ω) of the universal elliptic curve F(2, ω) over the moduli scheme Y (2, ω) over ℤ[1/2], classifying elliptic curves with full level-two-structure and an invariant regular non-vanishing relative differential, such that

• X(2, ω) and E(2, ω) are both smooth, projective and integral over Spec(ℤ[1/2]), E(2, ω) is flat over X(2, ω), and the fiber of E(2, ω) over any point of Spec(ℤ[1/2]) is a regular and rational variety. In any characteristic (other than two), the singular fibers of E(2, ω) over X(2, ω) are of type I_{2}, I_{0}^{∗} or I_{1}^{∗}, referring to the Kodaira-Néron classification. Moreover, E(2, ω) is minimal over X(2, ω) in the appropriate sense.

• The two-torsion sections of the generic fiber extend to regular sections of E(2, ω) over X(2, ω) which are contained in the smooth locus of E(2, ω) over X(2, ω), and the translations by two-torsion sections extend to regular automorphisms of E(2, ω).

• The actions of SL_{2}(𝔽_{2}) ≅ S_{3} and of 𝔾_{m} on F(2, ω) and Y (2, ω) induced from their actions on the underlying moduli problem extend to actions on E(2, ω) over X(2, ω).

Moreover, there is a sequence of morphisms

E(2, ω) → B → V → ℙ^{2}  (3.56)

over ℤ[1/2], where V is the blow-up of ℙ^{2} at four ℤ[1/2]-points in general position, and B is the pullback to V of the projective line bundle ℙ(O ⊕ O(1)) over ℙ^{2}. The first and the third morphism in (3.56) are birational morphisms, and the middle morphism has two regular sections over ℤ[1/2]. The group S_{4} (as well as 𝔾_{m}) acts on all schemes in (3.56), compatibly with the morphisms and the sections.

3.4 Applications to Visibility

We are now in the position to combine the ideas Mazur used in proving Theorem 1 with Theorem 4 to prove the following

Theorem 5 Let E be an elliptic curve over a number field K, and σ ∈ H^{1}(G_{K}, E)[2] any element in the Weil-Châtelet group of E of order two (here, G_{K} denotes the absolute Galois-group of K). Then there are infinitely many abelian surfaces in which σ is visible.

PROOF. Recall from the Introduction that in order to construct an abelian surface as required, we need to find an elliptic curve F over K such that F [2] ≅ E[2] ≅ M as Galois-modules, and such that a lift η ∈ H^{1}(G_{K}, E[2]) ≅ H^{1}(G_{K}, F [2]) of σ is mapped to 0 ∈ H^{1}(G_{K}, F)[2]

– then, σ is visible in the abelian surface A from the exact sequence

0 → M →^{Δ} E × F → A → 0,

where Δ denotes the diagonal embedding. We are thus reduced to finding elliptic curves F with the above two properties.

Denote the base changes of E(2, ω) and X(2, ω) to K by E_{K} and X_{K}, as we did in Section 2.6. We start by twisting E_{K} and X_{K} by the (obvious adjustment to E_{K} and X_{K} of the) cocycles c_{E} and c′_{E} of Proposition 6, obtained from the G_{K}-action on E[2], to get the varities E_{K}^{∗} over X_{K}^{∗}; it follows from Proposition 6 that E_{K}^{∗}[2] ≅ E[2]_{X_{K}^{∗}}, and that in fact every elliptic curve F over K with F [2] ≅ E[2] as Galois-modules appears as the fiber of E_{K}^{∗} over a K-rational point of X_{K}^{∗}. Note that the lift η ∈ H^{1}(G_{K}, E[2]) of σ gives rise to a cohomology class η′ ∈ H^{1}(G_{K}, Aut(E_{K}^{∗})); we twist E_{K}^{∗} by η′ to obtain a variety Ẽ_{K} over X_{K}^{∗}. (We think of the second twist as “twisting each (smooth) fiber of E_{K}^{∗}/X_{K}^{∗} by η′′”.)

We now note that a K-rational point P in a smooth fiber F of Ẽ_{K} over X_{K}^{∗} gives us an elliptic curve as desired: F is a genus one curve over K with a K-rational point, hence an elliptic curve; going through our construction, we see that F is hence an elliptic curve with F [2] ≅ E[2] as Galoismodules whose twist induced by the cohomology class η ∈ H^{1}(G_{K}, E[2]) ≅ H^{1}(G_{K}, F [2]) is trivial. Our aim is thus to prove that there are “many” K-rational points on Ẽ_{K}.

To this end, note that Ẽ_{K} is a twist of E_{K} by the automorphism group S_{4} generated by the action of S_{3} on the level-two-structure and the translations by two-torsion sections. We can hence take the base changes to K of all the schemes in the sequence of morphisms (3.56) from Theorem 4 and twist them by the corresponding cocycles to obtain a sequence of morphisms of varieties over K

Ẽ_{K} → B̃_{K} → Ṽ_{K} → ℙ̃_{K}^{2}.  (3.57)

A theorem proved independently by A. Skorobogatov (see [Sk]), N. Shepherd-Barron (see [SB]) and H. P. F. Swinnerton-Dyer now says that any twist of ℙ^{2} (over some field k) blown up at four points in general position has a k-rational point. In our case, we deduce that Ṽ_{K} has a K-rational point.

From the third map in (3.57), we obtain that ℙ̃_{K}^{2} has a K-rational point. But any Brauer-Severi variety which has a point over the underlying field is actually isomorphic to the corresponding projective space (see [Ch]), so ℙ̃_{K}^{2} ≅ ℙ_{K}^{2} over K; in particular, the K-rational points are Zariski-dense in ℙ̃_{K}^{2}. Since the third morphism in (3.57) is a birational map over K, the K-rational points are hence also Zariski-dense in Ṽ_{K}.

A priori, B̃_{K} is a conic bundle over Ṽ_{K}. Since the ℙ^{1}-bundle B_{K} → V_{K} has two sections over K, though, which are compatible with the S_{4}-actions by which we twisted, the same still holds for B̃_{K} → Ṽ_{K}; hence, B̃_{K} is still a ℙ^{1}-bundle over Ṽ_{K}, and it follows that the K-rational points are Zariski-dense in B̃_{K}, too. Finally, since the first map in (3.57) is a birational morphism over K, the K-rational points are also Zariski-dense in Ẽ_{K}, which finishes the proof of the theorem.

We conclude this section with two remarks:

1. Reformulation of Theorem 5. In the proof of Theorem 5, the elliptic curve E only appeared through its two-torsion Galois-module E[2]. It might hence be appropriate to reformulate the theorem as follows:

Let K be a number field with absolute Galois group G_{K}, and let M be a two-dimensional vector space over 𝔽_{2} with a continuous G_{K}-action; moreover, let η ∈ H^{1}(G_{K}, M) be any cohomology class. Then there are infinitely many elliptic curves F over K such that F [2] ≅ M as G_{K}-modules, and under the map on cohomology

H^{1}(G_{K}, M) → H^{1}(G_{K}, F)

induced by the embedding M ≅ F [2] ↪ F, η is mapped to 0.

Note that any two-dimensional 𝔽_{2}-vector space M with a continuous G_{K}-action, as in the statement above, does appear as the two-torsion module of some elliptic curve over K.

