The Birch and Swinnerton-Dyer formula for modular abelian varieties of

analytic rank zero by

Amod Sadanand Agashe

B. Tech. (Indian Institute of Technology, Bombay) 1991 M. S. (Stanford University) 1993

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy in Mathematics in the

GRADUATE DIVISION

of the

UNIVERSITY of CALIFORNIA at BERKELEY

Committee in charge: Professor Kenneth A. Ribet, Chair

Professor Hendrik W. Lenstra Professor David A. Forsyth

Spring 2000 The dissertation of Amod Sadanand Agashe is approved:

Chair Date \overline{Date} \overline{Date}

University of California at Berkeley Spring 2000 The Birch and Swinnerton-Dyer formula for modular abelian varieties of

analytic rank zero

Copyright 2000 by Amod Sadanand Agashe

Abstract

The Birch and Swinnerton-Dyer formula for modular abelian varieties of analytic rank zero

by

Amod Sadanand Agashe Doctor of Philosophy in Mathematics

University of California at Berkeley

Professor Kenneth A. Ribet, Chair

Let N be a positive integer and let A be an optimal quotient of J_{0}(N) such that its special L-value, L_{A}(1), is non-zero. Let Ω_{A} be the volume of A(R) calculated using a set of generators of the group of invariant differentials on the Néron model of A. Denote the Shafarevich-Tate group of A by � _{A} and the dual abelian variety of A by Â. Let c_{p}(A) be the order of the arithmetic component group of A at the prime p. Then the Birch and Swinnerton-Dyer (BSD) conjecture (as generalized by Tate and Gross) asserts the formula:

\frac{L_{A}(1)}{Ω_{A}} = \frac{|� _{A}| · ∏_{p|N} c_{p}(A)}{|A(Q)| · |Â(Q)|} .

We express the ratio L_{A}(1)/Ω_{A} as a rational number when A is the quotient associated to a newform of weight 2 for Γ_{0}(N), and also, in the case of prime level, when A is the winding quotient of J_{0}(N). One would like to compare our expression for L_{A}(1)/Ω_{A} to the right-hand side of the BSD formula above. We present partial results in that direction and also outline a program to approach the problem. We compute L_{A}(1)/Ω_{A}, up to a bounded factor, using modular symbols. Assuming the BSD formula, when N = 1091 and A is the winding quotient, we find that there are elements of � _{A} that are visible neither in J_{0}(N) nor in J_{1}(N).

There are three numbers associated to a strong modular elliptic curve: the modular degree, the congruence number and the Manin constant. We prove a result about primes whose squares do not divide the conductor of the elliptic curve: if such a prime divides either the Manin constant or the congruence number, then it divides the modular degree as well. We also generalize the Manin constant to higher dimensional quotients of J_{0}(N) and prove some results about our generalization.

Professor Kenneth A. Ribet Dissertation Committee Chair

To

my parents, Kaustubh and Maneesha

Contents

Acknowledgements iv

A guide to this thesis v

1 Introduction: The Birch and Swinnerton-Dyer conjecture 1 1.1 The conjecture for abelian varieties . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The conjectural formula for modular abelian varieties . . . . . . . . . . . . 2

2 A generalization of the Manin constant and the relation between congruence primes and the modular degree 5 2.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 A generalization of the Manin constant . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Motivation and definition . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2 A finer definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Results and a conjecture . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Proofs of Theorems 2.1.6, 2.1.9 and 2.2.6 . . . . . . . . . . . . . . . . . . . . 10

3 The Birch and Swinnerton-Dyer conjecture for the winding quotient of prime level 15 3.1 Definition of the winding quotient . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 A formula for the ratio of the special L-value to the real volume . . . . . . 16 3.3 Calculations of certain factors from Section 3.2 . . . . . . . . . . . . . . . . 21

3.3.1 Algorithms using modular symbols . . . . . . . . . . . . . . . . . . . 21 3.3.2 Algorithms to calculate the factors . . . . . . . . . . . . . . . . . . . 25 3.3.3 Tables of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4 An approach to the Birch and Swinnerton-Dyer formula . . . . . . . . . . . 28

4 Formulas for the ratio of the special L-value to the real volume for certain other quotients 32 4.1 Quotients associated to newforms . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Introduction and results . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.1.2 Proof of Proposition 4.1.2 . . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.3 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 The winding quotient of level a product of two distinct primes . . . . . . . 36 4.2.1 Notations and results . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2.2 Proof of Theorem 4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3 Some other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 Detecting invisible elements of the Shafarevich-Tate group 42 5.1 Definitions and the result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 The strategy to detect invisible elements . . . . . . . . . . . . . . . . . . . . 43 5.3 Discovery of invisible elements . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 49

List of Symbols 54

Index 57

Acknowledgements

I started my graduate research with Prof. L. Merel, and most of the work in this thesis was done under his guidance. In particular, Chapter 3, Section 4.2, and Section 5.3 owe a lot to him. I would like to express my heartfelt gratitude to him for patiently spending a lot of time in discussions with me and for giving me direction during my graduate research. I am also deeply grateful to Prof. K. Ribet for acting as my official advisor, for his constant support, and for his mathematical input at various points in this thesis. Special thanks go to William Stein for the great collaboration that shaped the latter part of my graduate research. Finally, I am grateful to Prof. B. Mazur for answering several mathematical questions.

There are a lot of other people to whom I am indebted for several mathematical discussions and general advice during my graduate research. They are too many to name here, and also I am afraid I might miss out someone, so I hope they excuse me for not mentioning their names; my gratitude to them is no less than to anyone else: their help will never be forgotten. In any case, I would like to at least thank Ahmed Abbes, Matt Baker, Jim Borger, Kevin Buzzard, Bas Edixhoven, Hendrik Lenstra, Martin Olsson, Bjorn Poonen, and Richard Taylor who provided direct mathematical input at certain points in this thesis.

I am very grateful to the Department of Mathematics at the University of California, Berkeley for supporting me financially with teaching assistantships and fellowships, including the Raymond H. Sciobereti fellowship. Also, I would like to thank the Department of Mathematics at the Université de Paris 6 for their excellent hospitality during my stay there. Part of the research on this thesis was supported by NSF grant #DMS 99 70593, for which I am very grateful.

A guide to this thesis

There are three different themes in this thesis:

1) The ratio of the special L-value to the real volume: As mentioned in the abstract, if A is an abelian variety over Q, then the Birch and Swinnerton-Dyer (BSD) formula gives a conjectural formula for the ratio of the special L-value of A, denoted L_{A}(1), to the real volume of A, denoted Ω_{A}. We first describe the conjectural formula in Chapter 1. In Chapter 3, we give a formula that expresses the ratio mentioned above as a rational number when A is the winding quotient of prime level. In Chapter 4, we give a similar formula for this ratio for some other quotients of J_{0}(N), including the quotient associated to a newform by Shimura. Such formulas were known for strong modular elliptic curves, but not for any higher dimensional quotients. Chapters 3 and 4 depend on Chapter 1, but otherwise can be read fairly independently of the rest of the thesis.

2) The generalized Manin constant, congruence primes and the modular degree: In Chapter 2, we obtain new results on the Manin constant, state a generalization of its definition, and give some new relations between congruence primes and the modular degree. This chapter can be read independently of the rest of the thesis.

3) Invisible elements of the Shafarevich-Tate group: Mazur introduced the notion of visible elements of the Shafarevich-Tate group of optimal modular elliptic curves. In Chapter 5, we generalize some of his results to higher dimensional abelian varieties. This chapter can be read fairly independently of the rest of the thesis.

At the beginning of each of Chapters 2, 3, and 5, and the beginnings of the sections of Chapter 4, we summarize the contents of those particular parts of the thesis. So the reader can read those summaries for a more detailed outline. Also, Chapter 1 serves as an introduction and motivation for the entire thesis.

We have tried to write this thesis so that it can be easily used as a reference. Even though the chapters appear in a sequence demanded by logical progression, they need not be read in a linear order. At the beginning of each chapter (or section), we mention the prerequisites for that chapter and its relation to the rest of the thesis. Also, in each chapter we try to recall the notation from before that will be used, or give cross-references for the notation. Moreover, there is a list of symbols and an index of definitions at the end of the thesis. So the readers can safely jump directly into any chapter they are interested in.

The Theorems, Lemmas, Remarks, etc., are numbered consecutively and by section. Thus Theorem 2.1.6 is the 6th result in Section 1 of Chapter 2. The equations are labeled consecutively and by chapter, but independent of the Theorems, Lemmas, etc.; however, the equations are usually only referred to within the proof or the section that they appear in.

The two main new ideas in this thesis are: the trick used to cancel the “discriminant” from the numerator and the denominator of L_{A}(1)/Ω_{A} that allows us to express this ratio as a rational number (see the Claim in the proof of Theorem 3.2.2 on p. 21), and the idea of using a “conjugate” isogeny to obtain some new information about the Manin constant and about the relation between congruence primes and the modular degree (see the latter half of the proof of Proposition 2.3.5 on p. 13).

Chapter 1

Introduction: The Birch and Swinnerton-Dyer conjecture

1.1 The conjecture for abelian varieties

Now that the Shimura-Taniyama-Weil conjecture has been proved, one of the main outstanding problems in number theory is the Birch and Swinnerton-Dyer (BSD) conjecture. This conjecture was made more than thirty years ago by Birch and Swinnerton-Dyer [BSD63] [BSD65] for elliptic curves; it was then extended to abelian varieties by Tate [Tat95], and to motives by Deligne, Beilinson, Bloch and Kato [BK90]. As Cassels remarks [Cas63], a fundamental problem of number theory is: given a set of polynomial equations with rational coefficients, find all of its rational solutions and investigate their structure. In many cases, the BSD conjecture predicts the existence of such solutions and describes their structure without actually finding the solutions. Thus the BSD conjecture addresses some basic questions in number theory. It also gives a relation among several fundamental invariants of an abelian variety defined over a number field. In particular, it proposes a formula for the order of the Shafarevich-Tate group of the abelian variety, a mysterious invariant that arises in the calculation of the Mordell-Weil group using descent, and elsewhere. A lot of progress has been made on the BSD conjecture by the works of Coates, Wiles, Gross, Zagier, Rubin, Kolyvagin, Kato, et al.; but a large part of the conjecture is still not proved (for a summary of results and references, see [IR90, Chap. 20, §5], [Dar97, §4], and [Rub98, Thm. 8.6, Cor. 8.9]).

We now describe the conjecture briefly. Let A be an abelian variety defined over Q. Attached to A is a complex-valued function L_{A}(s), defined on the part of the complex plane where Re(s) is sufficiently large. It is called the L-function of A and is obtained by packaging information about the number of points of A over finite fields (see [Lan91, § III.5, p. 95] for the precise definition). Suppose, as conjectured, that the function L_{A}(s) extends to an analytic function on the entire complex plane. Let r denote the rank of the finitely generated abelian group A(Q).

The first part of the Birch and Swinnerton-Dyer conjecture says:

Conjecture 1.1.1. L_{A}(s) has a zero of order r at s = 1.

Let W denote the Z-module of invariant differentials on the Néron model of A. Then rank(W) = d, where d = dim(A), and ∧^{d}W is a free Z-module of rank 1 contained in H^{0}(A, Ω_{A/Q}^{d}). Let D be a generator of ∧^{d}W. Let {ω_{1}, …, ω_{d}} be a basis of H^{0}(A, Ω_{A/Q}) over Q. Then D = c · ∧_{j}ω_{j} for some c ∈ Q^{∗}. If G is a group with an action of complex conjugation, then we denote the subgroup of elements of G invariant under this action by G^{+}. Let {γ_{1}, …, γ_{d}} be a basis of H_{1}(A, Z)^{+} and let c_{∞}(A) denote the number of connected components of A(R). Then the quantity |c_{∞}(A) · c · det(∫_{γ_{i}} ω_{j})| depends only on A; we call it the real volume of A, and denote it by Ω_{A}.

Let Â denote the dual abelian variety of A. We define the regulator of A, denoted R_{A}, by R_{A} = | det(h(P_{i}, P′_{j}))|, where {P_{1}, …, P_{r}} is a basis of A(Q)/A(Q)_{tor}, {P′_{1}, …, P′_{r}} is a basis of Â(Q)/Â(Q)_{tor}, and h denotes the Néron pairing (see [Lan91, III.1.7]).

The Shafarevich-Tate group of A, denoted � _{A}, consists of equivalence classes of principal homogeneous spaces of A that are locally trivial everywhere; it is conjectured to be finite. Let A denote the Néron model of A over Z and let A^{0} denote the largest open subgroup scheme of A in which all the fibers are connected. If p is a prime number, then let c_{p}(A) = [A_{F_{p}}(F_{p}) : A_{F_{p}}^{0}(F_{p})].

If G is a finite group, then |G| denotes its order. Assume that the Shafarevich-Tate group is finite, and that the first part of the BSD conjecture is true. Then the second part of the BSD conjecture, as generalized by Tate, and reformulated by Gross (e.g., see [Lan91, III.5]) gives the formula:

Conjecture 1.1.2.

\frac{1}{r!} \frac{L_{A}^{(r)}(1)}{Ω_{A}} =^{?} \frac{|� _{A}| · R_{A} · ∏_{p} c_{p}(A)}{|A(Q)_{tor}| · |Â(Q)_{tor}|},

where L_{A}^{(r)}(s) denotes the rth derivative of L_{A}(s), and the rest of the terms are as defined above.

Here, and in the future, the symbol =^{?} denotes an equality that is conjectured but not proved. The formula above shall be called the BSD formula.

Note that the formula says that the ratio L_{A}^{(r)}(1)/Ω_{A}, which a priori is a complex number, is in fact rational; moreover, the formula expresses this rational number in terms of certain arithmetic invariants of A.

1.2 The conjectural formula for modular abelian varieties

In this thesis, we will consider the BSD formula only for a certain class of abelian varieties that we define now. In this section, we also gather some results and notation that will be used in Chapters 3 and 4. We continue to us the notation of the previous section.

Let N be a positive integer and let X_{0}(N) be the modular curve over Q associated with the problem of parametrizing elliptic curves with a cyclic subgroup of order N. Let J_{0}(N) denote the Jacobian of X_{0}(N); it is an abelian variety defined over Q. The integer N will be referred to as the level. The Hecke algebra, denoted by T, is the subring of endomorphisms of J_{0}(N) generated by the Hecke operators T_{ℓ} for ℓ∤N and U_{p} for p|N. See, for example, [DDT94, §1.3, §1.5] for detailed definitions of all of the above. Let I be an ideal of the Hecke algebra such that T/I is torsion-free, and let A be the quotient abelian variety J_{0}(N)/IJ_{0}(N), defined over Q. We call such abelian varieties modular abelian varieties. Then the L-function of A, L_{A}(s), has analytic continuation to the entire complex plane (by [Shi94, Thm. 7.14], completed by [Car86]). The order of vanishing of L_{A}(s) at s = 1 is called the analytic rank of A. This definition is motivated by Conjecture 1.1.1. In this thesis, we consider only modular abelian varieties A whose analytic rank is zero, i.e., for which L_{A}(1) is non-zero. We call L_{A}(1) the special L-value of A. So assume that A has analytic rank zero. Then, by [KL89] (which uses [GZ86], and was completed independently in [BFH90] and [MM91]), the Mordell-Weil group, A(Q), and the Shafarevich-Tate group, � _{A}, are finite. In particular, the first part of the BSD conjecture (Conjecture 1.1.1) is true in this case, and also the regulator, R_{A}, is equal to 1. Since Â, the dual abelian variety of A, is isogenous to A, the Mordell-Weil group Â(Q) is also finite.

The BSD formula becomes

Conjecture 1.2.1.

\frac{L_{A}(1)}{Ω_{A}} =^{?} \frac{|� _{A}| · ∏_{p|N} c_{p}(A)}{|A(Q)| · |Â(Q)|}.

It is known that L_{A}(1)/Ω_{A} is a rational number [Shi77] and when A is an elliptic curve, there are formulas to calculate this rational number as well as all the invariants in the BSD formula except |� _{A}| (e.g., see [Cre97]). Also, one can use Euler systems to bound the order of � _{A} from above in terms of the order given by Conjecture 1.2.1 (staying away from certain primes), as in the work of Kolyvagin and of Kato (e.g., see [Rub98, Thm. 8.6]). However, as Swinnerton-Dyer remarked in [SD67], there was no known formula for calculating L_{A}(1)/Ω_{A} when A is not an elliptic curve. In this thesis, we give such formulas for certain quotients of J_{0}(N) that need not be elliptic curves (see Sections 3.2, 4.1.1, and 4.2.1); the first such formula appeared in [Aga99] (although it was not stated explicitly there).

Now we discuss the left-hand side of the BSD formula in Conjecture 1.2.1 in more detail. Let S_{2}(Γ_{0}(N), C) denote the space of cusp forms of weight 2 for Γ_{0}(N). By the Fourier expansion of a cusp form, we mean its Fourier expansion at the cusp ∞ (e.g., see [DDT94, §1.2]); the coefficients of this expansion are called the Fourier coefficients of the cusp form. Let S_{2}(Γ_{0}(N), Z) be the subgroup of cusp forms with integral Fourier coefficients. If R is a ring, let S_{2}(Γ_{0}(N), R) = S_{2}(Γ_{0}(N), Z)⊗R. The Hecke algebra T acts on S_{2}(Γ_{0}(N), R) (see, e.g., [DDT94, §1.3, §1.5] for details). If M is a T-module, then M[I] denotes the submodule of elements of M that are killed by every element of I. The map

S_{2}(Γ_{0}(N), C) → H^{0}(X_{0}(N), Ω_{X_{0}(N)/C}),

given by f ↦ ω_{f} = 2πif(z)dz induces a canonical isomorphism

S_{2}(Γ_{0}(N), Q) ⟶^{≅} H^{0}(X_{0}(N), Ω_{X_{0}(N)/Q})

(for example, by [DDT94, Thm. 1.33]). We have the standard immersion X_{0}(N) → J_{0}(N) obtained by sending the cusp ∞ to 0; pulling back differentials along this map gives us an isomorphism (e.g., see [Mil86c, Prop. 2.2])

H^{0}(J_{0}(N), Ω_{J_{0}(N)/Q}) ≅ H^{0}(X_{0}(N), Ω_{X_{0}(N)/Q}).

Using the two isomorphisms above, we get an isomorphism:

S_{2}(Γ_{0}(N), Q) ⟶^{≅} H^{0}(J_{0}(N), Ω_{J_{0}(N)/Q}).  (1.1)

Let d = dim(A). A Q-basis for H^{0}(A, Ω_{A/Q}) is given by the differentials corresponding to a set of generators of the Z-module S_{A} = S_{2}(Γ_{0}(N), Z)[I] under the isomorphism (1.1); denote this basis by {ω_{1}, …, ω_{d}}. As in Section 1.1, let D be a generator of ∧^{d}H^{0}(A, Ω_{A/Z}^{d}), where again A is the Néron model of A over Z. Then D = c · ∧_{j}ω_{j} for some c ∈ Q^{∗}. The absolute value of c is depends only on A; denote this absolute value by c_{A}. We call the constant c_{A} the generalized Manin constant and study it in Chapter 2. The generalized Manin constant is an integer (Theorem 2.2.2) and the only primes that can divide it are the prime 2 and the primes whose squares divide N (Theorem 2.2.3).

If ⟨ , ⟩ : M ×M′ → C, is a pairing between two Z-modules M and M′, each of the same rank m, and {α_{1}, …, α_{m}} and {β_{1}, …, β_{m}} are some bases of M and M′ (respectively), then the discriminant of the pairing ⟨ , ⟩ is defined as the absolute value of det(⟨α_{i}, β_{j}⟩), and denoted by disc(M × M′ → C). We have a pairing

⟨ , ⟩ : (H_{1}(X_{0}(N), Z)^{+} ⊗ C) × S_{2}(Γ_{0}(N), C) → C

given by (γ, f) ↦ ⟨γ, f⟩ = ∫_{γ} ω_{f}. This induces a pairing H_{1}(A, Z)^{+} × S_{A} → C. From the definition of Ω_{A} in Section 1.1 and the discussion above, we get

Ω_{A} = c_{A} · c_{∞}(A) · disc(H_{1}(A, Z)^{+} × S_{A} → C).

In this thesis, we shall study only the left-hand side of the BSD formula in Conjecture 1.2.1 (except in Section 3.4, where we discuss the right-hand side briefly). We may rewrite the left-hand side as

\frac{1}{c_{A}} · \frac{L_{A}(1)}{c_{∞}(A) · disc(H_{1}(A, Z)^{+} × S_{A} → C)} .  (1.2)

Now we are in a position to describe in more detail how the various chapters are related to the BSD formula. In Chapter 2, we study the constant c_{A} in (1.2) above. We prove formulas expressing the term

\frac{L_{A}(1)}{c_{∞}(A) · disc(H_{1}(A, Z)^{+} × S_{A} → C)}

appearing in (1.2) as a rational number when A is the winding quotient of prime level (in Chapter 3) and for certain other quotients (in Chapter 4). The only relation of the BSD formula to Chapter 5 is that we use it to get some information on |� _{A}| (assuming the conjectural formula) by using the formula for (1.2) that we derive in Chapter 3.

Chapter 2

A generalization of the Manin constant and the relation between congruence primes and the modular degree

We extend the techniques of a paper of Abbes and Ullmo [AU96] to show that if E is a strong modular elliptic curve and if p is a prime such that p does not divide the modular degree of E, and p^{2} does not divide the conductor of E, then p does not divide the Manin constant of E either. We also show that except for primes whose squares divide the conductor, the congruence primes associated to the newform corresponding to E divide the modular degree of E. Finally we generalize the notion of the Manin constant to arbitrary quotients of J_{0}(N) under the action of ideals of the Hecke algebra and mention the corresponding results and a conjecture for it. This chapter may be read independently of the rest of the thesis. Only Theorem 2.2.3 and Conjecture 2.2.8 from this section will be referred to later.

2.1 Introduction and results

We recall the definition of the Manin constant and what is known about it. Let N be a positive integer and recall that X_{0}(N) is the modular curve over Q that classifies elliptic curves with a given cyclic subgroup of order N. Let J_{0}(N) be the Jacobian of X_{0}(N) and let f be a newform of weight 2 for Γ_{0}(N) with integral Fourier coefficients. Recall that the Hecke algebra, denoted T, is the sub-ring of endomorphisms of J_{0}(N) generated by the Hecke-operators T_{ℓ} for primes ℓ∤N and by U_{p} for p|N. Let I_{f} be the annihilator of f under the action of T, and let E = J_{0}(N)/I_{f}J_{0}(N) be the indicated quotient, an elliptic curve over Q. We call such an elliptic curve the strong modular elliptic curve associated to f. Composing the quotient map J_{0}(N) → E with the standard immersion X_{0}(N) → J_{0}(N) obtained by sending the cusp ∞ to 0, we get a surjective map of curves

ϕ_{E} : X_{0}(N) → E.

The degree of the map ϕ_{E}, denoted degϕ_{E}, is called the modular degree of E.

Let E_{Z} be the Néron model of E. Then the Z-module of invariant differentials on E_{Z} is free of rank one; let ω be one of its two generators. Pulling it back to X_{0}(N), we get a differential form ϕ∗_{E}ω on X_{0}(N). Now the newform f gives another differential 2πif(z)dz on X_{0}(N). By the multiplicity one theorem, we have ϕ∗_{E}ω = c · 2πif(z)dz for some c ∈ Q^{∗}. The absolute value of c depends only on E; we denote this absolute value by c_{E}. It is called the Manin constant of E. The Manin constant is of interest because it plays a role in the BSD conjecture (see Sections 1.2 and 2.2.1).

Conjecture 2.1.1 (Manin). c_{E} = 1.

The following results are known towards this conjecture:

Theorem 2.1.2 (Edixhoven [Edi91, Prop. 2]). c_{E} ∈ Z.

Theorem 2.1.3 (Mazur [Maz78, Cor. 4.1]). If p is a prime such that p|c_{E}, then p^{2} |N or p = 2.

Theorem 2.1.4 (Raynaud [AU96, Prop. 3.1]). If 2^{2} ∤N, then 2^{2} ∤c_{E}.

Theorem 2.1.5 (Abbes-Ullmo [AU96, Thm. A]). If p is a prime such that p|c_{E}, then p|N.

Also B. Edixhoven has unpublished results (see [Edi89]) which say that the only primes that can possibly divide c_{E} are 2, 3, 5, 7; he also gives bounds on the valuation of c_{E} (independent of E) with respect to these primes. Thus, from what is known so far, if a prime p divides c_{E}, then either p = 2 and 2 | N, or p = 3, 5, or 7 and p^{2} | N; moreover, there are bounds (independent of E) on the powers of these primes that can divide c_{E}.

To this, we add the following theorem, whose proof builds on the techniques of [AU96].

Theorem 2.1.6. If p is a prime such that p|c_{E}, then p^{2} |N or p|degϕ_{E}.

In view of what was known before, the only new information is that if 2|N, but 2^{2} ∤N and 2∤degϕ_{E}, then 2∤c_{E}. For example, for the elliptic curve E = 46A1 of [Cre97, Table 1], earlier one could only conclude that c_{E} is 1 or 2. But, from [Cre97, Table 5], we find that the modular degree of E is 5, and so by Theorem 2.1.6, c_{E} = 1.

Let r_{E} be the largest integer such that there exists a modular form g that has integral Fourier coefficients, that is orthogonal to f with respect to the Petersson inner product, and that satisfies g ≡ f mod r_{E}. It is called the congruence number of E and primes dividing it are called the congruence primes of E.

Theorem 2.1.7 (Ribet [Zag85, Thm. 3], [AU96, Lem 3.3] ). degϕ_{E} | r_{E}; moreover, if N is prime, then r_{E} = degϕ_{E}.

The congruence number and the modular degree are quantities of great interest: congruence primes have been studied by Hida and Ribet, among others, and played a role in Wiles’ proof of Fermat’s last theorem, and the modular degree plays a role in a reformulation of the abc-conjecture, among other things.

K. Ribet pointed out to us that results regarding the relationship between congruence primes and the modular degree can be obtained from the techniques in the proofs of multiplicity one theorems as in [Maz77, II.9] and [Wil95, §2.1]. Using these techniques, one can show that if p is a prime such that p| \frac{r_{E}}{degϕ_{E}}, then p|N (see [AS99a] for details). This result also follows independently from [AU96, Prop. 3.3 and Prop. 3.4].

Frey and Muller [FM99] raised the question whether r_{E} = degϕ_{E}. The answer, however, is no. For example, consider the elliptic curve 54B1 of [Cre97]; call it F. By [Cre97, Table 5]), degϕ_{F} = 2; calculations of W. Stein show that r_{F} = 6. Thus r_{F} ≠ degϕ_{F}, and 3| \frac{r_{F}}{degϕ_{F}}. The problem, of course, is that 3^{2} divides the level, which is 54. In all the examples where W. Stein found that r_{E} ≠ degϕ_{E}, it is always the case that if p is a prime such that p| \frac{r_{E}}{degϕ_{E}}, then p^{2} |N. So one can reformulate the question of Frey and Muller as:

Question 2.1.8. If p is a prime such that p^{2} ∤N, then is it true that p∤ \frac{r_{E}}{degϕ_{E}}?

As an outcome of the proof of Theorem 2.1.6, we get the result:

Theorem 2.1.9. Let p be a prime such that p^{2} ∤N. If p| \frac{r_{E}}{degϕ_{E}} then p|degϕ_{E}; in particular, if p|r_{E}, then p|degϕ_{E} also.

Thus if N is squarefree, then the congruence primes divide the modular degree. For example, from [Cre97, Table 3] (and using [Stu87]), we find that the eigenform corresponding to the elliptic curve 26A shares a 2-congruence with the form corresponding to the elliptic curve 26B; also we have 2^{2} ∤ 26. So we conclude that 2 divides the modular degrees of the elliptic curves 26A and 26B. Indeed, on looking up [Cre97, Table 5], we find that the modular degrees of 25A and 25B are in fact equal to 2.

In Section 2.2, we generalize the notion of the Manin constant to quotients of J_{0}(N) of arbitrary dimension associated to ideals of the Hecke algebra, and indicate which of the above results apply in the general situation. Finally, in Section 2.3, we prove all the new theorems mentioned in Sections 2.1 and 2.2.

2.2 A generalization of the Manin constant

Again let N be a positive integer. For simplicity of notation, we will often denote the Jacobian of X_{0}(N), usually denoted J_{0}(N), by just J. Let I be an ideal of T such that T/I is torsion-free and let A be the quotient abelian variety J/IJ defined over Q. If R is a Dedekind domain with field of fractions Q, and B is an abelian variety over Q, then let B_{R} denote its Néron model over R. If R is a subring of C, let S_{2}(R) denote the subgroup of S_{2}(Γ_{0}(N), C) consisting of modular forms whose Fourier expansions (at the cusp ∞) have coefficients in R (this is the same as the definition of S_{2}(Γ_{0}(N), R) in Section 1.2: for a proof, see [DDT94, §1.5]).

2.2.1 Motivation and definition

In the Birch and Swinnerton-Dyer conjecture for A, one considers the ratio of a certain L-value to the real volume of A (see Section 1.1). This real volume is calculated by using a set of generators for H^{0}(A_{Z}, Ω_{A/Z}^{1}), which is the group of global differentials on the Néron model of A. But in calculations (see [AS99b]), or while proving formulas regarding the BSD conjecture (see Sections 3.2, 4.1.1, and 4.2.1), it is easier to work with the volume obtained by using a set of generators of S_{2}(Z)[I], which is the group of cusp forms with integral Fourier expansion annihilated by I.

We recall the definition of the generalized Manin constant from Section 1.2. Let d = dim(A). A Q-basis for H^{0}(A, Ω_{A/Q}) is given by the differentials corresponding to a set of generators of the Z-module S_{2}(Z)[I]; denote this basis by {ω_{1}, …, ω_{d}}. If D is a generator of ∧^{d}H^{0}(A_{Z}, Ω_{A/Z}^{1}), then there exists c ∈ Q^{∗} such that D = c · ∧_{j}ω_{j} in H^{0}(A_{Q}, Ω_{A/Q}^{1}). The absolute value of c depends only on A.

Definition 2.2.1. The generalized Manin constant c_{A} is the absolute value of the constant c defined above.

If the abelian variety A is an elliptic curve, then c_{A} is the usual Manin constant. In calculations or formulas regarding the Birch and Swinnerton-Dyer conjecture, one finds the real volume with respect to S_{2}(Z)[I] and corrects it by c_{A} (see Section 1.2) . Note that the generalized Manin constant in this form was considered by Gross [Gro82, (2.5) on p. 222] and Lang [Lan91, III.5, p.95].