2. Variant of Theorem 5 for two-torsion of the Shafarevich-Tate group. Without using the fact that any twist of the variety V_{K} has a K-rational point, we can still prove that any two-torsion element σ ∈ � (E/K)[2] of the Shafarevich-Tate group of any elliptic curve E/K (K again an arbitrary number field) is visible in infinitely many abelian surfaces, by a simple variant on the proof we gave for Theorem 5, along the exact lines of Mazur’s argument for proving Theorem 1:

Note that in the setup given by the sequence of morphisms in (3.57), all we need is a K-rational point on ℙ̃_{K}^{2}. We observe that by Proposition 6 in the previous chapter, the original elliptic curve E is contained in the first twist E_{K}^{∗} as the fiber over a K-rational point Q, say, of X_{K}^{∗} (there are, in fact, infinitely many choices for Q). The fiber over Q in the second twist Ẽ_{K}, then, is the principal homogeneous space C_{σ} of E corresponding to σ. But if σ is locally trivial everywhere, C_{σ} has a K_{v}-rational point for every completion K_{v} of K – and thus the same is true for Ẽ_{K} and for ℙ̃_{K}^{2} (since from (3.57) we obtain a K-rational morphism Ẽ_{K} → ℙ̃_{K}^{2}). But by the local-to-global principle for Brauer-Severi-varieties, we deduce that ℙ̃_{K}^{2} has a K-rational point; the rest of the argument goes through as above.

We reiterate that W. Stein’s argument which we alluded to in the Introduction of this dissertation already gives us that any σ ∈ � (E/K)[2] is visible in some abelian surface, since the splitting field of σ has degree two over K. The argument we just outlined establishes the existence of infinitely many such surfaces, and extends to any twotorsion element of the Weil-Châtelet group without any conditions on local triviality, as formulated in Theorem 5.

Finally, again for lack of a suitable reference, we briefly recall the argument for establishing the local-to-global principle for Brauer-Severi varieties over a number field K:

Let S be a twist of ℙ^{n} over K, having a K_{v}-rational point for every completion K_{v} of K. From the theory of twisting algebraic varieties (cf. [Ser2, Chapter 3]), it follows that S corresponds to a cohomology class c_{S} ∈ H^{1}(G_{K}, P GL_{n}(K̄)) (where P GL_{n}(K̄) = AutK̄(ℙ^{n}); see, e. g., [Har, Section II.7., Example 7.1.1.]). Since S has a K_{v}-rational point for every completion K_{v} of K, it follows from [Ch], as above, that S is isomorphic to ℙ^{n} over every K_{v}; hence, c_{S} is mapped to 0 under all restriction maps H^{1}(G_{K}, P GL_{n}(K̄)) → H^{1}(G_{K_{v}}, P GL_{n}(K̄_{v})).

We have the diagram of short exact sequences

0 → K̄^{∗} → GL_{n}(K̄) → P GL_{n}(K̄) → 0
↓ ↓ ↓
0 → ∏_{v} K̄_{v}^{∗} → ∏_{v} GL_{n}(K̄_{v}) → ∏_{v} P GL_{n}(K̄_{v}) → 0

giving rise to the corresponding long exact sequences of non-abelian Galois-cohomology (cf. [Ser1, Appendix to Chapter VII]); using that H^{1}(Gal(k̄/k), GL_{n}(k̄)) = 0 for any field k (Hilbert’s Theorem ’90), we hence get a commutative diagram

H^{1}(G_{K}, P GL_{n}(K̄))  ↪  H^{2}(G_{K}, K̄^{∗}) = B_{K}
↓  ↓
∏_{v} H^{1}(G_{K_{v}}, P GL_{n}(K̄_{v})) ↪ ∏_{v} H^{2}(G_{K_{v}}, K̄_{v}^{∗}) = ∏_{v} B_{K_{v}}.

By global class field theory, the map B_{K} → ∏_{v} B_{K_{v}} is injective. Since c_{S} gets mapped to 0 ∈ ∏_{v} H^{1}(G_{K_{v}}, P GL_{n}(K̄_{v})), we deduce from an easy diagram chase that in fact c_{S} = 0, so S ≅ ℙ^{n} over K; in particular, S has many K-rational points.

Visualizing Certain Elements of H^{1}(G_{K}, E(K̄))[2]

4.1 Introduction

Having established in Theorem 5 that, given any elliptic curve E over any number field K, any element σ ∈ H^{1}(G_{K}, E)[2] is visible in infinitely many abelian surfaces, we will now investigate whether in some specific cases we can impose some conditions on the abelian surfaces in which to visualize certain two-torsion elements of the Weil-Châtelet group. The case we will examine in this chapter is the following: We will assume that σ can be lifted to a cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩), where P is a K-rational point of order two on E, under the map on cohomology induced by ⟨P ⟩ ↪ E. These are, in a sense, the simplest elements of Weil-Châtelet groups to consider.

It turns out that the question of visualizing such a σ in an abelian surface can be anwered very explicitly – σ is not just visible in infinitely many abelian surfaces (as we already know, either by Theorem 5 or by W. Stein’s result mentioned in Chapter 1), but we can explicitly give several families of such surfaces, as described in the next section (note that the method we used in proving Theorem 5 established the existence of such surfaces in a non-explicit way). This should not be surprising, given that in a scenario as simple as ours, it is, e. g., not hard to give an explicit equation for the homogeneous space corresponding to σ – cf. [Sil1, Section X.3.].

As we mentioned in the first chapter of this dissertation, every abelian surface in which an element σ ∈ H^{1}(G_{K}, E) as above is visible is a quotient of E × F, for some elliptic curve F over K with certain properties. Instead of asking which additional properties we can demand for the abelian surface to satisfy, we might ask which additional conditions we can impose on F.

In this chapter, we will show that under certain hypotheses, we can find infinitely many elliptic curves F over K, with distinct j-invariants, such that σ is visible in a certain quotient of E × F, and such that we can specify the G_{K}-module structure of F [4], the four-torsion of F.

The precise statement of our result will be made below; we will briefly comment on our strategy of proof. It is not surprising that, in analogy to the proofs of Theorems 1 and 5, we will establish conditions under which we can guarantee the existence of “many global points” on certain twists of E(4)_{K}, the generalized universal elliptic curve over K with full level-four-structure. In our endavors we will make use of the fact that E(4) is the desingularized Kummer surface of the abelian surface E_{1} × E_{1}, where E_{1} is the elliptic curve given by the affine Weierstrass equation y^{2} = x^{3} +4x, as was observed by T. Shioda in his article [Sh3].

After the statement of our results, we will give a summary of the arithmetic of the double cover E_{1} × E_{1} −−→ E(4), before we proceed to giving proofs for our assertions. Finally, we will show how to generate examples in which the hypotheses of the theorem we prove are satisfied.

In analogy to one of our comments at the end of the previous chapter, we point out that in all that follows, the only data that will be used is the lift of σ ∈ H^{1}(G_{K}, E)[2] to a cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩) (where P is a K-rational two-torsion point of E, as stated above). Hence, we might as well forget about the original elliptic curve E and start instead with a one-dimensional vector space V over 𝔽_{2} with trivial G_{K}-action and a cohomology class η ∈ H^{1}(G_{K}, V), and ask for an elliptic curve F with a K-rational two-torsion point such that under the corresponding embedding V ↪ F, the induced map on cohomology sends η to 0 (see the next section for an argument why this is enough). The question is again which additional conditions on F we can demand to be satisfied. We will leave it to the reader to reformulate our results according to this philosophy.

4.2 Explicit Examples

The following should be seen as warming-up exercises in “visualizing cohomology classes explicitly”.