2.2.2 A finer definition

We essentially follow the ideas in [Edi91, Prop. 2]. Let M_{0}(N) denote the compactified coarse moduli scheme \overline{M}([Γ_{0}(N)]) (see [KM85, 8.6]) and let M_{0}(N)^{0} be the open part of M_{0}(N) where the projection to SpecZ is smooth. By the Néron property, we get maps

M_{0}(N)^{0} → J_{Z} → A_{Z}.

Call the composite map ϕ_{A} (we are abusing notation slightly here; however this map ϕ_{A} induces the map ϕ_{E} from Section 2.1 when A is an elliptic curve E). It follows from [KM85, Thm. 8.11.10] that the formal completion of M_{0}(N) along the (unramified) cusp ∞ is Spf(Z[[q]]). Thus we have a map

H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) → Z[[q]].

which we denote by q-exp. Its image is contained in S_{2}(Z), which is considered as a subgroup of Z[[q]] by taking the Fourier expansion of cusp forms (see [DDT94, Thm. 1.33], for example). We have the series of maps:

H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) ⟶^{ϕ_{A}^{∗}} H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) ⟶^{q-exp} Z[[q]].

The image of H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1}) under the composite is contained in S_{2}(Z)[I] ⊆ Z[[q]]. Let

C_{A} = \frac{S_{2}(Z)[I]}{q-exp(ϕ∗_{A}(H^{0}(A_{Z}, Ω_{A_{Z}/Z}^{1})))}

be the finite quotient group. It is in fact a module over T, and c_{A} is just the order of C_{A}. In particular, the above argument gives:

Theorem 2.2.2 (generalization of Thm. 2.1.2 of Edixhoven). c_{A} ∈ Z.

2.2.3 Results and a conjecture

At first, one might think that the generalized Manin constant should also be 1. But, when A = J_{0}(N), the constant c_{A} is just the order of the cokernel of the map

H^{0}(M_{0}(N)^{0}, Ω_{M_{0}(N)^{0}/Z}^{1}) ⟶^{q-exp} S_{2}(Z),  (2.1)

and this map is not surjective in general. The order of the cokernel can be calculated using methods in [DR73]. For example, B. Edixhoven informed us that for N = 33 the cokernel has order 3, and thus c_{J_{0}(33)} = 3. So the generalized Manin constant is not 1 in general.

B. Edixhoven informed us that if N is square-free, then the map (2.1) is surjective if and only if there are no old-spaces in S_{2}(Γ_{0}(N), C). In the formula for the ratio of the special L-value to the real volume for winding quotients of level a product of two distinct primes (see Theorem 4.2.1), primes dividing the level appear in the numerator, and if the BSD formula is true, then it is reasonable to expect that they cancel the Manin constant that appears in the denominator (this was found to be the case in an explicit example). However, in the corresponding formula for the winding quotient of prime level (see Theorem 3.2.2), there is no reason to expect that the prime level divides any factor in the numerator. All this led us to suspect that c_{A} might be 1 for quotients of J^{new}, where J^{new} = J/J_{old}, where, in turn, J_{old} is the subvariety of J = J_{0}(N) generated by the images of the degeneracy maps from J_{0}(M) to J_{0}(N) for all M |N.

We have the following results in that direction:

Theorem 2.2.3 (Stein [AS99a], generalization of Thm. 2.1.3 of Mazur). If A is a quotient of J^{new} and if p is a prime such that p|c_{A}, then p^{2} |N or p = 2.

Also, it is easy to see that [AU96, Prop. 3.1] generalizes to give:

Theorem 2.2.4 (Raynaud). If A is a quotient of J^{new} and 2^{2} ∤N, then 2^{(dim A+1)} ∤c_{A}.

Let A be a quotient of J^{new} and let ϕ_{2} denote the quotient map J → A. There is a canonical isomorphism J ≅ Ĵ (see [Mil86c, Thm. 6.6]) and we shall use this to implicitly identify Ĵ with J. Dualizing the map ϕ_{2}, we get the map ϕ_{1} : Â → J; recall that Â denotes the dual abelian variety of A. Then, by [Maz98, Prop. 8], the composite

Â ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A  (2.2)

is an isogeny. If G is a finite group, then by the exponent of G, we mean the smallest integer n such that multiplication by n kills every element of G.

Definition 2.2.5. The modular exponent of A, denoted n_{A}, is the exponent of the kernel of the isogeny (2.2).

We have the following theorem, which we prove in the next section:

Theorem 2.2.6. If A is a quotient of J^{new} and if p is a prime such that p^{2} ∤N, but p|c_{A}, then p|n_{A}.

Question 2.2.7. Is the direct generalization of Theorem 2.1.5 of Abbes and Ullmo true, i.e., if A is a quotient of J^{new} and p is a prime that divides c_{A}, then does p divide N?

The theorems above give good reasons why the generalized Manin constant may be 1 when the level is square-free. But what if the level is not square-free? Computations of [FpS^{+}99] involving Jacobians of genus 2 curves that were quotients of J^{new} found c_{A} = 1 in 28 cases, including cases with quotients at the following non-square-free levels: 3^{2} · 7, 3^{2} · 13, 5^{3}, 3^{3} · 5, 3 · 7^{2}, 5^{2} · 7, 2^{2} · 47, 3^{3} · 7.

All the above facts have led the author and W. Stein [AS99a] to make the following conjecture.

Conjecture 2.2.8. If A is a quotient of J^{new}, then c_{A} = 1.

Note that strong modular elliptic curves are quotients of J^{new} and so this conjecture is a generalization of the conjecture of Manin (Conjecture 2.1.1). The hard part is to settle the question whether primes whose squares divide N can divide the generalized Manin constant. One approach to this problem is to construct good models for X_{0}(N) when N is not necessarily square free; for example, this has been done in certain cases in [Edi89].

2.3 Proofs of Theorems 2.1.6, 2.1.9 and 2.2.6

We continue to us the notation introduced so far. In particular, A is the quotient of J = J_{0}(N) under the action of an ideal I of the Hecke algebra. Also we will use notation similar to the one in [AU96] since we will follow their techniques closely. If G is a finite group, then in this section, we denote its order by # G. Recall that we had the maps

Â ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A.

Pulling back differentials along ϕ_{2} and ϕ_{1}, we get the maps:

H^{0}(A_{C}, Ω_{A/C}^{1}) ⟶^{ϕ_{2}^{∗}} H^{0}(J_{C}, Ω_{J/C}^{1}) ⟶^{ϕ_{1}^{∗}} H^{0}(Â_{C}, Ω_{Â/C}^{1}).

Let m denote the largest square that divides the level N and let S = Spec(Z[\frac{1}{m}]). Then M_{0}(N)_{S} is smooth and semistable over S. Let Ω be the relative dualizing sheaf of M_{0}(N)_{S} over S. As in [AU96, § 2.1], we have an injection

q-exp : H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(Z[\frac{1}{m}]) ;

this is not an isomorphism in general, but it induces an isomorphism

q-exp : H^{0}(M_{0}(N)_{F_{p}}, Ω) ⟶^{≅} S_{2}(F_{p})  (2.3)

for each prime p that does not divide N (we are abusing notation slightly by calling different maps q-exp; however it will be clear from the context which map we are using). We have

H^{0}(M_{0}(N)_{S}, Ω) ↪ S_{2}(Z[\frac{1}{m}]) ↪ S_{2}(C) ≅ H^{0}(J_{C}, Ω_{J/C}^{1}).

Applying ϕ_{1}^{∗} to the first two groups, we get an injection

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ↪ ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])).

Denote the cokernel by C. It is a finite group and, by (2.3), the only primes that can divide its order are the primes that divide N. An easy generalization of [AU96, Prop. 3.2] gives

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ≅ H^{0}(Â_{S}, Ω_{Â/S}^{1}).

So we have an exact sequence

0 → H^{0}(Â_{S}, Ω_{Â/S}^{1}) → ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) → C → 0.

Taking the quotient by the pullback of H^{0}(A_{S}, Ω_{A/S}^{1}) under ϕ_{2} ∘ ϕ_{1}, we get

0 → \frac{H^{0}(Â_{S}, Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})} → \frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}]))}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})} → C → 0.  (2.4)

Now

# (\frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}]))}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) = (c_{A})_{m} · # (\frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}]))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}) ,  (2.5)

where, if i is an integer, then i_{m} denotes the largest divisor of i prime to m.

Let W (I) denote the orthogonal complement of S_{2}(Z)[I] in S_{2}(C) with respect to the Petersson inner product.

Definition 2.3.1. Consider the group

(\frac{S_{2}(Z)}{S_{2}(Z)[I] ⊕ (W (I) ∩ S_{2}(Z))}) .

Its order is called the congruence number of A, denoted r_{A}, and its exponent is called the congruence exponent of A, denoted m_{A}.

Note that this definition of the congruence number coincides with the definition in Section 2.1 when A is an elliptic curve. Theorem 2.1.7 generalizes (e.g., generalizing the proof of [AU96, Lem 3.3]) to gives us:

Theorem 2.3.2. n_{A} |m_{A}; moreover, if N is prime, then n_{A} = m_{A}. In particular, n_{A} |r_{A}.

As in the proof of [AU96, Prop. 3.3], we have the isomorphisms

(\frac{S_{2}(Z)}{S_{2}(Z)[I] ⊕ (W (I) ∩ S_{2}(Z))}) ⊗ Z[\frac{1}{m}] ⟶^{≅} \frac{S_{2}(Z[\frac{1}{m}])}{(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}]) ⊕ (W (I) ∩ S_{2}(Z[\frac{1}{m}]))}
⟶^{≅} \frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}.

Thus

# (\frac{ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])))}{ϕ_{1}^{∗}(S_{2}(Z)[I] ⊗ Z[\frac{1}{m}])}) = (r_{A})_{m}.

Putting this in (2.5) and then using (2.4), we get

(c_{A})_{m}(r_{A})_{m} = # (\frac{H^{0}(Â_{S}, Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S}, Ω_{A/S}^{1})}) #C.  (2.6)

Recall (Definition 2.2.5) that n_{A} was the smallest integer that annihilates the kernel of the composite isogeny

Â ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A.  (2.7)

If B and B′ are abelian varieties with an isogeny f : B → B′, and if n is an integer that kills kerf, then using the fact that kerf ⊆ ker n_{B}, where n_{B} denotes the multiplication by n map on B, we find that n_{B} factors as n_{B} = g_{n} ∘ f with g_{n} an isogeny B′ → B (see [Mil86a, §8]).

Definition 2.3.3. If B and B′ are abelian varieties with an isogeny f : B → B′, then a conjugate isogeny to f (or an isogeny conjugate to f) is an isogeny g : B′ → B such that g ∘ f is multiplication by the exponent of kerf on B. By the discussion above, given any isogeny, there always exists an isogeny conjugate to it.

Let ϕ′ be an isogeny conjugate to the isogeny (2.7). So the composite

Â ⟶^{ϕ_{1}} J ⟶^{ϕ_{2}} A ⟶^{ϕ′} Â

is multiplication by n_{A}. Using this, we see that some power of n_{A} kills (\frac{H^{0}(Â_{S},Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}).

Thus we have

Lemma 2.3.4. If p is a prime such that p divides # (\frac{H^{0}(Â_{S},Ω_{Â/S}^{1})}{ϕ_{1}^{∗}ϕ_{2}^{∗}H^{0}(A_{S},Ω_{A/S}^{1})}), then p|n_{A}.

We already remarked that a prime can divide #C only if it divides N. The main addition to the techniques of [AU96] is the following result that further controls the primes that can divide #C:

Proposition 2.3.5. If A is a quotient of J^{new} and p is a prime such that p^{2} ∤N, but p|#C, then p|n_{A}.

We will prove this shortly, but let us first use this to prove the theorems from Sections 2.1 and 2.2.

Proofs of Theorems 2.2.6, 2.1.6 and 2.1.9. Theorem 2.2.6 follows easily, since if p satisfies the hypothesis of the theorem, then p | (c_{A})_{m}; so by equation (2.6), Lemma 2.3.4 and Proposition 2.3.5, it divides n_{A}. Also if A = E is an elliptic curve, as in Section 2.1, then (e.g., see [AU96]):

n_{E} = degϕ_{E} = # (\frac{H^{0}(Ê_{S}, Ω_{Ê/S}^{1})}{ϕ_{2}^{∗}ϕ_{1}^{∗}H^{0}(E_{S}, Ω_{E/S}^{1})}) .

Equation (2.6) becomes

(c_{E})_{m}(r_{E})_{m} = (degϕ_{E})_{m}#C,

and, since degϕ_{E} |r_{E}, we get

(c_{E})_{m}(r_{E}/degϕ_{E})_{m} = #C.

Using Proposition 2.3.5, we again easily get theorems 2.1.6 and 2.1.9.

Also note that by equation (2.6), Lemma 2.3.4 and Proposition 2.3.5, we get the following generalization of Theorem 2.1.9:

Theorem 2.3.6. Let p be a prime such that p^{2} ∤N. If p|r_{A}, then p|n_{A}.

It remains to give:

Proof of Proposition 2.3.5. We have the exact sequence

0 → ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) → ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) → C → 0.  (2.8)

We want to show that certain primes do not divide #C. We already know that the only primes that can divide #C are those that divide N. So let p be a prime that divides N. Then considering the multiplication by p map applied to each term of the sequence of maps (2.8) and using the snake lemma, we get:

0 → C[p] → ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} → ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p} → C ⊗ F_{p} → 0.

Suppose p ∤ n_{A}. Then to show that p ∤ #C, i.e., that C[p] is trivial, all we have to show is that the map

\begin{matrix}ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} \\ ↓ q-exp \\ ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p} \end{matrix}  (2.9)

is injective.

The key idea is to use the conjugate isogeny ϕ′ (defined just after Definition 2.3.3). Let

ϕ′′ = ϕ′ ∘ ϕ_{2}. Then we have maps Â →^{ϕ_{1}} J →^{ϕ′′} Â such that the composite is multiplication by n_{A}. Pulling back differentials, we get the maps

H^{0}(Â_{C}, Ω_{Â/C}^{1}) ⟶^{ϕ′′^{∗}} H^{0}(J_{C}, Ω_{J/C}^{1}) ⟶^{ϕ_{1}^{∗}} H^{0}(Â_{C}, Ω_{Â/C}^{1}),

where the composite is again multiplication by n_{A}. Note that these maps extend to the Néron models over S and preserve the Z[\frac{1}{m}]-integral structure. Hence, applying these maps to (2.9), we get:

ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} ⟶^{ϕ′′^{∗}} H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} ⟶^{ϕ_{1}^{∗}} ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p}
↓ q-exp  ↓ q-exp  ↓ q-exp
ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p}  ⟶^{ϕ′′^{∗}}  S_{2}(Z[\frac{1}{m}]) ⊗ F_{p}  ⟶^{ϕ_{1}^{∗}}  ϕ_{1}^{∗}(S_{2}(Z[\frac{1}{m}])) ⊗ F_{p}

Suppose x is an element of the group ϕ_{1}^{∗}(H^{0}(M_{0}(N)_{S}, Ω)) ⊗ F_{p} in the top left corner of the diagram above that is in the kernel of the map in (2.9), i.e., the left-most q-exp map above. Then its image y = (ϕ′′^{∗})(x) in the group H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} above maps to 0 in S_{2}(Z[\frac{1}{m}]) ⊗ F_{p} under the middle q-exp map (by commutativity of the first square). But we have H^{0}(M_{0}(N)_{S}, Ω) ⊗ F_{p} ≅ H^{0}(M_{0}(N)_{F_{p}}, Ω). Suppose p^{2} ∤N. Then M_{0}(N)_{F_{p}} consists of two irreducible components. Now q-exp(y) = 0 means that y ∈ H^{0}(M_{0}(N)_{F_{p}}, Ω) is zero on the component that contains the cusp ∞. Since A is a quotient of J^{new}, the element x is an eigenvector for the Atkin-Lehner involution W_{N}, and hence so is y. But W_{N} is an involution that swaps the two components of M_{0}(N)_{F_{p}}. Hence y is zero on all of M_{0}(N)_{F_{p}}; i.e., y = 0. Looking at the top line in the diagram above, we find that x maps to 0 under the composite. But its image under this composite is n_{A}x; so n_{A}x = 0. Since p ∤ n_{A}, this means that x = 0, i.e., the map (2.9) is injective, which is what was left to prove.