4.2.1 Two-torsion of the Weil-Châtelet Group

Recall the situation which we described in the previous section: We are given an elliptic curve E over a number field K, with a non-trivial K-rational twotorsion point P, and a cohomology class σ ∈ H^{1}(G_{K}, E(K̄))[2] which can be lifted to an element η ∈ H^{1}(G_{K}, ⟨P ⟩) under the map on cohomology induced by ⟨P ⟩ ↪ E. Note that H^{1}(G_{K}, ⟨P ⟩) is canonically isomorphic to K^{∗}/(K^{∗})^{2}; let t ∈ K^{∗} be such that it represents the image of η under this isomorphism. (We may, of course, assume that η ≠ 0, so t can be taken not to be a square in K.)

Note also that in order to visualize σ in abelian surfaces, we are looking for elliptic curves F over K with a non-trivial K-rational two-torsion point P′, such that η is mapped to zero under

H^{1}(G_{K}, ⟨P ⟩) ˜→H^{1}(G_{K}, ⟨P′⟩) → H^{1}(G_{K}, F),

where the first map is the isomorphism induced by the isomorphism ⟨P ⟩ ˜→⟨P′⟩ of group schemes, and the second map is induced by the inclusion ⟨P′⟩ ↪ F. For each such elliptic curve F, σ is visible in the abelian surface (E × F)/(P, P′), by an argument analogous to the one indicated in Chapter 1 for lifts of elements of W C(E/K)[m] to elements in H^{1}(G_{K}, E[m]).

The conditions on F can easily be seen to be equivalent to the following: Fix a square root \sqrt{t} of t in K̄, and let τ be the generator of Gal(K(\sqrt{t})/K); then we are looking for an elliptic curve F over K, with a non-trivial Krational two-torsion point P′, and a point Q defined over K(\sqrt{t}), such that Q^{τ} − Q = P′ – in other words, such that the cocycle c_{t} : G_{K} → ⟨P′⟩ given by c_{t}(τ′) = O if τ′(\sqrt{t}) = \sqrt{t}, c_{t}(τ′) = P′ if τ′(\sqrt{t}) = −\sqrt{t}, for all τ′ ∈ G_{K}, becomes a coboundary on F. We will say in this case that Q trivializes η on F. As we remarked at the end of the introduction, we can safely forget everything about our original elliptic curve E at this point and just remember the “cocycle data” t ∈ K^{∗}/(K^{∗})^{2} – in particular, the curves F which we will exhibit below will only depend on t.

It quickly turns out that for every (non-square) t ∈ K^{∗} we can not just easily find elliptic curves F with a K-rational two-torsion point P′ and a point Q trivializing η as just described, but that we even have a choice, e. g., whether we want Q to be a two-torsion point of F, or a four-torsion point of F such that 2Q = P′, or a point of F of infinite order. We will briefly describe these cases in turn. Note that the case “Q a point of order two” can safely be pronounced very easy; we mention it here, because it has a nice analogue in the case of three-torsion of W C(E/K).

Q a point of order two.

Given a non-square t ∈ K^{∗}, let F_{t}^{(1)} be the elliptic curve given by the affine Weierstrass equation

F_{t}^{(1)} : y^{2} = x(x^{2} − t),

where we let P′ = (0, 0); then it is clear that Q = (\sqrt{t}, 0) trivializes η on F_{t}^{(1)} in the set-up just discussed. Note that all the F_{t}^{(1)} have the same j-invariant 1728.

Q a point of order four.

For t ∈ K^{∗}, let F_{t}^{(2)} be the elliptic curve given by

F_{t}^{(2)} : y^{2} = x^{3} + 7tx^{2} + t^{2}x,

where again we let P′ = (0, 0), and Q = (t, 3t\sqrt{t}). It is readily checked that the above equation will indeed give a non-singular curve, that 2Q = P′, and that Q trivializes η on F_{t}^{(2)}.

For a family of examples of a similar but slightly different kind, see Section 4.4.1.

Q a point of infinite order.

Finally, for t ∈ K^{∗}, let F_{t}^{(3)} be the elliptic curve given by

F_{t}^{(3)} : y^{2} = x^{3} + tx^{2} − tx;

once again, we let P′ = (0, 0), and we set Q = (\sqrt{t}, t). It is an easy check that the above family gives a generically non-singular curve, that Q has generically infinite order, and that Q trivializes η on F_{t}^{(3)} as described above if F_{t}^{(3)} is non-singular. Noting that t only matters up to squares in K^{∗}, it follows that for any scenario of the kind we are considering, we can find a suitable elliptic curve F_{t}^{(3)} with Q of infinite order trivializing η.

4.2.2 Three-torsion of the Weil-Châtelet Group

There is an analogue of the case “Q a point of order two” of the previous subsection for special elements of the Weil-Châtelet group of order three of certain elliptic curves:

As usual, let E be an elliptic curve over a number field K; this time, we assume that K contains the set μ_{3} of third roots of unity. Let P be a nontrivial K-rational point of order three on E, and suppose σ ∈ H^{1}(G_{K}, E)[3] can be lifted to a cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩). We assume that η ≠ 0.

We note that H^{1}(G_{K}, ⟨P ⟩) ≅ Hom(G_{K}, ⟨P ⟩) ≅ K^{∗}/(K^{∗})^{3} in a canonical way; suppose that η corresponds to the class of a ∈ K^{∗} in K^{∗}/(K^{∗})^{3} under this isomorphism. Let τ′ be a generator of Gal(K(a^{1/3})/K) such that η((τ′)^{j}) = jP for all j, treating η as a homomorphism from Gal(K(a^{1/3})/K) to ⟨P ⟩. We also pick ω to be the third root of unity in K such that τ′(a^{1/3}) = ωa^{1/3} (we assume that we have fixed some third root a^{1/3} of a in K̄).

Let F be the elliptic curve given by the homogeneous cubic

X^{3} + Y^{3} − aZ^{3} = 0

in ℙ^{2}; here, we choose the point O = (1 : −1 : 0) as the identity of the group law on F. It is easily checked that T = (1 : −ω^{2} : 0) is a point of order three on F, defined over K by our initial assumptions. We hence obtain a cohomology class η′ ∈ H^{1}(G_{K}, ⟨T ⟩) from η by transport, such that η′((τ′)^{j}) = jT, this time viewing η′ as a homomorphism from Gal(K(a^{1/3})/K) to ⟨T ⟩. Also, let Q = (0 : a^{1/3} : 1); Q is also a point of order three on F, and a straightforward calculation shows that for all τ ∈ Gal(K(a^{1/3})/K), Q^{τ} − Q = η′(τ). It is now clear that the cohomology class η′ is in the kernel of the map H^{1}(G_{K}, ⟨T ⟩) → H^{1}(G_{K}, F)[3] on cohomology induced by the map ⟨T ⟩ ↪ F; η′ is trivialized by Q. Note that, again, the elliptic curve F we construct only depends on a ∈ K^{∗}/(K^{∗})^{3}, and not on E. We summarize our result in the following

Proposition 10 Let E be an elliptic curve over a number field K containing μ_{3}, let P be a non-trivial K-rational three-torsion point on E, and let σ be a cohomology class in H^{1}(G_{K}, E)[3] which can be lifted to an element of H^{1}(G_{K}, ⟨P ⟩) under the map on cohomology induced by ⟨P ⟩ ↪ E. Then σ is visible in an abelian surface.