Chapter 3

The Birch and Swinnerton-Dyer conjecture for the winding quotient of prime level

We first define the winding quotient of J_{0}(N) in Section 3.1. Then, in Section 3.2, we give a formula that expresses the ratio of the L-value to the real volume for the winding quotient of prime level as a rational number that can be computed. In Section 3.3 we present some calculations using the formula just mentioned. Finally, in Section 3.4, we outline a program to prove the BSD formula for the winding quotient. This chapter depends on Chapter 1.

3.1 Definition of the winding quotient

Let N be a positive integer and again let X_{0}(N) denote the usual modular curve of level N. Let H denote the complex upper half plane, and let {0, i∞} denote the projection of the geodesic path from 0 to i∞ in H ∪ P^{1}(Q) to X_{0}(N)(C). We have an isomorphism

H_{1}(X_{0}(N), Z) ⊗ R ⟶^{≅} Hom_{C}(H^{0}(X_{0}(N), Ω^{1}), C),

obtained by integrating differentials along cycles (see [Lan95, § IV.1]). Let e be the element of H_{1}(X_{0}(N), Z)⊗R that corresponds to the map ω ↦ − ∫_{{0,i∞}} ω under this isomorphism. It is called the winding element and was introduced by Mazur [Maz77, §II.18]. Let T denote the Hecke algebra (as in Section 1.2); it acts on H_{1}(X_{0}(N), Z)⊗R. Let I_{e} be the annihilator of e with respect to this action. The quotient abelian variety J_{e}(N) = J_{0}(N)/I_{e}J_{0}(N) is called the winding quotient of level N. It was introduced by L. Merel in his proof of the uniform boundedness conjecture [Mer96a].

In this chapter, we take N to be a prime number p and denote J_{e}(p) by J_{e} for simplicity of notation. Also, we use notation from Chapter 1.

3.2 A formula for the ratio of the special L-value to the real volume

One can easily check that L_{J_{e}}(1) is non-zero and hence, by the work of Kolyvagin and Logachev, J_{e}(Q) is finite; so the first part of the Birch and Swinnerton-Dyer conjecture is valid in this case. Also, the order of the Shafarevich-Tate group, � _{J_{e}}, is known to be finite. See [Mer96a, Prop. 1] and [KL89] for details of all of the above.

The Birch and Swinnerton-Dyer formula (Conjecture 1.2.1) becomes:

Conjecture 3.2.1.

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} =^{?} \frac{|� _{J_{e}}| · c_{p}(J_{e})}{|J_{e}(Q)| · |Ĵ_{e}(Q)|} .

Let H = H_{1}(X_{0}(p), Z), H_{e} = H[I_{e}], Î_{e} = Ann_{T}I_{e}, Ĥ_{e} = H[Î_{e}], n = num((p − 1)/12), and again let c_{Je} denote the generalized Manin constant of J_{e}. Let ℑ denote the annihilator of the divisor (0) − (∞), considered as an element of J_{0}(N)(C), under the action of T; it is called the Eisenstein ideal and was introduced by Mazur ([Maz77, §II.9]). It follows from [Maz77, II.18.6] that ℑe ⊆ H_{e}^{+}. We will show presently that the groups H^{+}/(Ĥ_{e}^{+} + H_{e}^{+}) and H_{e}^{+}/ℑe are finite. The main result of this section is the following formula, which expresses the left-hand side of the formula in Conjecture 3.2.1 as a rational number:

Theorem 3.2.2. With notation as above,

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} = \frac{1}{c_{Je}} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{n} .

It was L. Merel’s idea that a formula like the one above should hold. Before stating the proof of this proposition, we state some preparatory results.

First, we give some information on certain subgroups and quotients of H. If f is a newform of weight 2 for Γ_{0}(p), then let [f] denote its orbit under the action of Gal(\overline{Q}/Q). Let S_{[f]} be the Q-subspace consisting of forms in ⊕_{g∈[f]}Cg that have rational Fourier coefficients. The space S_{[f]} is stable under the action of T ⊗ Q; let T_{[f]} denote the image of T ⊗ Q acting on S_{[f]}.

Lemma 3.2.3. The natural projection map ϕ : T ⊗ Q → ⊕_{[f]}T_{[f]} is an isomorphism of rings. The image of I_{e} ⊗ Q under ϕ is ⊕_{[f]:⟨e,f⟩=0}T_{[f]} and the image of Î_{e} ⊗ Q under ϕ is ⊕_{[f]:⟨e,f⟩≠0}T_{[f]}

Proof. The fact that the map T ⊗ Q → ⊕_{[f]}T_{[f]} is an isomorphism is well known (e.g., see [DDT94, §1.6]). The statement about the image of Î_{e} ⊗ Q under ϕ follows from the other statements and the fact that T has no nilpotents. So we only have to prove that the image of I_{e} ⊗ Q under ϕ is ⊕_{[f]:⟨e,f⟩=0}T_{[f]}. Suppose t ∈ I_{e} ⊗ Q. Then te = 0. Let f be a newform, and let its eigenvalue for t be λ. Then ⟨te, f⟩ = 0, hence λ⟨e, f⟩ = 0. If ⟨e, f⟩ ≠ 0, then this means λ = 0, i.e. tf = 0. This shows that the image of I_{e} ⊗ Q is contained in ⊕_{[f]:⟨e,f⟩=0}T_{[f]}. All we have to do is show the reverse containment. So suppose t is an element of T such that ϕ(t) ∈ ⊕_{[f]:⟨e,f⟩=0}T_{[f]}. If f is a newform such that ⟨e, f⟩ ≠ 0 and g ∈ S_{[f]}, then by definition tg = 0, so ⟨te, g⟩ = 0. If f is a newform such that ⟨e, f⟩ = 0, and g ∈ S_{[f]}, then ⟨te, g⟩ = ⟨e, tg⟩ = 0 (because tg can be written as the linear combination of the action of tf_{i}, where f_{i} are the the Galois conjugates of f, each of which pair to 0 with e). So te pairs to 0 with every element of S_{[f]} for all newforms f. Since the S_{[f]}’s constitute the entire space of cusp forms, and the de Rham pairing is non-degenerate, we have te = 0, i.e., t ∈ I_{e} ⊗ Q. This finishes the proof.

Corollary 3.2.4. dim_{Q}T ⊗ Q = dim_{Q}(T ⊗ Q)[I_{e}] + dim_{Q}(T ⊗ Q)[Î_{e}].

Proof. Immediate from Lemma 3.2.3 and the fact that T ⊗ Q has no nilpotents.

Corollary 3.2.5. The natural map (I_{e} ⊗ Q) ⊕ (Î_{e} ⊗ Q) → T ⊗ Q given by (a, b) ↦ a + b is an isomorphism.

Proof. Immediate from Lemma 3.2.3, using the isomorphism ϕ.

Remark 3.2.6.

1) S_{2}(Γ_{0}(N), Q) is a free T ⊗ Q-module of rank 1 (e.g., see [DI95, Prop. 12.4.14]).

2) H ⊗ Q is a free T ⊗ Q-module of rank 2 (e.g. see [DDT94, Lemma 1.37]).

3) H^{+} ⊗Q is a free T⊗Q-module of rank 1 (e.g., see the proof of Lemma 1.37 in [DDT94]).

Let π denote the quotient map J_{0}(p) → J_{e}; since its kernel is connected, it induces a sujection π_{∗} : H → H_{1}(J_{e}, Z).

Lemma 3.2.7. The kernel of the surjection H ⟶^{π_{∗}} H_{1}(J_{e}, Z) is Ĥ_{e}.

Proof. Let K denote the kernel of π_{∗}.

Claim 1: Ĥ_{e} ⊆ K. Proof of the claim. Since J_{e} = J_{0}(p)/I_{e}J_{0}(p), all we have to show is that Ĥ_{e} pairs to zero with every cusp form killed by I_{e} under the de Rham pairing (which is a perfect pairing). Suppose h ∈ Ĥ_{e}, considered as an element of H ⊗ Q. By Corollary 3.2.5, there are elements î_{e} ∈ Î_{e} ⊗ Q and i_{e} ∈ I_{e} ⊗ Q such that î_{e} + i_{e} = 1 in T ⊗ Q. Thus h = (î_{e} + i_{e})h = i_{e}h. Since the de Rham pairing is equivariant under the action of T, this shows that h pairs to 0 with every cusp form killed by I_{e}.

Claim 2: dim_{Q}Ĥ_{e} ⊗ Q = dim_{Q}K ⊗ Q. Proof of the claim. We have

dim J_{e} = dim_{Q}H^{0}(J_{e}, Ω_{J_{e}/Q}) = dim_{Q}S_{2}(Γ_{0}(N), Q)[I_{e}] = dim_{Q}(T ⊗ Q)[I_{e}],

where the last equality follows by Part 1 of Remark 3.2.6. Hence

dim_{Q}K ⊗ Q = 2(dim J_{0}(p) − dim J_{e}) = 2(dim_{Q}T ⊗ Q − dim_{Q}(T ⊗ Q)[I_{e}]).

Next, dim_{Q}Ĥ_{e} ⊗ Q = 2 · dim_{Q}(T ⊗ Q)[Î_{e}], by Part 2 of Remark 3.2.6. The claim now follows from Corollary 3.2.4.

Lemma 3.2.7 follows from Claims 1 and 2 above.

If A be an abelian variety over R, then let c denote the action of complex conjugation on A(C) as well as the induced action on H_{1}(A(C), Z). The following result is probably well known, but we could not find a suitable reference; the proof given below was provided by H. Lenstra.

Proposition 3.2.8. If A is an abelian variety over R, then the group of connected components of A(R) is canonically isomorphic to H^{1}(Gal(C/R), H_{1}(A, Z)) and its order is equal to the order of the 2-group H_{1}(A, Z)^{+}/(1 + c)H_{1}(A, Z).

Proof. Denote H_{1}(A, Z) by L and let V = L ⊗ R (so V = H_{1}(A, R)). Now consider the exact sequence

0 → L → V → A(C) → 0

of ⟨c⟩-modules, and take its Tate cohomology sequence. The group V is uniquely divisible, hence multiplication by any integer is an isomorphism on V, and so it is an isomorphism on the Tate cohomology groups of V as well. At the same time, by [AW67, §6, Cor. 2], these cohomology groups are finite, hence are killed by multiplication by the order. Thus V has trivial cohomology. So the long exact sequence gives us:

0 → Ĥ^{0}(⟨c⟩, A(C)) → H^{1}(⟨c⟩, L) → 0.

But Ĥ^{0}(⟨c⟩, A(C)) is, by definition, equal to A(C)^{+}/(1 + c)A(C), and A(C)^{+} = A(R). So we get an isomorphism A(R)/(1 + c)A(C) ⟶^{≅} H^{1}(⟨c⟩, L). Now A(C) is compact and connected, so its continuous image (1 + c)A(C) is compact (hence closed) and connected as well. Since (1 + c)A(C) is closed, A(R)/(1 + c)A(C) is an Hausdorff group (for example, using [Bou66, Prop. 18 of § III.2.6]); and since (1 + c)A(C) is connected, A(R)/(1 + c)A(C) has the same group of components as A(R) itself. But the group A(R)/(1 + c)A(C) is also finite, since H^{1}(⟨c⟩, L) is finite (using [AW67, §6, Cor. 2]). So A(R)/(1 + c)A(C), being Hausdorff and finite, is discrete and equal to its own component group. Thus, the group of components of A(R) is canonically isomorphic to H^{1}(⟨c⟩, L).

So all we have to show is that the order of H^{1}(⟨c⟩, L) is equal to the order of L^{+}/(1+c)L. Observe that the latter group is Ĥ^{0}(⟨c⟩, L), so it suffices to prove that the Herbrand quotient of L is equal to 1. Now, by the semilinearity of the action of c on V (with respect to the complex structure on V), it follows that c has equally many eigenvalues +1 as −1 on V. Now use [AW67, §8, Prop. 12] and the fact that L⊗Q ≅ Z⟨c⟩^{dim A} ⊗Q as ⟨c⟩-representation spaces (which can be checked by looking at traces), to conclude that the Herbrand quotient of L is the same as that of Z⟨c⟩^{dim A}. But Z⟨c⟩^{dim A} has Herbrand quotient 1. Hence so does L, and that finishes the proof.

Corollary 3.2.9. J_{0}(p)(R) is connected.

Proof. By [Mer96b, Prop. 5], H_{1}(J_{0}(p), Z)^{+} = (1 + c)H_{1}(J_{0}(p), Z). Then by Proposition 3.2.8, J_{0}(p)(R) has only one connected component.

Lemma 3.2.10.

|\frac{(H/Ĥ_{e})^{+}}{H^{+}/Ĥ_{e}^{+}} | = c_{∞}(J_{e}).

Proof. Since the level is prime, by [Mer96b, Prop. 5], we have H^{+} = (1 + c)H. Now, by Lemma 3.2.7, H_{1}(J_{e}, Z) ≅ H/Ĥ_{e}, and so

(1 + c)H_{1}(J_{e}, Z) ≅ \frac{(1 + c)H}{(1 + c)Ĥ_{e}} ≅ \frac{H^{+}}{Ĥ_{e}^{+}} .

Thus

|\frac{(H/Ĥ_{e})^{+}}{H^{+}/Ĥ_{e}^{+}} | = |\frac{H_{1}(J_{e}, Z)^{+}}{(1 + c)H_{1}(J_{e}, Z)}| = c_{∞}(J_{e}),

by Proposition 3.2.8.

As mentioned earlier, ℑe is a subgroup of H_{e}^{+}.

Lemma 3.2.11. The group H_{e}^{+}/ℑe is finite.

Proof. All we have to show is that dim_{Q}ℑe ⊗ Q = dim_{Q}H_{e}^{+} ⊗ Q. By [Maz77, Prop. 9.7], T/ℑ ≅ Z/nZ; so dim_{Q}ℑe⊗Q = dim_{Q}Te⊗Q = dim_{Q}T/I_{e} ⊗Q = dim_{Q}T⊗Q−dim_{Q}I_{e} ⊗Q, and dim_{Q}H_{e}^{+} ⊗ Q = dim_{Q}(T ⊗ Q)[I_{e}]. The lemma now follows from Lemma 3.2.3 and Corollary 3.2.4.

Also, note that using Corollary 3.2.4 and Part 3 of Remark 3.2.6, we find that the group H^{+}/(Ĥ_{e}^{+} + H_{e}^{+}) is finite.

Let S_{e} = S_{2}(Γ_{0}(p), Z)[I_{e}]. We have a perfect pairing

T × S_{2}(Γ_{0}(p), Z) → Z  (3.1)

which associates to (T, f) the first Fourier coefficient a_{1}(f | T) of the modular form f | T (see [Rib83, (2.2)]); this induces a pairing

ψ : T/I_{e} × S_{e} → Z.

Lemma 3.2.12. The pairing ψ above is a perfect pairing.

Proof. Both T/I_{e} and S_{e} are free Z-modules; moreover by Lemma 3.2.3 and Part 1 of Remark 3.2.6, they have the same rank. So it suffices to prove that the induced maps S_{e} → Hom(T/I_{e}, Z) and T/I_{e} → Hom(S_{e}, Z) are injective. The injectivity of the first map follows from the perfectness of the pairing (3.1). Suppose the image of T ∈ T in T/I_{e} maps to the trivial element of Hom(S_{e}, Z). Then a_{1}(f |T) = 0 ∀f ∈ S_{e}. Now if f is in S_{e}, then so is f |T_{n} for any Hecke operator T_{n} (including U_{p}). But then a_{n}(f |T) = a_{1}((f |T_{n})|T) = 0 ∀n. Thus f = 0. Hence f | T = 0 ∀f ∈ S_{e}. By Lemma 3.2.3, this shows that T ∈ I_{e}. Thus the map T/I_{e} → Hom(S_{e}, Z) is injective and we are done.

Finally, we are ready to give:

Proof of Theorem 3.2.2. Recall that we have the pairing ⟨ , ⟩ : (H^{+}⊗C)×S_{2}(Γ_{0}(p), C) → C given by (γ, f) ↦ ⟨γ, f⟩ = ∫_{γ} 2πif(z)dz. In the rest of this proof, at various points, we will consider pairings between two Z-modules; unless otherwise stated, each such pairing is obtained in a natural way from this pairing.

For simplicity of notation, we denote c_{∞}(J_{e}) simply by c_{∞} in this proof. Using Lemma 3.2.7 and the discussion in Section 1.2, we get

Ω_{J_{e}} = c_{Je} · c_{∞} · disc((H/Ĥ_{e})^{+} × S_{e} → C).

Next, by [Shi94, Thm. 7.14] (completed by [Car86]), we have L_{J_{e}}(1) = ∏⟨e, f⟩, where, in this proof, the product symbol denotes the product over all elements f belonging to the normalized eigenform basis for the C-vector space S_{2}(Γ_{0}(p), C)[I_{e}].

Using all this, we get

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} = \frac{1}{c_{Je}} · \frac{∏⟨e, f⟩}{c_{∞} · disc((H/Ĥ_{e})^{+} × S_{e} → C)}.  (3.2)

We are going to replace (H/Ĥ_{e})^{+} by another lattice. We need some preparation before doing this. From Corollary 3.2.5, T ⊗ Q ≅ (I_{e} ⊗ Q) ⊕ (Î_{e} ⊗ Q). Thus if an element of H is killed by both I_{e} and Î_{e}, then it is killed by T, and then by Part 2 of Remark 3.2.6, it is the trivial element. Thus H_{e} ∩ Ĥ_{e} = ϕ. So the the homomorphism H_{e}^{+} → (H/Ĥ_{e})^{+} is an injection. The homomorphism

H^{+} → (H/Ĥ_{e})^{+}/H_{e}^{+}  (3.3)

has kernel (Ĥ_{e}^{+} + H_{e}^{+}). But the map (3.3) is not surjective; consider the following map to its cokernel:

(H/Ĥ_{e})^{+} → \frac{(H/Ĥ_{e})^{+}/H_{e}^{+}}{H^{+}/(Ĥ_{e}^{+} + H_{e}^{+})}.

It is surjective and has kernel H^{+}/Ĥ_{e}^{+}. Hence

\frac{(H/Ĥ_{e})^{+}}{H^{+}/Ĥ_{e}^{+}} ≅ \frac{\frac{(H/Ĥ_{e})^{+}}{H_{e}^{+}}}{(\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} )}  (3.4)

Now we are ready to perform some change of lattices:

\frac{∏⟨e, f⟩}{disc((H/Ĥ_{e})^{+} × S_{e} → C)} = \frac{∏⟨e, f⟩}{disc(H_{e}^{+} × S_{e} → C)} · |\frac{(H/Ĥ_{e})^{+}}{H^{+}/Ĥ_{e}^{+}} | · |\frac{H^{+}}{Ĥ_{e}^{+} + H_{e}^{+}} |

= \frac{∏⟨e, f⟩}{disc(ℑe × S_{e} → C)} · c_{∞} · |\frac{H^{+}}{Ĥ_{e}^{+} + H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |

= \frac{∏⟨e, f⟩}{disc(Te × S_{e} → C)} · c_{∞} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{|\frac{Te}{ℑe} |} ,  (3.5)

where we got the first equality using (3.4) and the second equality by Lemma 3.2.10. Note that

|\frac{Te}{ℑe}| = |\frac{T}{ℑ}| = n,  (3.6)

where the latter equality is from [Maz77, II.9.7].

The following claim is the key step of the proof.

Claim:

\frac{∏⟨e, f⟩}{disc(Te × S_{e} → C)} = 1.  (3.7)

Proof of the claim. The perfect pairing ψ of Lemma 3.2.12 defines t_{e} ∈ T/I_{e} ⊗ C characterized by ⟨e, f⟩ = a_{1}(t_{e}f) ∀f ∈ S_{e}. On the one hand,

∏⟨e, f⟩ = ∏ a_{1}(t_{e}f) = ( det_{S_{e}⊗C}t_{e}) ∏ a_{1}(f) = det_{T/I_{e}⊗C}t_{e}.

On the other hand, by the perfectness of the pairing ψ and the canonical isomorphism Te ≅ T/I_{e}, the discriminant of the pairing Te × S_{e} → C that associates to (te, f) the complex number ⟨te, f⟩ coincides with the discriminant of the pairing T/I_{e}×Hom(T/I_{e}, Z) → C that associates to (t, ψ) the complex number ψ(t_{e}t), obtained by extending ψ by C-linearity. The latter discriminant is also equal to det_{T/I_{e}⊗C}t_{e}.

Putting (3.7) and (3.6) in (3.5) and then using (3.2), we get

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} = \frac{1}{c_{Je}} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{n} .

That proves Theorem 3.2.2.

3.3 Calculations of certain factors from Section 3.2

We continue using notation from Sections 3.1 and 3.2. With the idea of studying the conjectural Birch and Swinnerton-Dyer formula, we did computations (with the help of a computer) to calculate the terms |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | and |\frac{H_{e}^{+}}{ℑe} | appearing in Theorem 3.2.2, for various primes p. For the former, we did calculations up to p = 397, and for the latter, up to p = 1447. The computations were done using the theory of modular symbols; we describe this theory in §3.3.1. In §3.3.2, we describe algorithms to calculate |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | and |\frac{H_{e}^{+}}{ℑe} |.

The results of the calculations are mentioned in §3.3.3. These were used in [Aga99], and will be used in Section 3.4 and Proposition 5.3.3. Only §3.3.3 will be referred to later, so the reader primarily interested in the rest of the thesis can safely skip this section (Section 3.3). In this section, we assume that the prime p under consideration is greater than 3 (if p ≤ 3, the calculations are vacuous anyway).

3.3.1 Algorithms using modular symbols

Let N be any positive integer. The theory of modular symbols gives a presentation of the group H_{1}(X_{0}(N), Z) together with a description of the action of Hecke operators on the set of generators. The real vector spaces S_{2}(Γ_{0}(N), R) and H_{1}(X_{0}(N), R) are dual in a way that make the Hecke operators self dual; hence we can use modular symbols to obtain information about S_{2}(Γ_{0}(N), R) as a T-module (see [Cre97, §2.1.2]). Thus they are of vital use in computations involving modular forms. We mention only the part of the theory that we need for our calculations; in particular, we will take N to be a prime number. For details, see [Ste00] and the references therein.

Let H denote the complex upper half plane. If α, β ∈ P^{1}(Q), then let {α, β} denote the class in H_{1}(X_{0}(p), cusps; Z) of the image in X_{0}(p)(C) of the geodesic path from α to β in H ∪ P^{1}(Q). It is called a modular symbol. This generalizes the definition of {0, i∞} in Section 3.1. Following [Man72], we have a map

Γ_{0}(p) \ SL_{2}(Z) → H_{1}(X_{0}(p), cusps; Z)

given by g ↦ {g0, g∞}. The image of this map generates H_{1}(X_{0}(p), cusps; Z) as a Z-module. Next we have a bijection

Γ_{0}(p) \ SL_{2}(Z) ⟶^{≅} P^{1}(Z/pZ)

given by

Γ_{0}(p) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ (c : d).

So we have a map

P^{1}(Z/pZ) → H_{1}(X_{0}(p), cusps; Z).  (3.8)

The image of F_{p}^{×} \ {−1, 1} under this map is contained in H_{1}(X_{0}(p); Z), which sits inside H_{1}(X_{0}(p), cusps; Z) canonically; this image generates H_{1}(X_{0}(p); Z) as a Z-module. If x ∈ F_{p}, then let [x] denote the image of x in H_{1}(X_{0}(p); Z) under (3.8). The symbol [x] is called a Manin symbol.

We obtain the following presentation of the group H_{1}(X_{0}(p); Z) (see [Mer96b, §1.3], or [Cre97, §2.2]):

generators:[x] for x ∈ F_{p}^{×} \ {−1, 1}, where [x] is just a symbol relations: [x] + [−1/x] = 0 ∀x : x ≢ (−1/x) mod p

[x] = 0 ∀x : x ≡ (−1/x) mod p

[x] + [−1 − 1/x] + [−1/(1 + x)] = 0 ∀x : x ≢ (−1 − 1/x) mod p [x] = 0 ∀x : x ≡ (−1 − 1/x) mod p [−2] + [−1/2] = 0 where all the calculations in the brackets are done modulo p

We can express the winding element in terms of the Manin symbols using the formula (see [Mer96b, Prop. 11 and Lem. 3]):

(p − 1)e = − ∑_{x=2}^{p−2} F (x)[x],  (3.9)

where

F (x) = (p − 1)/2 − 2 ∑_{q=1}^{(p−1)/2} (⌊\frac{q(x + 1)}{p}⌋ − ⌊\frac{q(x − 1)}{p}⌋),

and where ⌊y⌋ denotes the integer part of y.

If i is a positive integer, then define σ_{1}(i) = ∑_{d|i,d>0} d. There are formulas for the action of the Hecke operators on the Manin symbols. For our purposes, we only need the formula below (see [Mer96a, Lem. 2]):

(T_{i} − σ_{1}(i))e = − ∑_{a,b,c,d∈Z;a>b≥0;d>c>0;ad−bc=i}[c/d].  (3.10)

The set over which the sum is taken in (3.10) is finite and there is an algorithm to generate it based on [Mer94, §3.3], which we indicate now (this was communicated to us by L. Merel). First, consider all integers δ such that δ | i and δ > 1. For each δ, consider k = 1, …, (δ − 1). For each such pair (δ, k), construct matrices

( \begin{matrix}a_{q} & b_{q} \\ c_{q} & d_{q} \end{matrix} )

inductively as follows. Choose

( \begin{matrix}a_{0} & b_{0} \\ c_{0} & d_{0} \end{matrix} ) = ( \begin{matrix}i/δ & 0 \\ k & δ \end{matrix} ) .

Given the matrix for q = j, let t be the smallest integer greater than or equal to d_{j}/c_{j}. Then define

( \begin{matrix}a_{j+1} & b_{j+1} \\ c_{j+1} & d_{j+1} \end{matrix} ) = ( \begin{matrix}a_{j} & b_{j} \\ c_{j} & d_{j} \end{matrix} ) ( \begin{matrix}t & 1 \\ −1 & 0 \end{matrix} ) .

At each stage, the value of c_{q} will decrease. We keep doing the inductive process while c_{q} > 0 (thus the matrix for which c_{q} = 0 is excluded from the set). Now repeat the procedure for every pair (δ, k) as above. The set of all a_{q}, b_{q}, c_{q}, d_{q} we obtain in the process are the a, b, c, d (respectively) that satisfy the conditions a, b, c, d ∈ Z; a > b ≥ 0; d > c > 0; ad − bc = i in (3.10) above. For example, for i = 3, the matrices we get are

( \begin{matrix}1 & 0 \\ 1 & 3 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 2 & 3 \end{matrix} ) , and ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) .

So (T_{3} − σ_{1}(3))e = [1/3] + [2/3] + [1/2], where the calculations in the brackets are done modulo p.

To do calculations involving the Hecke algebra or an ideal of the Hecke algebra, we need a finite set of generators for the Hecke algebra as a Z-module. This is achieved in Proposition 3.3.4. Before that, we briefly discuss the generation of the Hecke algebra over certain fields. For Lemma 3.3.1 and Proposition 3.3.2, we take the level to be any positive integer N, not necessarily prime.

Lemma 3.3.1. Let K denote the field Q or the field F_{ℓ} where ℓ is a prime. If r is a positive integer such that T_{1}, …, T_{r} do not generate T ⊗ K as a K-vector space, then there exists a non-zero cusp form over K whose first r Fourier coefficients are zero.

Proof. Recall the definition of S_{2}(Γ_{0}(N), K) from Section 1.2. We have a perfect pairing (T ⊗ K) × S_{2}(Γ_{0}(N), K) → K given by (T, f) ↦ a_{1}(f | T) (see [Rib83, (2.2)]). Since T_{1}, …, T_{r} do not generate T ⊗ K as a K-vector space, there exists a non-zero cusp form f over K such that f is orthogonal to T_{1}, …, T_{r} under this pairing, i.e., such that its first r Fourier coefficients are zero.

Let g denote the genus of X_{0}(N). We may assume g ≥ 1, since otherwise the theory is vacuous.

Proposition 3.3.2. Let K denote Q or F_{ℓ} where ℓ ∤ N. Then T ⊗ K is generated as a K-vector space by T_{1}, …, T_{2g−1}.

Proof. Suppose not. Then, by Lemma 3.3.1, there is a cusp form f whose first (2g − 1) coefficients vanish. But that gives a differential on X_{0}(N)_{K} that vanishes to order at least (2g − 1) at ∞, which is forbidden by the Riemann-Roch theorem.

Let us revert to the assumption that the level N is a prime p. In that case, we can improve on Proposition 3.3.2 to get the following result, communicated to us by M. Baker:

Proposition 3.3.3. T ⊗ Q is generated as a Q-vector space by T_{1}, …, T_{g}.

Proof. We may assume that g ≥ 2, since for g = 1, the result is already contained in Proposition 3.3.2. Suppose the result is false. Then by Lemma 3.3.1, there is a cusp form f over Q whose first g coefficients vanish. But that gives a differential on X_{0}(p) that vanishes to order at least g at ∞. This shows that ∞ is a Weierstrass point [Atk67] on X_{0}(p). But that cannot happen since the level is a prime, by [Ogg78]. This contradiction proves the lemma.

Our calculations showed that T_{1}, …, T_{g} do not generate T over Z in general (e.g., when the level is 53, we found that T_{1}, …, T_{g} generate a subgroup of T of index 3). Let r denote the integer part (or floor) of (p + 1)/6. A generating set for T over Z is provided by the the following result (communicated to us by K. Ribet; for a generalization, see [Ste00]).

Proposition 3.3.4. The Hecke algebra T is generated as a Z module by T_{1}, …, T_{r}.