We remark that when we combine this result with the argument in Mazur’s proof of Theorem 1, as indicated in Chapter 1, we obtain again that there are in fact infinitely many abelian surfaces in which a cohomology class σ as in Proposition 10 is visible.

4.3 Statement of Visibility Results

We will now state the main result of this chapter; first, however, we will establish one point of notation:

Recall that we denote by E_{1} the elliptic curve over K having affine Weierstrass equation E_{1} : y^{2} = x^{3} + 4x. Since E_{1} has j-invariant 0, its twists are parametrized by K^{∗}/(K^{∗})^{4}. We will denote the twist of E_{1} corresponding to the class of an element b ∈ K^{∗} in K^{∗}/(K^{∗})^{4} by E_{b}; E_{b} can then easily be seen to be given by the affine Weierstrass equation E_{b} : y^{2} = x^{3} + 4bx. We also note that for b = (b′)^{2} a square in K^{∗}, an alternative affine equation for E_{(b′)^{2}} is given by E_{(b′)^{2}} : b′y^{2} = x^{3} + 4x. For an exposition of these facts, cf. [Sil1, Section X.5.].

Theorem 6 Let E be an elliptic curve over a number field K with a nontrivial K-rational two-torsion point P. Suppose that σ ∈ H^{1}(G_{K}, E(K̄))[2] can be lifted to a cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩) under the map on cohomology induced by the injection ⟨P ⟩ ↪ E. We note that H^{1}(G_{K}, ⟨P ⟩) ≅ Hom(G_{K}, ⟨P ⟩) ≅ K^{∗}/(K^{∗})^{2}, canonically; let η correspond to the class of a ∈ K^{∗} in K^{∗}/(K^{∗})^{2} under this isomorphism.

Then, if for some b ∈ K^{∗} the elliptic curves

E_{ab^{2}} : by^{2} = x^{3} + 4ax and E_{a^{3}b^{2}} : by^{2} = x^{3} + 4a^{3}x

both have positive rank over K, the following holds:

There are infinitely many elliptic curves F over K with distinct j-invariants such that

F (K̄)[4] ≅ ℤ/4ℤ × μ_{4}

as G_{K}-modules, and if we denote (0, −1) ∈ ℤ/4ℤ × μ_{4} by P′, then under the map on cohomology

H^{1}(G_{K}, ⟨P ⟩) ˜→H^{1}(G_{K}, ⟨P′⟩) → H^{1}(G_{K}, F (K̄))[2]

induced by the map ⟨P ⟩ ˜→⟨P′⟩ ↪ F, the cohomology class η ∈ H^{1}(G_{K}, ⟨P ⟩) is mapped to 0 ∈ H^{1}(G_{K}, F (K̄))[2]. In particular, σ is visible in the abelian surface (E × F)/(P, P′) for any of these curves F.

We remark that since the elliptic curve we construct in Theorem 6 has full rational four-torsion over the field K(i), the situation cannot in general correspond to one of the cases “η is trivialized by a point of order two” or “η is trivialized by a point of order four” which we discussed in Section 4.2.1.

We also note that it is not immediately clear how to find examples for which the rank condition in Theorem 6 can be satisfied (twisting E_{a} and E_{a^{3}} by the same quadratic character with both resulting twists having positive rank). We will exhibit a family of examples in which the rank conditions are satisfied in the last section of this chapter.

4.4 The Universal Elliptic Curve E(4)

As we already pointed out at the beginning of this chapter, we will make use in the proof of Theorem 6 of some arithmetic information about E(4) which we obtain from its representation as the desingularized Kummer surface of a certain abelian surface. In this section we will give an exposition of the relevant facts – explicit equations and formulae for E(4) as well as for its representation as a Kummer surface, and a discussion of lifting certain automorphisms of E(4) to automorphisms of the covering abelian surface.

4.4.1 Explicit Equations and Formulae

In what follows, E(4) will denote the characteristic zero fiber of the universal elliptic curve with full level-four-structure over the corresponding modular curve X(4) representing the (characteristic zero) moduli problem of classifying generalized elliptic curves E such that E[4] ≅ ℤ/4ℤ × μ_{4}, inducing the identity on the second exterior products using the Weil-pairing. Note that by a standard genus computation (cf. [Shi, Section 1.6.]), X(4) has genus zero and is, in fact, isomorphic to ℙ^{1} over ℚ, as will be seen shortly.

In [Sh2], T. Shioda shows that E(4) is birationally equivalent to the Jacobi quartic J

J : y^{2} = (1 − σ^{2}x^{2})(1 − x^{2}/σ^{2});

here, the base ℙ^{1} is parametrized by the affine coordinate σ, and the six cusps of X(4) correspond to σ = 0, ∞, ±1, ±i. J and E(4) are isomorphic outside their singular fibers over ℙ^{1}.

In order to describe sections and automorphisms of E(4), it will be more convenient to work with the normalization J′ of J over Spec(ℚ(σ)); J′ is given by the intersection of the two quadrics

z_{3}^{2} = (z_{0} − σ^{2}z_{2})(z_{0} − z_{2}/σ^{2})
z_{1}^{2} = z_{0}z_{2}

in ℙ^{3} with homogeneous coordinates (z_{0} : z_{1} : z_{2} : z_{3}), where the relationship between affine and projective coordinates is given by

(z_{0} : z_{1} : z_{2} : z_{3}) = (1 : x : x^{2} : y) and (x, y) = (z_{1}/z_{0}, z_{3}/z_{0}).

If we choose P_{0} : (z_{0} : z_{1} : z_{2} : z_{3}) = (1 : 0 : 0 : 1) to be the zero-section of our family of elliptic curves, then one can easily compute that the sections

Q_{1} : (z_{0} : z_{1} : z_{2} : z_{3}) = (σ^{2} : −σ : 1 : 0)
Q_{2} : (z_{0} : z_{1} : z_{2} : z_{3}) = (1 : −i : −1 : σ + 1/σ)

give a basis of the four-torsion E(4)[4] of E(4) in the non-singular fibers which establishes the isomorphism E(4)[4] ≅ ℤ/4ℤ × μ_{4} as group schemes over ℙ^{1}. The two-torsion sections (over the non-singular fibers) of E(4) are then given by

P_{1} = 2Q_{1} : (z_{0} : z_{1} : z_{2} : z_{3}) = (1 : 0 : 0 : −1)
P_{2} = 2Q_{2} : (z_{0} : z_{1} : z_{2} : z_{3}) = (0 : 0 : 1 : −1)
P_{3} = P_{1} + P_{2} : (z_{0} : z_{1} : z_{2} : z_{3}) = (0 : 0 : 1 : 1),

and we easily deduce that, in terms of affine coordinates for J, multiplication by −1 on E(4) is given by

(x, y) ↦ (−x, y).

We remark that the non-singular fibers of J′ are embedded into ℙ^{3} as degree four curves by the linear system of the equivalence class of the divisor P_{0} + P_{1} + P_{2} + P_{3} (which is equivalent to the divisor 4P_{0}). Since this divisor class is stable under addition of four-torsion points and under the action of SL_{2}(ℤ/4ℤ) on E(4), it follows automatically that all these automorphisms are given on the non-singular fibers by projective linear transformations of the ambient projective space ℙ^{3} (with coefficients in ℚ(i)(σ)).