Proof. Suppose not. Let ℓ be a prime that divides the order of the quotient T/⟨T_{1}, …, T_{r}⟩, where ⟨T_{1}, …, T_{r}⟩ denotes the sub Z-module of T generated by T_{1}, …, T_{r}. By a simple group theoretic argument that uses the fact that T is a finitely generated abelian group, this implies that T/ℓT is not generated as a F_{ℓ}-vector space by T_{1} mod ℓ, …, T_{r} mod ℓ. Then, by Lemma 3.3.1, there exists a non-zero cusp form over F_{ℓ} whose first r Fourier coefficients are zero. But then, by [Stu87, Thm. 1], f ≡ 0 mod ℓ, i.e., f = 0, giving a contradiction.

It is not clear if T_{1}, …, T_{2g−1}, which generate T ⊗ Q over Q, generate T over Z in general. This might be useful for speeding up calculations since 2g − 1 ≤ r; however r − (2g − 1) ≤ 3.

3.3.2 Algorithms to calculate the factors

We now describe how to compute |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | and |\frac{H_{e}^{+}}{ℑe} | using modular symbols.

First we describe the calculation of |\frac{H_{e}^{+}}{ℑe} |. Note that since H^{+}/H_{e}^{+} is torsion free, H_{e}^{+}/ℑe is just the torsion subgroup of H^{+}/ℑe.

Note that complex conjugation on the C-valued points of X_{0}(N) (a variety over Q) is the same as the involution on X_{0}(N)(C) induced by the involution ∗ : z ↦ −\overline{z} on H, under the quotient map H∪P^{1}(Q) → X_{0}(N)(C). This is because the rational structure on X_{0}(N) is given by modular functions that have rational Fourier coefficients, and ∗ acts as complex conjugation on q(z) = exp(2πiz). Using this, one can show that if c denotes complex conjugation on H, then c takes [x] to [−x]. Also, in our case (prime level), H^{+} = (1 + c)H (by [Mer96b, Prop. 5]). Thus H^{+} is the subgroup of H generated by [x] + [−x]. To find the generators of the subgroup ℑe, we use the Lemma below.

Lemma 3.3.5. The ideal ℑ is generated as a Z-module by the set consisting of the element n and the elements (T_{i} − σ_{1}(i)) for i = 2, …, r.

Proof. Let ℑ′ denote the Z-submodule of T generated by n and the elements (T_{i} − σ_{1}(i)) for i = 2, …, r. Then, by [Maz77, § II.9], ℑ′ ⊆ ℑ. Consider the map of Z-modules ϕ : Z → T/ℑ′ given by 1 ↦ T_{1}. Now T_{1}, T_{2} − σ_{1}(2), …, T_{r} − σ_{1}(r) generate T as a Zmodule and so T/ℑ′ is generated as a Z-module by T_{1} = ϕ(1); thus the map ϕ is surjective. Also, we have nZ ⊆ kerϕ. Thus we a sequence of surjective maps Z →^{ϕ} T/ℑ′ → T/ℑ. The kernel of the first map contains nZ, but on the other hand, the kernel of the composite is nZ, by [Maz77, II.9.7]. Hence ℑ′ = ℑ.

So ℑe, as a subgroup of H, is generated by the set consisting of the element ne and the elements (T_{i} − σ_{1}(i)) · e for i = 1, …, r. To express these elements in terms of the Manin symbols, we use (3.9) and (3.10) from § 3.3.1. Note that H is a free Z-module and that we can divide out formula (3.9) by (p − 1)/n to calculate ne only if we write the right-hand side of (3.9) in terms of a basis for H.

So, to calculate H^{+}/ℑe, we first find a basis for H; then we quotient out by the relation for ne obtained from (3.9), and by the relations (3.10) for i = 2, …, r; call the resulting quotient G. Finally, H^{+}/ℑe is the subgroup of G generated by the images of the elements [x] + [−x] for all x. All this can be done using algorithms for the Smith normal form (for example, see [Coh00, §2.4.4]; these algorithms were pointed out to us by B. Sturmfels). We used Maple and the LiDIA library (in C++) to do the calculations.

In [Aga99, Thm. 1], we used the result that 7 divides |\frac{H_{e}^{+}}{ℑe} | for p = 1091, which could not be justified there due to lack of space. To prove this, one can simplify the algorithm using the two facts:

1) We have H^{+} ↪ H/H_{−} with cokernel of order a power of 2, where H_{−} is the subgroup generated by [x] − [−x] for all x, and

2) If ℑ′′ is the Z-submodule of T generated by the set consisting of (p − 1) and the elements (T_{i} −σ_{1}(i)) for i = 2, …, r, then ℑ′′ ⊆ ℑ with index dividing (p−1)/n, and the only primes that can divide (p − 1)/n are 2 and 3.

Using this, and some simplifications in the relations, we have that for the prime p = 1091, the group given by:

generators:[x] for x ∈ F_{p}^{×} \ {−1, 1}, where [x] is just a symbol relations: [x] + [−1/x] = 0 ∀x

[x] + [−1 − 1/x] + [−1/(1 + x)] = 0 ∀x

[−2] + [−1/2] = 0

[x] − [−x] = 0 ∀x ∑_{x=1}^{p−1} F (x)[x] = 0 ∑_{a,b,c,d∈Z;a>b≥0;d>c>0;ad−bc=i}[c/d] = 0 for i = 2, …, r, where all the calculations in the brackets are done modulo p,

has torsion subgroup that differs from H_{e}^{+}/ℑe only by powers of 2 and 3. Hence we can use it to check that 7 divides the factor |\frac{H_{e}^{+}}{ℑe} |.

Next, we can calculate |\frac{H^{+}}{Ĥ_{e}^{+}+ℑe} | as follows. We already know the generators of ℑe. Next we find the generators of the kernel of the map T ⊗ Q → H ⊗ Q given by t ↦ te. For this, we use the fact that T_{1}, T_{2}−σ_{1}(2), …, T_{g} −σ_{1}(g) generate T⊗Q (by Proposition 3.3.3) and do calculations using formulas (3.9) and (3.10). Then we clear the denominators of the set of generators of the kernel; call the resulting set S. Let G be the subgroup of H generated by th, where t runs over all elements of S and h runs over the set of generators of H. Now G is a subgroup of Ĥ_{e} with finite index, but not necessarily equal to Ĥ_{e}. To remove this problem, use the fact that H/Ĥ_{e} is torsion free: thus |\frac{H^{+}}{Ĥ_{e}^{+}+ℑe} | is the ratio of the order of

the torsion subgroup of \frac{H^{+}}{G^{+}} to the order of the torsion subgroup of \frac{H^{+}}{G^{+}+ℑe}, both of which can be computed.

Finally, note that we have an exact sequence

0 → \frac{H_{e}^{+}}{ℑe} → \frac{H^{+}}{Ĥ_{e}^{+} + ℑe} → \frac{H^{+}}{Ĥ_{e}^{+} + H_{e}^{+}} → 0.

Hence to get the factor |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | appearing in Theorem 3.2.2, we only have to divide

|\frac{H^{+}}{Ĥ_{e}^{+}+ℑe} | by |\frac{H_{e}^{+}}{ℑe} |.

3.3.3 Tables of calculations

We computed the structure of H_{e}^{+}/ℑe for prime levels up to 1447. The results appear in Table 3.1; in that table, a sequence of integers n_{1}, …, n_{r} in the second column denotes the abelian group Z/n_{1}Z × ⋯ × Z/n_{r}Z. Only the levels for which the group was found to be non-trivial are reported. Thus, for example, Table 3.1 tells us that the group H_{e}^{+}/ℑe is trivial for level 17, and for level 997, it is isomorphic to Z/3Z × Z/24Z.

We calculated |\frac{H^{+}}{Ĥ_{e}^{+}+ℑe} | for prime levels up to 397. The only level for which an odd

prime divided |\frac{H^{+}}{Ĥ_{e}^{+}+ℑe} | was 389, where 5^{2} was a factor. So the odd part of |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} |

is 5^{2}/5 = 5.

We will interpret the results of our calculations in the next section.

Table 3.1: Nontrivial H_{e}^{+}/ℑe for prime levels ≤ 1447.

level structure
359 2, 2
389 10
433 7
563 13
571 12
643 8
709 11
821 4
887 4, 4
911 4, 4, 4
997 3, 24
1061 151
1091 49
1163 4
1171 22
1229 16
1231 4, 4, 4
1283 25
1361 4, 4, 4
1429 25
1433 2, 2

3.4 An approach to the Birch and Swinnerton-Dyer formula

Recall the notation from Sections 3.1 and 3.2. In this section, we outline a program to prove the BSD formula for quotients of J_{0}(N) whose special L-value is non-zero. Any such quotient is a quotient of J_{e}(N); so the winding quotient is the first interesting quotient from the point of view of the BSD formula for quotients of analytic rank zero. Hence we focus on the winding quotient J_{e}(N) and for simplicity assume that N is a prime. The contents of this section are highly speculative and should be taken with a grain of salt!

Let p be a prime and again let J_{e} = J_{e}(p). If we compare Theorem 3.2.2 to the BSD formula (Conjecture 3.2.1), then we get:

\frac{|� _{J_{e}}| · c_{p}(J_{e})}{|J_{e}(Q)| · |Ĵ_{e}(Q)|} =^{?} \frac{1}{c_{Je}} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{n} .  (3.11)

Thus the BSD formula for J_{e}(p) reduces to formula (3.11) above. In the rest of the section, we describe a plan to prove this formula by examining the terms that appear in it.

• Mordell-Weil groups:

Recall that since J_{e}(Q) is finite, so is Ĵ_{e}(Q). The following result was pointed out to us by L. Merel:

Proposition 3.4.1 (Merel). |Ĵ_{e}(Q)| = n.

Proof. We have Ĵ_{e}(Q) ⊆ J_{0}(p)(Q)_{tor}. By [Maz77, § II.11 and III.1.2], J_{0}(p)(Q)_{tor} is cyclic of order n and is generated by the divisor class of c = (0) − (∞). So it suffices to show that c ∈ Ĵ_{e}(Q). Now J_{e} = J_{0}(p)/I_{e}J_{0}(p). Hence Ĵ_{e}(\overline{Q})[n] can be seen as the group of elements in J_{0}(p)(\overline{Q})[n] which are orthogonal to (I_{e} · J_{0}(p))(\overline{Q})[n] under the Weil pairing. Since the Hecke action is adjoint with respect to the Weil pairing, all we have to show is that I_{e}c = 0; but this follows because I_{e} annihilates e, and e maps to c under the universal covering map H ⊗ R → J_{0}(p)(C) (see [Maz77, § II.18.5]).

Also, L. Merel communicated to us the following result (see [Oes]):

Proposition 3.4.2 (Merel, Oesterlé). |J_{e}(Q)| = n.

Sketch of the proof. Using [Maz77, II.18.10] (see also a complement given in [Mer96b, §5.1]), we get I_{e} ⊆ ∩_{r≥1}ℑ^{r}. Using this condition, one observes that the proof of [Maz77, III.1.4] generalizes to give J_{0}(p)(Q)_{tor} ≅ J_{e}(Q). Then the result follows from the fact that |J_{0}(p)(Q)_{tor}| = n (again by [Maz77, III.1.2, § II.11]).

• Generalized Manin constant:

We already conjectured (Conjecture 2.2.8) that c_{Je} = 1. The only prime that can divide c_{Je} is 2 (Theorem 2.2.3). So let us assume c_{Je} = 1 for now; in any case the results will be valid if we stay away from the prime 2.

• Congruence primes:

Let S = S_{2}(Γ_{0}(p), Z). Recall from Definition 2.3.1 that the congruence number of J_{e} is r_{Je} = #(\frac{S}{S[I_{e}]⊕(W (I_{e})∩S)}), where W (I_{e}) is the orthogonal complement of S[I_{e}] in S_{2}(Γ_{0}(p), C) with respect to the Petersson inner product. The primes that divide r_{Je} are called the congruence primes of J_{e}.

Lemma 3.4.3.

r_{Je} = #(\frac{S}{S[I_{e}] ⊕ S[Î_{e}]}).

Proof. Clearly S[Î_{e}] ⊆ (W (I_{e}) ∩ S). Counting dimensions using Corollary 3.2.4 and the fact that S ⊗ Q is a free T ⊗ Q module of rank 1 (Part 1 of Remark 3.2.6), we get S[Î_{e}] = (W (I_{e}) ∩ S). The lemma now follows from the definition of r_{Je}.

Lemma 3.4.4. If a prime ℓ is not a congruence prime of J_{e}, then

T ⊗ Z_{ℓ} ≅ (I_{e} ⊗ Z_{ℓ}) ⊕ (Î_{e} ⊗ Z_{ℓ}).

Proof. By Corollary 3.2.5, I_{e} ∩ Î_{e} = {0}. It follows, for example from [Dia89, §2], that

#(\frac{S}{S[I_{e}] ⊕ S[Î_{e}]}) = #(\frac{T/I_{e} ⊕ T/Î_{e}}{T} ),  (3.12)

and that we have a canonical isomorphism

\frac{T}{I_{e} ⊕ Î_{e}} ⟶^{≅} \frac{T/I_{e} ⊕ T/Î_{e}}{T} ,  (3.13)

induced by the projection maps. The lemma follows from equations (3.12) and (3.13), and Lemma 3.4.3.

Lemma 3.4.5. H^{+} ⊗ Z_{ℓ} ≅ T ⊗ Z_{ℓ} for every prime ℓ ≠ 2.

Proof. By [Maz77, II.15.1, II.16.3] (see also [Til97, §3]), H_{1}(X_{0}(p), Z)_{𝔪} is free of rank 2 over T_{𝔪} for every maximal ideal 𝔪 of T with residue characteristic unequal to 2. The lemma now follows from the discussion in §II.18 of [Maz77].

• Order of the arithmetic component group: Recall that if A is an abelian variety over Q and p is a prime number, then c_{p}(A) = [A_{F_{p}}(F_{p}) : A_{F_{p}}^{0}(F_{p})], where A denotes the Néron model of A over Z and A^{0} denotes the largest open subgroup scheme of A in which all the fibers are connected. The group A_{F_{p}}/A_{F_{p}}^{0} is called the component group of A at p and is denoted by Φ_{p}(A). The group of F_{p}-valued points of Φ_{p}(A) is called the arithmetic component group of A at p. We have a short exact sequence of group schemes

0 → A_{F_{p}}^{0} → A_{F_{p}} → Φ_{p}(A) → 0,

which gives us the long exact sequence of Galois cohomology groups:

0 → A_{F_{p}}^{0}(F_{p}) → A_{F_{p}}(F_{p}) → Φ_{p}(A)(F_{p}) → H^{1}(Gal(\overline{F}_{p}/F_{p}), A_{F_{p}}^{0}(\overline{F}_{p})).

By [Lan56], the group H^{1}(Gal(\overline{F}_{p}/F_{p}), A_{F_{p}}^{0}(\overline{F}_{p})) is trivial. Hence c_{p}(A) is just the order of the arithmetic component group of A at p.