Further calculation shows the following: In terms of affine coordinates (x, y) on J, translations by two-torsion sections on E(4) are given by the rational maps

t_{P_{1}} : (x, y) ↦ (−x, −y)  (4.1)
t_{P_{2}} : (x, y) ↦ (1/x, −y/x^{2});  (4.2)

translations by four-torsion sections, in terms of homogeneous coordinates (z_{0} : z_{1} : z_{2} : z_{3}) on J′, are given by

t_{Q_{1}} : (z_{0} : z_{1} : z_{2} : z_{3}) ↦ (σz_{2} − σ^{3}z_{0} : σ^{2}z_{3} : σ^{3}z_{2} − σz_{0} : (1 − σ^{4})z_{1})
t_{Q_{2}} : (z_{0} : z_{1} : z_{2} : z_{3}) ↦ (−i(σ + 1/σ)z_{1} + z_{3} : −i(z_{0} + z_{2}) :
−i(σ + 1/σ)z_{1} − z_{3} : (σ + 1/σ)(z_{0} − z_{2})).

Finally, there is the action of SL_{2}(ℤ/4ℤ) on E(4) over X(4) induced by the action on the universal level structure; to be precise, we have to specify an ordered basis of E(4)[4], which in our case will be (Q_{1}, Q_{2}). Note that SL_{2}(ℤ/4ℤ) is generated by the two matrices

M_{1} = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) and M_{2} = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

The action of M_{1} on E(4), in terms of affine coordinates on J, is then given by the rational map

s_{M_{1}} : (σ; x, y) ↦ (iσ; −ix, y),  (4.3)

and the action of M_{2} is described (in terms of homogeneous coordinates on J′) by

s_{M_{2}} : (σ; (z_{0} : z_{1} : z_{2} : z_{3})) ↦ (\frac{σ + 1}{σ − 1}; (σ(z_{0} + z_{3} − z_{2}) : i(σ^{2} − 1)z_{1} :
σ(z_{3} − z_{0} + z_{2}) : 2σ(z_{0} + z_{2}))).

Note that since E(4) is a separated variety, it is sufficient to give automorphisms by specifying rational maps on the set of non-singular fibers, which form a Zariski-dense open set. All maps in this subsection are given over ℚ or ℚ(i), as should be clear from the formulae.

Remark. We note that the action of s_{M_{1}}^{2} on X(4) is given (in terms of the coordinate σ) by σ ↦ −σ; hence, any twist of X(4) by a cohomology class in H^{1}(G_{K}, ⟨s_{M_{1}}^{2}⟩) ≅ K^{∗}/(K^{∗})^{2} is again isomorphic to ℙ^{1}. As a consequence, for any a ∈ K^{∗}, there are infinitely many elliptic curves F over K whose four-torsion F [4] is isomorphic (as a G_{K}–module) to the twist Z_{a} of ℤ/4ℤ × μ_{4} induced by the cocycle in

H^{1}(G_{K}, ⟨(\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix})⟩) ≅ Hom(G_{K}, ⟨(\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix})⟩) ≅ K^{∗}/(K^{∗})^{2}

corresponding to the class of a in K^{∗}/(K^{∗})^{2} (here, (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) ∈ SL_{2}(ℤ/4ℤ)

acts on ℤ/4ℤ × μ_{4} by sending (1, 1) ⟼ (1, 1) and (0, i) ⟼ (2, i)).

Now take V to be a one-dimensional vector space over 𝔽_{2} with trivial G_{K}- action, let η denote the cohomology class in H^{1}(G_{K}, V) corresponding to the class of a ∈ K^{∗} under the canonical isomorphism H^{1}(G_{K}, V) ≅ K^{∗}/(K^{∗})^{2}, and embed V into an elliptic curve F as in the previous paragraph by sending the non-trivial element to the point P ∈ Z_{a} ⊂ F which corresponds to the point (2, 1) ∈ ℤ/4ℤ × μ_{4} (which remains unaffected by the twist). We see easily that there are then eight choices for a point Q in F of order four which trivializes η, and such that 2Q ≠ P. This can be seen as a variant of Theorem 6 which we get “for free”, without having to consider the arithmetic of E(4), and as another example in the spirit of Section 4.2.1.

4.4.2 E(4) as a Desingularized Kummer Surface

The elliptic curve E_{1}.

In [Sh3], Shioda shows that E(4) is isomorphic (over ℚ(i)) to the desingularized Kummer surface of the abelian surface E_{1} × E_{1}, where we recall that E_{1} is the elliptic curve given over ℚ by the affine quartic equation

E_{1} : y^{2} = x^{4} − 1,

or alternatively by the affine Weierstrass equation Y^{2} = X^{3} + 4X. Note that E_{1} is the unique elliptic curve over ℚ with j-invariant 1728 which has a rational four-torsion point. We are here presenting those arithmetic facts about the elliptic curve E_{1} and the double cover E_{1} × E_{1} −−→ E(4) which are direct consequences of the material in [Sh3] and which we will use later on.

Note that E_{1} has complex multiplication by the full ring of Gaussian integers ℤ[i]. If we pick the point (x, y) = (1, 0) on the quartic affine representation of E_{1} as the identity O of the group law, then the other three points with y = 0 are the three non-trivial two-torsion points of E_{1}. Let ν be the point of order two given by (x, y) = (−1, 0); then, translation by ν on E_{1} is given by

t_{ν} = t_{(−1,0)} : (x, y) ↦ (−x, −y).

Multiplication by −1 is described by

m_{−1} : (x, y) ↦ (x, −y),

and multiplication by i on E_{1} can be given by

m_{i} : (x, y) ↦ (1/x, iy/x^{2});

in particular, m_{i} has O and ν as fixed points. Finally, let ν′ be the point of order two with coordinates (x, y) = (i, 0); then, translation by ν′ on E_{1} is given by

t_{ν′} : (x, y) ↦ (ix, −iy/x^{2}).

The double cover of E(4).

Let p_{1} denote the natural projection

p_{1} : Π_{0} = E_{1} × E_{1} → (E_{1} × E_{1})/t_{(ν,ν)} = Π_{1},

where t_{(ν,ν)} is translation by the point (ν, ν) on E_{1} × E_{1}, and let p_{2} : Π_{0} = E_{1} × E_{1} → E_{1} × E_{1} = Π_{2} denote the isogeny given by (A_{1}, A_{2}) ↦ (A_{1} − A_{2}, (1 + i)A_{2}), for A_{1}, A_{2} points on E_{1}. It is easy to see that both p_{1} and p_{2} have the same kernel {O, ν} on Π_{0}; we deduce that Π_{1} = (E_{1} ×E_{1})/t_{(ν,ν)} ≅ E_{1} ×E_{1} = Π_{2} over ℚ(i). We hence get the following commutative diagram

Π_{0} = E_{1} × E_{1}
p_{1} ↙  ↘ p_{2}
Π_{1} = (E_{1} × E_{1})/t_{(ν,ν)}  ≅  Π_{2} = E_{1} × E_{1};

note that p_{1} is defined over ℚ, whereas p_{2} and the isomorphism between Π_{1} and Π_{2} are defined over ℚ(i).

Let S be the desingularized Kummer surface of Π_{1}, and let p : Π_{1} −−→ S denote the rational map which is the composition of taking the quotient of the abelian surface Π_{1} by multiplication by −1 and blowing up the sixteen nodes of the quotient (coming from the sixteen points of order two on Π_{1}).

Shioda shows that S is, in fact, isomorphic to E(4) over ℚ, and that the rational map p ∘ p_{1} : Π_{0} −−→ E(4) can be given, in terms of affine coordinates ((x_{1}, y_{1}), (x_{2}, y_{2})) on the quartic representations of the two factors of Π_{0} = E_{1} × E_{1} and of affine coordinates (σ; x, y) on the Jacobi quartic J, by the equations

p ∘ p_{1} : σ = x_{2}/x_{1} , x = x_{1}x_{2} , y = y_{1}y_{2}.  (4.4)

Note, again, that the maps p and p_{1} are defined over ℚ, but that Π_{1} is isomorphic to E_{1} × E_{1} only over ℚ(i).

4.4.3 Lifting Automorphisms

In proving Theorem 6 in the next section, we are going to need some arithmetic information about certain twists of E(4). In view of our representation of E(4) as a Kummer surface from the previous subsection, we will obtain this information by considering corresponding twists of Π_{0} and Π_{2}. In order to achieve this, as a first step we will show how to lift some automorphisms of E(4) to automorphisms of Π_{0} under the rational map p ∘ p_{1}.

Lemma 8 Translation by two-torsion sections on E(4), as well as the action s_{M_{1}} of the element M_{1} of SL_{2}(ℤ/4ℤ) (defined in Section 4.4.1), can be lifted to automorphisms of Π_{0} = E_{1} × E_{1} under the rational map

p ∘ p_{1} : E_{1} × E_{1} −−→ E(4)

described in the previous subsection. By this we mean the following: For ϕ any of the above automorphisms of E(4), there is an automorphism ϕ′ of Π_{0} = E_{1} × E_{1} such that the diagram of rational maps

E_{1} × E_{1} −−→^{p∘p_{1}} E(4)
ϕ′ ↓  ↓ ϕ
E_{1} × E_{1} −−→^{p∘p_{1}} E(4)

commutes.

PROOF. Using equation (4.4) describing the rational map p ∘ p_{1} : E_{1} × E_{1} −−→ E(4), we can lift the maps mentioned in the lemma explicitly. We will use affine coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}) on the quartic representations of the first and second factor of Π_{0} = E_{1} × E_{1}, respectively, and affine coordinates (σ; x, y) on the Jacobi quartic J. Note that in order to prove the lemma for a given automorphism ϕ of E(4), it is enough to exhibit an automorphism ϕ′ of Π_{0} = E_{1} × E_{1} which commutes with multiplication by −1 and with translation by the point (ν, ν) on Π_{0} and which induces a rational map from E(4) to itself which coincides with ϕ on a Zariski-dense open (e. g. on all the non-singular fibers!).

• Recall that translation by the two-torsion section P_{1} on J is given by

t_{P_{1}} : (σ; x, y) ↦ (σ; −x, −y).

From equation (4.4), it follows that the automorphism of Π_{0} = E_{1}×E_{1} given by

((x_{1}, y_{1}), (x_{2}, y_{2})) ↦ ((ix_{1}, y_{1}), (ix_{2}, −y_{2}))

lifts t_{P_{1}} through p ∘ p_{1}. From the formulae in Section 4.4.2 it follows that this corresponds to the automorphism

(A_{1}, A_{2}) ↦ (ν′ − iA_{1}, ν′ + iA_{2})

of Π_{0} = E_{1} × E_{1}, for A_{1} and A_{2} points on E_{1}.

• From Section 4.4.1, translation by the two-torsion section P_{2} on J is

given by

t_{P_{2}} : (σ; x, y) ↦ (σ; 1/x, −y/x^{2}).

Thus, t_{P_{2}} lifts through p ∘ p_{1} to the automorphism of Π_{0} = E_{1} × E_{1} given by

((x_{1}, y_{1}), (x_{2}, y_{2})) ↦ ((1/x_{2}, iy_{2}/x_{2}^{2}), (1/x_{1}, iy_{1}/x_{1}^{2})),

and this corresponds to the map on Π_{0} = E_{1} × E_{1} given by

(A_{1}, A_{2}) ↦ (iA_{2}, iA_{1})

for points A_{1} and A_{2} on E_{1}.

• From the two previous points it follows that t_{P_{3}} = t_{P_{1}} ∘t_{P_{2}}, translation by the two-torsion section P_{3} on E(4), also lifts through p ∘ p_{1} to an automorphism of Π_{0} = E_{1} × E_{1}.

• Finally, recall from Section 4.4.1 that the action of M_{1} ∈ SL_{2}(ℤ/4ℤ)

on J is given by

s_{M_{1}} : (σ; x, y) ↦ (iσ; −ix, y)

and hence lifts through p ∘ p_{1} to the automorphism

((x_{1}, y_{1}), (x_{2}, y_{2})) ↦ ((−ix_{1}, y_{1}), (x_{2}, y_{2}))

of Π_{0} = E_{1} × E_{1}, which can also be described, for A_{1} and A_{2} points on E_{1}, by

(A_{1}, A_{2}) ↦ (i(ν′ − A_{1}), A_{2}).

This completes the proof of the lemma.

Remarks.

• It follows easily from Lemma 8 and its proof that all the automorphisms ϕ of E(4) mentioned there do lift through p to automorphisms of the abelian surface Π_{1} (so they also lift to automorphisms of Π_{2}).

• Note that the action of (s_{M_{2}})^{2} on E(4), as defined in Section 4.4.1, is multiplication by −1 on E(4) and can be lifted through p∘p_{1} by an argument analogous to the arguments we just gave in proving Lemma 8. s_{M_{2}}, the action of the element M_{2} ∈ SL_{2}(ℤ/4ℤ) on E(4), cannot be lifted through p ∘ p_{1}; it can, however, be shown that s_{M_{2}} can be lifted through p to an automorphism of the abelian surface Π_{1}. The automorphism of Π_{2} = E_{1} × E_{1} it induces (via the isomorphism Π_{1} ≅ Π_{2} from Subsection 4.4.2) can be described by

(A_{1}, A_{2}) ↦ (A_{2} + ν′, iA_{1} + ν′)

for A_{1}, A_{2} points on E_{1}.

• As it turns out, the translations by four-torsion sections on E(4) do not lift even through p to automorphisms of the abelian surface Π_{1}; this fact effectively destroys our naive hopes of using the set-up of this chapter to deduce anything about visibility in abelian surfaces of elements of the Weil-Châtelet group of exact order four. One way to see that translations by four-torsion sections do not lift is as follows: Any automorphism of E(4) that can be lifted through p clearly must preserve the set of sixteen lines on E(4) resulting from blowing up the sixteen nodes of (E_{1} × E_{1})/(m_{−1} × m_{−1}). It can be shown, however, that those sixteen lines consist of the four two-torsion sections on E(4) and of two lines in each singular fiber of E(4) over the six cusps of X(4)

– and this set is, of course, not preserved by translations by sections of exact order four on E(4).

4.5 Proof of Theorem 6

We now proceed to prove Theorem 6 as stated above. The idea of proof is the following: In order to show that global points are Zariski-dense in a certain twist of E(4), we show that this twist is dominated by (a family of) corresponding twists of the abelian surface E_{1} × E_{1}. We can then ensure the existence of enough global points on one of these twists by imposing rank conditions on the components of certain decomposable abelian surfaces covering them.

In what follows, we will write E(4) for its base change E(4) ×_{ℚ} K to the number field K; similar conventions will hold for other varieties which we use. We will split the proof into four parts, according to the program laid out above.

K-rational points on a twist of E(4).