Recall the discussion around Definition 2.3.3: composing the quotient map π : J_{0}(p) → J_{e} with its dual map π̂ : Ĵ_{e} → J_{0}(p) (where we identify { \atop J_{0}(p)} with J_{0}(p) using the usual canonical isomorphism, e.g., see [Mil86c, Thm. 6.6]), we get maps Ĵ_{e} \underrightarrow{π̂} J_{0}(p) →^{π} J_{e} such that the composite, call it ψ, is an isogeny. Recall that, by definition, n_{Je} is the exponent of ker ψ. Then using the fact that ker ψ ⊆ ker n_{Je}, where n_{Je} also denotes the multiplication by n_{Je} map on Ĵ_{e}, we find that n_{Je} factors as n_{Je} = ψ′ ∘ ψ with ψ′ an isogeny J_{e} → Ĵ_{e} (see [Mil86a, §8]). We can easily check by the definition of ψ′ that ψ ∘ ψ′ is the multipli-

cation by n_{Je} map on J_{e}. Thus the composite J_{e} →^{ψ′} Ĵ_{e} \underrightarrow{π̂} J_{0}(p) →^{π} J_{e} is multiplication by n_{Je}. This induces maps Φ_{p}(J_{e}) → Φ_{p}(J_{0}(p)) →^{π} Φ_{p}(J_{e}) such that the composite is multiplication by n_{Je}. Thus multiplication by n_{Je} kills the cokernel Φ_{p}(J_{e})/π(Φ_{p}(J_{0}(p))). By Theorem 2.3.2, the only primes that can divide n_{Je} are the congruence primes of J_{e}. Thus, “away” from the congruence primes of J_{e}, the map Φ_{p}(J_{0}(p)) → Φ_{p}(J_{e}) is surjective.

If E is an optimal modular elliptic curve, it is known that the map Φ_{p}(J_{0}(p)) → Φ_{p}(E) is surjective. There are two approaches to this: one strategy is to use Ribet’s levellowering theorem (communication of K. Ribet); another strategy (see [MO89, Cor. 2, p. 183]) is to use explicit formulas for the order of the component group given by Ribet (letter to J.-F. Mestre) (these formulas have been generalized to higher dimensional quotients by W. Stein [Ste00]). Also, calculations of W. Stein always found that the map Φ_{p}(J_{0}(p)) → Φ_{p}(A_{f}) was surjective, where A_{f} denotes the quotient of J_{0}(p) associated to a newform f (see Section 4.1.1). By generalizing the proof for elliptic curves, one hopes to show that the map Φ_{p}(J_{0}(p)) →^{π} Φ_{p}(J_{e}) is surjective. Since |Φ_{p}(J_{0}(p))| = n, where again n = num(\frac{p−1}{12}), this would show that c_{p}(J_{e}) divides n. In any case, this is true “away” from the congruence primes of J_{e}.

For simplicity, assume that c_{p}(J_{e}) = n and c_{Je} = 1. Then according to the BSD formula, the order of the Shafarevich-Tate group of J_{e} is the product of |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | and

|\frac{H_{e}^{+}}{ℑe} |. Let us examine the nature of these two factors.

• The factor |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} |: From Lemmas 3.4.4 and 3.4.5, we find that the only primes

that can divide the factor |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | are the prime 2 and the congruence primes of J_{e}.

For example, the only prime level less than 397 for which our calculations (Section 3.3.3) detected an odd prime that divided |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | was the level 389, where 5 divided this factor;

and in this case, we indeed find that 5 is a congruence prime of J_{e} (see [AS99b]).

Mazur [Maz98] showed that if E is an elliptic curve such that its Mordell-Weil rank is zero and such that the newform corresponding to E is congruent modulo a prime ℓ to the newform associated to another elliptic curve whose Mordell-Weil rank is non-zero, then, under certain mild hypotheses, ℓ divides |� _{E}|. In [AS99b], this result is extended to quotients of higher dimension in certain cases. For example, one finds that 5 divides |� _{J_{e}(389)}|; so |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | divides |� _{J_{e}}| away from the prime 2, for level 389. In view of this, one hopes to show that the factor |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | divides |� _{J_{e}}|.

• The factor |\frac{H_{e}^{+}}{ℑe} |: This factor is quite mysterious and deserves further study. In fact,

the mystery behind the BSD formula probably lies in this factor. We computed |\frac{H_{e}^{+}}{ℑe} | for several primes p (see Section 3.3.3). We found that for all prime levels up to 1447, except for the levels 1091, 1283 and 1429, whenever a prime divided |\frac{H_{e}^{+}}{ℑe} |, it was a congruence prime of J_{e}. For the level 1091, there was a congruence between the newform corresponding to J_{e}(1091) and a newform at level 2·1091. Similar calculations for the levels 1283 and 1429 have not been done yet.

So it is possible that the primes that divide the second factor are either congruence primes of J_{e} or primes of congruence between a modular form killed by I_{e} and forms of higher levels (that have L-value equal to zero). Thus, by using the techniques mentioned above for the factor |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} |, one may be able to prove that |\frac{H_{e}^{+}}{ℑe} | divides |� _{J_{e}}|.

• Upper and lower bounds on |� _{J_{e}}|: Our strategy above might show that the conjectural order of � _{J_{e}} as predicted by the BSD formula divides the (actual) order of � _{J_{e}}. See [AS99b] for some results in this direction. For example, one can show that the odd part of the BSD-conjectured order of � _{J_{e}(389)}, which is 25, divides the actual order of � _{J_{e}(389)}.

As mentioned in the introduction, one can use Euler systems to bound the p-primary part of the order of � _{J_{e}} from above in terms of the BSD conjectural order, for almost every prime p (see [Rub98, Cor. 8.9]: the results generalize to abelian varieties; instead of Prop. 8.3 in [Rub98], one has to use the results in [Rib97]).

So one hopes to show that the BSD conjectural order of � _{J_{e}} bounds the (actual) order of � _{J_{e}} from below as well as from above (in particular cases, for example) and thus prove the BSD formula.

Chapter 4

Formulas for the ratio of the special L-value to the real volume for certain other quotients

In this chapter, we generalize the techniques of Section 3.2 to prove formulas expressing the ratio of the special L-value to the real volume as a rational number for certain other quotients. We do this for the quotient associated to a newform in Section 4.1 and for the winding quotient of level a product of two distinct primes in Section 4.2. In Section 4.3, we indicate other possible extensions. Sections 4.1, 4.2 and 4.3 are independent of each other.

4.1 Quotients associated to newforms

We express as an easily computable rational number the ratio of the L-value to the real volume for the quotient of J_{0}(N) attached to a newform (by Shimura). This formula is used to do calculations regarding the Birch and Swinnerton-Dyer conjecture in [AS99b], which extend the calculations in [Cre97], for quotients that are elliptic curves, to quotients of higher dimension. This section depends on Section 1.2.

4.1.1 Introduction and results

Let N be a positive integer. Let f be a newform for Γ_{0}(N) and let a_{n}(f) denote its nth Fourier coefficient. Then the series ∑_{n=1}^{∞} \frac{a_{n}(f)}{n^{s}} converges absolutely in s for Re(s) > 3/2 and can be analytically continued to the entire complex plane [Shi94, Thm. 3.66]. It is called the L-function of f and is denoted L(f, s). Suppose L(f, 1) ≠ 0. Recall the definition of the Hecke algebra T from Section 1.2. Let I_{f} be the annihilator of f under the action of T. Let A_{f} denote the quotient abelian variety J_{0}(N)/I_{f}J_{0}(N) over Q, which was introduced by Shimura in [Shi94]. We call it the quotient of J_{0}(N) associated to the newform f . Recall the definition of the special L-value L_{A_{f}} (1), the Manin constant c_{A_{f}}, the real volume Ω_{A_{f}}, and c_{∞}(A_{f}) from Section 1.2. We are interested in giving a formula for L_{A_{f}} (1)/Ω_{A_{f}}, which is the left-hand side of the Birch and Swinnerton-Dyer formula (Conjecture 1.2.1).

If V is a finite dimensional vector space over R, then a lattice L ⊂ V is a free abelian group of rank equal to dimV such that RL = V. If L, M ⊂ V are lattices, the lattice index [L : M] is the absolute value of the determinant of an automorphism of V taking L isomorphically onto M.

Recall the definition of the winding element e from Section 3.1. Note that by the Manin-Drinfeld Theorem (see [Lan95, Chap. IV, Thm. 2.1] and [Man72]), e ∈ H_{1}(X_{0}(N), Z) ⊗ Q. Also, since complex conjugation on H_{1}(X_{0}(N), Z) is induced by the map z ↦ −\overline{z} on the complex upper half plane, we see that e is invariant under complex conjugation. Thus Te ⊆ H_{1}(X_{0}(N), Z)^{+} ⊗ Q. Let f_{1} = f, f_{2}, …, f_{d} be the Galois conjugates of f (for some integer d) and let Φ denote the map H_{1}(X_{0}(N), Z)^{+} ⊗ Q → C^{d} given by mapping a cycle γ to {∫_{γ} f_{1}, …, ∫_{γ} f_{d}}. Then Φ(H_{1}(X_{0}(N), Z)^{+}) and Φ(Te) are both invariant under complex conjugation, and are lattices in R^{d} ⊆ C^{d} ( Φ(Te) is a lattice since L(f, 1) ≠ 0).

We have the following formula, which was conjectured by W. Stein based on some calculations:

Theorem 4.1.1. With notation as above,

\frac{L_{A_{f}} (1)}{Ω_{A_{f}}} = \frac{[Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)]}{c_{A_{f}} · c_{∞}(A_{f})}

We prove this Theorem in § 4.1.3. The proof is an adaptation of the proof of Theorem 3.2.2. See also [AS99b] for a similar proof. This formula was used for doing computations regarding the Birch and Swinnerton-Dyer conjectural formula in [AS99b].

Let C denote the cuspidal subgroup, i.e., the subgroup of the degree zero divisors on X_{0}(N) supported on the cusps and let n denote its order (this is consistent with the definition of n in Section 3.2, since when N = p is a prime, the order of the cuspidal subgroup is num(\frac{p−1}{12})). We show:

Proposition 4.1.2. With notation as above,

n · [Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)] ∈ Z.

We prove this in the next section. This result ties in very well with the Birch and Swinnerton-Dyer conjectural formula. See [AS99b] for details and a stronger result which says that the number n above can be replaced by the order of the image in A_{f}(Q) of the point (0) − (∞).

4.1.2 Proof of Proposition 4.1.2

Let ℑ denote the annihilator of the divisor (0) − (∞), considered as an element of J_{0}(N)(C), under the action of T. An easy adaptation of [Maz77, II.18.6] shows that ℑe ∈ H_{1}(X_{0}(N), Z)^{+}. So we have

[Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)] = \frac{[Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(ℑe)]}{[Φ(Te) : Φ(ℑe)]},

where both the numerator and the denominator on the right-hand side are the usual indices of subgroups of groups, and hence are integers. So, to prove Proposition 4.1.2, it suffices to prove that [Φ(Te) : Φ(ℑe)] divides n.

We have the surjection T → Φ(Te)/Φ(ℑe) given by taking t ∈ T to the coset containing Φ(te). Clearly ℑ is in the kernel. Hence we get a surjection T/ℑ → Φ(Te)/Φ(ℑe). This shows that [Φ(Te) : Φ(ℑe)] divides the order of T/ℑ. So it suffices to prove that the order of T/ℑ divides the order of C.

We have the map T → C, given by taking t ∈ T to t((0) − (∞)). By the definition of ℑ, the kernel of this map is ℑ. Thus, we have an injection T/ℑ → C, showing that the order of T/ℑ divides the order of C, thus finishing the proof.

4.1.3 Proof of Theorem 4.1.1

We continue to use the notation from § 4.1.1. Theorem 4.1.1 follows immediately from Proposition 4.1.4 and Proposition 4.1.6 below.

We shall first replace the lattice index in Theorem 4.1.1 by another index. Recall that if f ∈ S_{2}(Γ_{0}(N), C), then ω_{f} is the differential on X_{0}(N)(C) given by 2πif(z)dz.

Lemma 4.1.3. The kernel of the composite

Te ↪ H_{1}(X_{0}(N), Z)^{+} ⊗ Q → (H_{1}(X_{0}(N), Z)^{+}/kerΦ) ⊗ Q

is I_{f}e, where the second map above is the natural projection map.

Proof. If t ∈ I_{f}, then ∫_{te} ω_{f_{i}} = ∫_{e} ω_{tf_{i}} = 0 ∀i, and thus I_{f}e is in the kernel. Conversely if t ∈ T is such that te is in the kernel, then ∫_{te} ω_{f} = 0. Now f is an eigenform for all Hecke operators; let the eigenvalue for t be λ. Then we have λ ∫_{e} ω_{f} = 0, which means that λ = 0, since ∫_{e} ω_{f} = L(f, 1) ≠ 0. Thus tf = λf = 0, i.e., t ∈ I_{f}, and so the kernel of the map mentioned above is contained in I_{f}e. That proves the lemma.

Thus we can think of Te/I_{f}e as a lattice in (H_{1}(X_{0}(N), Z)^{+}/kerΦ) ⊗ R. Let π denote the quotient map J_{0}(N)→A_{f}. Since its kernel is connected, it induces a surjection H_{1}(J_{0}(N), R)→H_{1}(A_{f}, R). The standard immersion X_{0}(N) → J_{0}(N) obtained by sending the cusp ∞ to 0, gives us an isomorphism H_{1}(X_{0}(N), R) ≅ H_{1}(J_{0}(N), R). Combining the two maps above, we get a surjection π_{∗} : H_{1}(X_{0}(N), R) → H_{1}(A_{f}, R).

Proposition 4.1.4. The map π_{∗} induces an isomorphism

H_{1}(X_{0}(N), Z)^{+}/kerΦ ≅ H_{1}(A_{f}, Z)^{+},

and we have

[Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)] = [H_{1}(A_{f}, Z)^{+} : π_{∗}(Te/I_{f}e)] ,

where H_{1}(A_{f}, Z)^{+} and π_{∗}(Te/I_{f}e) are considered as lattices in H_{1}(A_{f}, R)^{+}.

Proof. From Lemma 4.1.3, we have

[Φ(H_{1}(X_{0}(N), Z)^{+}) : Φ(Te)] = [H_{1}(X_{0}(N), Z)^{+}/kerΦ : Te/I_{f}e],  (4.1)

where the groups on the right-hand side are considered as lattices in the R-vector space (H_{1}(X_{0}(N), Z)^{+}/kerΦ) ⊗ R.

Let γ be an element of H_{1}(X_{0}(N), Z). Then γ ∈ ker π_{∗} ⟺ ∫_{π_{∗}γ} ω = 0 ∀ ω ∈ H^{0}(A_{f}, Ω_{A_{f} /C})

⟺ ∫_{γ} π^{∗}(ω) = 0 ∀ ω ∈ H^{0}(A_{f}, Ω_{A_{f} /C})

⟺ ∫_{γ} ω_{f_{i}} = 0 ∀ i
⟺ γ ∈ kerΦ.

Combining this with the fact that π_{∗} is surjective, we see that π_{∗} induces an isomorphism H_{1}(X_{0}(N), Z)^{+}/kerΦ ≅ H_{1}(A_{f}, Z)^{+}.

Applying π_{∗} to the right-hand side of (4.1), we get the proposition.

Let S_{f} = S_{2}(Γ_{0}(N), Z)[I_{f}]. Recall that we have a perfect pairing

T × S_{2}(Γ_{0}(p), Z) → Z  (4.2)

which associates to (T, f) the first Fourier coefficient a_{1}(f | T) of the modular form f | T (see [Rib83, (2.2)]); this induces a pairing

ψ : T/I_{f} × S_{f} → Z.

Lemma 4.1.5. The pairing ψ above is a perfect pairing.