Recall that we are given an elliptic curve E over a number field K with a non-trivial K-rational two-torsion point P, and a cohomology class σ ∈ H^{1}(G_{K}, E)[2] which comes from the cohomology class in η ∈ H^{1}(G_{K}, ⟨P ⟩) ≅ K^{∗}/(K^{∗})^{2} corresponding to the class of the element a ∈ K^{∗}.

Let P_{2} be the two-torsion section of E(4) as specified in Section 4.4.1; we have the natural isomorphism H^{1}(G_{K}, ⟨P_{2}⟩) ≅ K^{∗}/(K^{∗})^{2}, viewing ⟨P_{2}⟩ as a subgroup of the automorphism group of E(4). By our usual line of reasoning, we easily deduce that it is enough for the proof of Theorem 6 to establish the following:

Claim: Under the hypotheses of Theorem 6, the K-rational points are Zariski-dense in the twist E_{a} of E(4) induced by the cohomology class η′ ∈ H^{1}(G_{K}, ⟨P_{2}⟩) ≅ K^{∗}/(K^{∗})^{2} corresponding to a ∈ K^{∗}.

Covering the twist.

Recall that by Lemma 8, we can lift t_{P_{2}}, translation by the section P_{2} on E(4), through the rational map p ∘ p_{1} : E_{1} × E_{1} −−→ E(4) to the automorphism t̃_{P_{2}} of E_{1} × E_{1} given for points A_{1}, A_{2} of E_{1} by

t̃_{P_{2}} : (A_{1}, A_{2}) ⟼ (iA_{2}, iA_{1}).

Note that as Galois-modules, ⟨t_{P_{2}}⟩ ≅ μ_{2}, whereas ⟨t̃_{P_{2}}⟩ ≅ μ_{4}, and the automorphism of E_{1} × E_{1} corresponding to an element c ∈ μ_{4} ≅ ⟨t̃_{P_{2}}⟩ descends under p ∘ p_{1} to the automorphism of E(4) corresponding to the element c^{2} ∈ μ_{2} ≅ ⟨t_{P_{2}}⟩.

As we have pointed out before, from Kummer theory we have natural isomorphisms H^{1}(G_{K}, μ_{2}) ≅ K^{∗}/(K^{∗})^{2} and H^{1}(G_{K}, μ_{4}) ≅ K^{∗}/(K^{∗})^{4}. Consider the maps connecting the long exact sequences of Galois cohomology resulting from the rows of the following commutative diagram of G_{K}-modules

1 → μ_{4} ↪ K̄^{∗} →^{4} K̄^{∗} → 1
↓ 2 ↓ 2 ↓
1 → μ_{2} ↪ K̄^{∗} →^{2} K̄^{∗} → 1,

where the horizontal maps are the obvious maps which make the rows exact, and the vertical map in the last column is the identity. It follows that the map on the first Galois cohomology groups H^{1}(G_{K}, μ_{4}) → H^{1}(G_{K}, μ_{2}) induced by the squaring map μ_{4} → μ_{2} corresponds to the map

K^{∗}/(K^{∗})^{4} → K^{∗}/(K^{∗})^{2}
a(K^{∗})^{4} ↦ a(K^{∗})^{2}

for a ∈ K^{∗}. In particular, the map H^{1}(G_{K}, μ_{4}) → H^{1}(G_{K}, μ_{2}) is surjective.

We can hence lift the cohomology class η′ ∈ H^{1}(G_{K}, ⟨t_{P_{2}}⟩) by which we are twisting E(4) to a cohomology class η′′ ∈ H^{1}(G_{K}, ⟨t̃_{P_{2}}⟩), where we view ⟨t̃_{P_{2}}⟩ as a subgroup of the automorphism group (over K̄) of E_{1} × E_{1}. The twist A of E_{1} × E_{1} induced by η′′ is an abelian surface, and an easy calculation shows that from the rational map p ∘ p_{1} : E_{1} × E_{1} −−→ E(4) we obtain a rational dominant map (which is generically a double cover)

{ \atop p ∘ p_{1}} : A −−→ E_{a} over K.

We remark at this point that the lift of η′ to a cohomology class η′′ ∈ H^{1}(G_{K}, ⟨t̃_{P_{2}}⟩) is of course not unique – the preimage in K^{∗}/(K^{∗})^{4} of a(K^{∗})^{2} is only defined up to squares. This corresponds to the fact that the rational map { \atop p ∘ p_{1}} is insensitive to precomposition with multiplication by −1 on A, and hence we obtain a rational double cover from any quadratic twist of A to E_{a}. We capture this by specifying the lift of a(K^{∗})^{2} to K^{∗}/(K^{∗})^{4} ≅ H_{1}(G_{K}, ⟨t̃_{P_{2}}⟩) by writing η′′_{ab^{2}}, denoting the corresponding twist of E_{1} × E_{1} by A_{ab^{2}}, and reducing the proof of Theorem 6 to the following

Claim: Under the hypotheses of Theorem 6, there is some b ∈ K^{∗} such that the K-rational points are Zariski-dense on the twist A_{ab^{2}} of E_{1} × E_{1}.

Ensuring the existence of many K-rational points: Splitting up the twist.

Rather than computing the twist A_{ab^{2}} of E_{1} ×E_{1} directly, it turns out to be easier to represent it as a very simple twist of another product of elliptic curves and to then work with that representation.

Denote by m_{i} × m_{i} the automorphism of E_{1} × E_{1} described by

m_{i} × m_{i} : (A_{1}, A_{2}) ↦ (iA_{1}, iA_{2}),

for A_{1}, A_{2} points on E_{1}. The cohomology class η′′_{ab^{2}} ∈ H^{1}(G_{K}, ⟨t̃_{P_{2}}⟩) from above is represented by a cocycle (which we also denote by η′′_{ab^{2}}) of the form

η′′_{ab^{2}}(g) = t̃_{P_{2}}^{e(g)} for g ∈ G_{K} and for some function e(g) : G_{K} → ℤ/4ℤ. We define a map τ_{1} : G_{K} → ⟨m_{i} × m_{i}⟩ by

τ_{1}(g) = (m_{i} × m_{i})^{e(g)}.

τ_{1} is easily seen to be a cocycle giving rise to a cohomology class (which we also denote by τ_{1}) in H^{1}(G_{K}, ⟨m_{i} × m_{i}⟩). From [Sil1, Prop. 5.4., Section X.5.], the twist of E_{1} × E_{1} induced by this cohomology class is

A′ = E_{ab^{2}} × E_{ab^{2}},

where E_{ab^{2}} is the elliptic curve given by the affine Weierstrass equation y^{2} = x^{3} + 4ab^{2}x, as above.

Finally, we define the automorphism s of E_{ab^{2}} × E_{ab^{2}} by

s : (A′_{1}, A′_{2}) ↦ (A′_{2}, A′_{1})

for points A′_{1}, A′_{2} of E_{ab^{2}}. The cohomology class η′ ∈ H^{1}(G_{K}, ⟨t_{P_{2}}⟩) from above is represented by a cocycle (also called η′) of the form η′(g) = t_{P_{2}}^{f(g)} for g ∈ G_{K}, for a certain (continuous) homomorphism f : G_{K} → ℤ/2ℤ. Then the map τ_{2} from G_{K} to ⟨s⟩ given by

τ_{2}(g) = s^{f(g)}

is a cocycle giving rise to a cohomology class τ_{2} ∈ H^{1}(G_{K}, ⟨s⟩).

Another easy calculation shows that the twist of E_{ab^{2}} × E_{ab^{2}} induced by τ_{2} is the abelian surface A_{ab^{2}} for which we want to show that the K-rational points are Zariski-dense in it.

Ensuring the existence of many K-rational points: The final cover.

Recall that we want to show that the K-rational points are Zariski-dense in the twist of E_{ab^{2}} ×E_{ab^{2}} induced by the cohomology class τ_{2} ∈ H^{1}(G_{K}, ⟨s⟩) just defined. We have a canonical isomorphism H^{1}(G_{K}, ⟨s⟩) ≅ K^{∗}/(K^{∗})^{2}, and from the definitions of the previous part of the proof it follows that under this isomorphism, τ_{2} corresponds to the class of a ∈ K^{∗} in K^{∗}/(K^{∗})^{2}.

Let q : E_{ab^{2}} × E_{ab^{2}} → E_{ab^{2}} × E_{ab^{2}} be the isogeny given by

q : (A′_{1}, A′_{2}) ↦ (A′_{1} − A′_{2}, A′_{1} + A′_{2}),

again for A′_{1}, A′_{2} points on E_{ab^{2}}. It is easy to see that under q, the automorphism s can be lifted to the automorphism id × m_{−1} of E_{ab^{2}} × E_{ab^{2}} given by

id × m_{−1} : (A′_{1}, A′_{2}) ↦ (A′_{1}, −A′_{2}).

We again have the natural isomorphism H^{1}(G_{K}, ⟨id × m_{−1}⟩) ≅ K^{∗}/(K^{∗})^{2}; let τ_{3} be the cohomology class in H^{1}(G_{K}, ⟨id × m_{−1}⟩) corresponding to the class of a in K^{∗}/(K^{∗})^{2}. Viewing ⟨id×m_{−1}⟩ as a subset of the automorphism group of E_{ab^{2}} × E_{ab^{2}}, we can construct the twist A′_{ab^{2}} of E_{ab^{2}} × E_{ab^{2}} induced by τ_{3}. Analogous to the construction of the map { \atop p ∘ p_{1}} : A_{ab^{2}} −−→ E_{a} above, we obtain from the isogeny q an isogeny

q̃: A′_{ab^{2}} → A_{ab^{2}}

over K.

Finally, again by [Sil1, Prop. 5.4., Section X.5.], we deduce that over K,

A′_{ab^{2}} ≅ E_{ab^{2}} × E_{a^{3}b^{2}}.

Under the conditions of Theorem 6, there exists a b ∈ K^{∗} such that both factors of A′_{ab^{2}} have positive rank; hence, the set of K-rational points is Zariski-dense in A′_{ab^{2}}, and thus in A_{ab^{2}} and in E_{a}, which finishes our proof.

4.6 An Example

We conclude this chapter by giving an example in which the rank conditions of Theorem 6 are satisfied. We will prove the following

Proposition 11 Let V be a one-dimensional vector space with trivial G_{ℚ}-action. Let η ∈ H^{1}(G_{ℚ}, V) be a cohomology class which under the canonical isomorphism H^{1}(G_{ℚ}, V) ≅ ℚ^{∗}/(ℚ^{∗})^{2} corresponds to the class of an element in ℚ^{∗} of the form a = λ(λ + 1)(λ + 2)(λ + 3), for some λ ∈ ℚ^{∗}. Then there are infinitely many elliptic curves F over ℚ such that

F [4] ≅ ℤ/4ℤ × μ_{4}

as G_{ℚ}-modules, and such that under the embedding V ↪ F sending the non-zero element of V to the point (0, −1) ∈ F [4] ⊂ F, η is mapped to 0 by the induced map on cohomology H^{1}(G_{ℚ}, V) → H^{1}(G_{ℚ}, F).

PROOF. Note that modulo (ℚ^{∗})^{2}, a is congruent to a′ = \frac{(λ+1)(λ+3)}{λ(λ+2)}; by Theorem 6, all we have to do is to find for each a′ ∈ ℚ^{∗} of this form an element b ∈ ℚ^{∗} such that the two elliptic curves

E_{a′b^{2}} : by^{2} = x^{3} + 4a′x and E_{(a′)^{3}b^{2}} : by^{2} = x^{3} + 4(a′)^{3}x

both have positive rank over ℚ. If a′ = −1, take b = −3, e. g., and note that the point (x, y) = (1, 1) has infinite order on E_{a′b^{2}} = E_{(a′)^{3}b^{2}}, by [Sil1, Section X.6., Proposition 6.1.]. Otherwise, note that

\frac{(a′)^{3} + 1}{a′ + 1} = (a′)^{2} − a′ + 1 = [(λ + 1)a′ + λ]^{2}

is a square in ℚ^{∗}; consequently, if we pick x = 2 in the equations for E_{a′b^{2}} and E_{(a′)^{3}b^{2}}, the ratio of the right hand sides \frac{8+8(a′)^{3}}{8+a′} = (a′)^{2} − a′ + 1 is a rational square, and it is thus easy to find a b ∈ ℚ^{∗} such that both E_{a′b^{2}} and E_{(a′)^{3}b^{2}} have a ℚ-rational point of infinite order with x-coordinate equal to 2, again by [Sil1, Section X.6., Proposition 6.1.].

For completeness, we give a family of elliptic curves E with a nontrivial ℚ-rational two-torsion point such that the cohomology class η in Proposition 11 is not mapped to 0 by the map on cohomology induced by V ↪ E (so we obtain non-trivial elements of the Weil-Châtelet group of E which are visualized in a quotient of E × F, with F as in Proposition 11). To this end, consider the elliptic curve E_{−p/4} given by the affine Weierstrass equation

E_{−p/4} : y^{2} = x^{3} − \frac{p}{4}x,

where p is a rational prime such that either p ≡ 7 or 11 (mod 16), or such that p ≡ 1 (mod 8) and 2 is not a quartic residue modulo p. On each E_{−p/4} we have the rational two-torsion point P = (0, 0) which gives rise to an embedding V ↪ E_{−p/4}. By [Sil1, Section X.6.], we have the exact sequence

0 ⟶ ⟨P ⟩ ⟶ E_{−p/4} ⟶ E_{p} ⟶ 0

(where E_{p} is the elliptic curve with affine Weierstrass equation y^{2} = x^{3} + px and rational two-torsion point P′ = (0, 0)), giving rise to the long exact sequence of G_{ℚ}-cohomology

… → E_{−p/4}(ℚ) → E_{p}(ℚ) → ℚ^{∗}/(ℚ^{∗})^{2} → H^{1}(G_{ℚ}, E_{−p/4}(ℚ̄)) → ….

But note that by [Sil1, Section X.6.; Prop. 6.1., Cor. 6.2.1., Prop. 6.5.], E_{p} has rank zero over ℚ, and the only ℚ-rational points on E_{p} are O and P′; since P′ ∈ E_{p} is the image of the point (i\sqrt{p}/2, 0) ∈ E_{−p/4}, we conclude that the only non-trivial class in ℚ^{∗}/(ℚ^{∗})^{2} which is in the image of E_{p}(ℚ) is the class of −p, which by exactness of the above long sequence is thus the only class which is mapped to 0 ∈ H^{1}(G_{ℚ}, E_{−p/4}). Hence for any choice of η as in Proposition 11, there are infinitely many curves E_{−p/4} in which η gives rise to a non-trivial element of the Weil-Châtelet group.

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