Proof. Both T/I_{f} and S_{f} are free Z-modules of the same rank. So it suffices to prove that the induced maps S_{f} → Hom(T/I_{f}, Z) and T/I_{f} → Hom(S_{f}, Z) are injective. The injectivity of the first map follows from the perfectness of the pairing (4.2). Suppose the image of T ∈ T in T/I_{f} maps to the trivial element of Hom(S_{f}, Z). Then a_{1}(f | T) = 0. But f is an eigenform for T; suppose the eigenvalue is λ. Then 0 = a_{1}(f |T) = λa_{1}(f) = λ. Thus f | T = 0, i.e., T ∈ I_{f}. Thus the map T/I_{f} → Hom(S_{f}, Z) is injective and we are done.

Next, we relate the new lattice index in Proposition 4.1.4 to the special L-value:

Proposition 4.1.6.

\frac{L_{A_{f}} (1)}{Ω_{A_{f}}} = \frac{[H_{1}(A_{f}, Z)^{+} : π_{∗}(Te/I_{f}e)]}{c_{A_{f}} · c_{∞}(A_{f})}.

Proof. The proof is similar to the proof of Theorem 3.2.2. Recall that we have the pairing (H_{1}(X_{0}(N), Z)^{+} ⊗ C) × S_{2}(Γ_{0}(N), C) → C given by (γ, f) ↦ ⟨γ, f⟩ = ∫_{γ} ω_{f}. In the proof, at various points, we will consider pairings between two Z-modules; unless otherwise stated, each such pairing is obtained in a natural way from this pairing.

Now L_{A_{f}} (s) = ∏_{i} L(f_{i}, s), by [Shi94, Thm. 7.14] and [Car86]. Recall from Section 1.2 that we have Ω_{A_{f}} = c_{A_{f}} · c_{∞}(A_{f}) · disc(H_{1}(A_{f}, Z)^{+} × S_{f} → C),. Hence

c_{A_{f}} · c_{∞}(A_{f}) · \frac{L_{A_{f}} (1)}{Ω_{A_{f}}} = \frac{∏_{i}⟨e, f_{i}⟩}{disc(H_{1}(A_{f}, Z)^{+} × S_{f} → C)}
= \frac{∏_{i}⟨e, f_{i}⟩}{disc(Te/I_{f}e × S_{f} → C)} [H_{1}(A_{f}, Z)^{+} : π_{∗}(Te/I_{f}e)] .

The proposition now follows from: Claim 1:

\frac{∏_{i}⟨e, f_{i}⟩}{disc(Te/I_{f}e × S_{f} → C)} = 1.

Proof of Claim 1. We first make another claim: Claim 2: The map T → Te given by t ↦ te induces an isomorphism T/I_{f} ⟶^{≅} Te/I_{f}e.

Proof of Claim 2. It is clear that the map T → Te/I_{f}e given by t ↦ te is surjective. All we have to show is that the kernel of this map is I_{f}. It is clear that the kernel contains I_{f}. Conversely, if t is in the kernel, then te ∈ I_{f}e; let i ∈ I_{f} be such that te = ie. Then (t − i)e = 0, and thus ∫_{(t−i)e} ω_{f} = 0, i.e., ∫_{e} ω_{(t−i)f} = 0. If the eigenvalue of f under (t − i) is λ, then this means λL(f, 1) = 0, i.e., λ = 0. Thus (t − i) ∈ I_{f}, i.e., t ∈ I_{f}. This proves Claim 2.

In what follows, i, j, k, and ℓ are indices running from 1 to d. Let {g_{k}} be a Z-basis of S_{f} and let {t_{j}} be the dual basis of T/I_{f} under the perfect pairing ψ in Lemma 4.1.5 above. Then by Claim 2, {t_{j}e} is a basis for Te/I_{f}e. Now g_{k} = ∑_{k} a_{ki}f_{i} for some {a_{ki} ∈ C}. Let A be the matrix having (k, i)-th entry a_{ki}, and let (a^{−1})_{iℓ} denote the (i, ℓ)-th element of the inverse of A. Then

disc(Te/I_{f}e × S_{f} → C) = det{⟨t_{j}e, g_{k}⟩} = det{⟨e, g_{k} |t_{j}⟩} = det{⟨e, (∑_{i} a_{ki}f_{i})|t_{j}⟩} = det{⟨e, ∑_{i} a_{ki}a_{1}(f_{i} |t_{j})f_{i}⟩} (since f_{i}’s are eigenvectors) = det{⟨e, ∑_{i} a_{ki} ∑_{ℓ}(a^{−1})_{iℓ}a_{1}(g_{ℓ} |t_{j})f_{i}⟩} (using f_{i} = ∑_{ℓ}(a^{−1})_{iℓ}g_{ℓ})

= det{⟨e, ∑_{i} a_{ki}(a^{−1})_{ij}f_{i}⟩} (using a_{1}(g_{ℓ} |t_{j}) = δ_{ℓj})
= det{∑_{i} a_{ki}(a^{−1})_{ij}⟨e, f_{i}⟩} = det{∑_{i} a_{ki}⟨e, f_{i}⟩(a^{−1})_{ij}}
= det(AΔA^{−1}) (where Δ = diag(⟨e, f_{i}⟩))
= det(Δ) = ∏_{i}⟨e, f_{i}⟩.

This proves Claim 1.

With that, we are done proving Proposition 4.1.6.

4.2 The winding quotient of level a product of two distinct primes

Let p and q be two distinct primes and let J_{e} be the winding quotient at level pq. We give a formula that expresses the ratio of L_{J_{e}}(1) to the real volume as a rational number and interpret this formula in terms of the Birch and Swinnerton-Dyer conjecture. The proof of this formula uses a generalization of the techniques of Section 3.2. This section depends on Chapter 1 and Section 3.1.

A. Brumer reminded us that if our main interest is to prove the Birch and Swinnerton-Dyer formula for any quotient of J_{0}(N), then it suffices to prove it for all new quotients (at all levels that divide N). This is because any such a quotient is isogenous to a product of new quotients (at various levels dividing N), and if the BSD formula is true for an abelian variety, then it is true for an isogenous abelian variety (see [Mil86b, Thm. 7.3]). However, since the BSD formula is not known to be true even for new quotients, the results of this section are still of interest. Also, these results led to the investigation of the generalized Manin constant in Chapter 2.

4.2.1 Notations and results

Let p and q be two distinct primes and recall, from Section 3.1, the definitions of the winding element e, the winding ideal I_{e} and the winding quotient J_{e}(N) at level N = pq. For simplicity of notation, we denote J_{e}(pq) by just J_{e} in this section. Also, if f is a modular form, and m is an integer, let a_{m}(f) denote the mth Fourier coefficient of f. Recall the definitions from Chapter 1, especially the definitions of the special L-value L_{J_{e}}(1), the Manin constant c_{Je}, the real volume Ω_{J_{e}}, and c_{∞}(J_{e}). Let H = H_{1}(X_{0}(pq), Z), H_{e} = H[I_{e}], Î_{e} = Ann_{T}I_{e}, Ĥ_{e} = H[Î_{e}], ℑ = Ann_{T}((0) − (∞)). If M is a positive integer, then let S_{M} denote the set of newforms f of level M such that L(f, 1) ≠ 0. Then we have:

Theorem 4.2.1. With notation as above,

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} = \frac{1}{|\frac{H^{+}/Ĥ_{e}^{+}}{(1+c)(H/Ĥ_{e})} |} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{∏_{f∈S_{p}}(1 + q − a_{q}(f)) · ∏_{g∈S_{q}}(1 + p − a_{p}(g)) · |\frac{Te}{ℑe} |} · \frac{q^{#S_{p}} · p^{#S_{q}}}{c_{Je}} .

Compare this to Theorem 3.2.2. Note that |\frac{H^{+}/Ĥ_{e}^{+}}{(1+c)(H/Ĥ_{e})} | is a power of 2. We will prove the formula above in the next section, but first let us compare it to the Birch and Swinnerton-Dyer formula (Conjecture 1.2.1). We will prove in Section 4.2.2 that L_{J_{e}}(1) ≠ 0, and so by [KL89] (which uses [GZ86], and was completed independently in [BFH90] and [MM91]), J_{e}(Q) and the Shafarevich-Tate group � _{J_{e}} are finite. The BSD formula then says that

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} =^{?} \frac{|� _{J_{e}}| · c_{p}(J_{e}) · c_{q}(J_{e})}{|J_{e}(Q)| · |Ĵ_{e}(Q)|} ,  (4.3)

where c_{p}(J_{e}) and c_{q}(J_{e}) are the orders of the arithmetic component group of J_{e} at p and q respectively, and Ĵ_{e} is the dual abelian variety of J_{e}. Compare formula (4.3) to Theorem 4.2.1.

Example 4.2.2. Consider the case when p = 3 and q = 11. It turns out that J_{0}(33) has analytic rank zero, so I_{e} = 0 and J_{e} = J_{0}(33). There are no newforms of level 3, and there is one newform of level 11, call it g. From [Cre97, Table 3], we find that a_{3}(g) = −1. So 1 + p − a_{p}(g) = 1 + 3 − (−1) = 5. Thus, for level 33, Theorem 4.2.1 says that

\frac{L_{J_{0}(33)}(1)}{Ω_{J_{0}(33)}} = \frac{|\frac{H^{+}}{ℑe} |}{5 · c_{∞}(J_{0}(33)) · |\frac{Te}{ℑe} |} · \frac{3}{c_{J_{0}(33)}} .  (4.4)

Also using [Lig75, Lem. 3.2.15] we find that the order of (0) − (∞) is 10, so the only primes that can divide |Te/ℑe| are 2 and 5. Moreover, by an independent calculation (using the fact that J_{0}(33) is isogenous to a product of elliptic curves), W. Stein found that

\frac{L_{J_{0}(33)}(1)}{Ω_{J_{0}(33)}} = \frac{1}{2^{2}5^{3}} · \frac{3}{c_{J_{0}(33)}} ,  (4.5)

which is in accord with the BSD formula (4.4).

Compare formulas (4.4) and (4.5) to what the BSD formula (4.3) predicts. Calculations of W. Stein showed that c_{3} = 2, c_{11} = 10 and 2 · 5 | |J_{0}(33)(Q)| | 2^{3}5^{2}. Now |� _{J_{0}(33)}| is a square; so the only way the number 3 can appear in formulas (4.4) and (4.5) is if 3 divides c_{J_{0}(33)}. This observation actually led to an investigation of the generalized Manin constant for which results are reported in Chapter 2. In fact, as mentioned in Section 2.2.3, we find that c_{J_{0}(33)} = 3.

In view of this example, and looking back at Theorem 4.2.1, we pose the following question:

Question 4.2.3. Is c_{Je(pq)} = q^{#S_{p}} · p^{#S_{q}} ?

4.2.2 Proof of Theorem 4.2.1

The proof is a generalization of the proof of Theorem 3.2.2. Let S_{e} = S_{2}(Γ_{0}(pq), Z)[I_{e}]. Just as in that proof, we get

\frac{L_{J_{e}}(1)}{Ω_{J_{e}}} = \frac{L_{J_{e}}(1)}{c_{∞}(J_{e}) · c_{Je} · disc(H_{1}(J_{e}, Z)^{+} × S_{e} → C)}

= \frac{L_{J_{e}}(1)}{disc(Te × S_{e} → C)} · \frac{|\frac{(H/Ĥ_{e})^{+}}{H^{+}/Ĥ_{e}^{+}} |}{c_{∞}(J_{e})} · \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |}{|\frac{Te}{ℑe} |} · \frac{1}{c_{Je}} ,  (4.6)

where again the pairings above are obtained in a natural way from the pairing

⟨ , ⟩ : (H_{1}(X_{0}(pq), Z)^{+} ⊗ C) × S_{2}(Γ_{0}(pq); C) → C

given by (γ, f) ↦ ⟨γ, f⟩ = ∫_{γ} 2πif(z)dz; also note that to get the statement of Lemma 3.2.3 in this situation, we use [Par99, Lem. 3.10].

We focus on the factor \frac{L_{Je}(1)}{disc(Te×S_{e}→C)} appearing above. If M is a positive integer, then let T_{M} denote the set of Galois orbits of newforms f of level M such that L(f, 1) ≠ 0. If f is a newform of level M, let A_{f} denote the quotient of J_{0}(M) associated to f by Shimura (as in Section 4.1.1).

We have the isogeny (see [Par99, §3.8])

J_{e} ∼ ∏_{f∈T_{p}} A_{f}^{2} · ∏_{g∈T_{q}} A_{g}^{2} · ∏_{h∈T_{pq}} A_{h}.

Hence

L_{J_{e}}(1) = ∏_{f∈T_{p}} L_{A_{f}} (1)^{2} · ∏_{g∈T_{q}} L_{A_{g}}(1)^{2} · ∏_{h∈T_{pq}} L_{A_{h}}(1)
= ∏_{f∈S_{p}}⟨e, f⟩^{2} · ∏_{g∈S_{q}}⟨e, g⟩^{2} · ∏_{h∈S_{pq}}⟨e, h⟩,  (4.7)

where the second equality follows from by [Shi94, Thm. 7.14] (completed by [Car86]). Note that in the formula above we should really be taking ⟨e, f⟩ “at level p”, but it is the same as taking it “at level pq” by the functoriality of the (de Rham) pairing (ditto for the ⟨e, g⟩’s). Hence L_{J_{e}}(1) ≠ 0.

Just as in Section 3.2, there is a perfect pairing T/I_{e}×S_{e} → Z which associates to (T, f) the first Fourier coefficient a_{1}(f | T) of the modular form f | T. This defines t_{e} ∈ T/I_{e} ⊗ C characterized by ⟨e, f⟩ = a_{1}(f | t_{e}) ∀f ∈ S_{e}. By the perfectness of this pairing and the canonical isomorphism Te ≅ T/I_{e}, the discriminant of the pairing Te × S_{e} → C that associates to (te, f) the complex number ⟨te, f⟩ coincides with the discriminant of the pairing T/I_{e} × Hom(T/I_{e}, Z) → C that associates to (t, ψ) the complex number ψ(t_{e}t) (obtained by extending ψ by C-linearity). The latter discriminant is equal to det_{T/I_{e}⊗C}t_{e} = det_{S_{e}⊗C}t_{e}.

So we need to find the action of t_{e} on S_{e} ⊗ C. Note that a basis for S_{e} ⊗ C is given by {h : h ∈ S_{pq}} ∪ {f(z), f(qz) : f ∈ S_{p}} ∪ {g(z), g(pz) : g ∈ S_{q}}. If f is a newform of some level M dividing pq, let V_{f} denote the space spanned by f(z) and f((pq/M)z). Then S_{e} ⊗ C = ⊕_{{h:h∈S_{pq}}}V_{h} ⊕_{{f:f∈S_{p}}} V_{f} ⊕_{{g:g∈S_{q}}} V_{g}. So we have

disc(Te × S_{e} → C) = ∏_{h∈S_{pq}} det_{V_{h}} t_{e} · ∏_{f∈S_{p}} det_{V_{f}} t_{e} · ∏_{g∈S_{q}} det_{V_{g}} t_{e}.  (4.8)

From (4.7) and (4.8), we get

\frac{L_{J_{e}}(1)}{disc(Te × S_{e} → C)} = ∏_{h∈S_{pq}} \frac{⟨e, h⟩}{det_{V_{h}} t_{e}} · ∏_{g∈S_{q}} \frac{⟨e, g⟩^{2}}{det_{V_{g}} t_{e}} · ∏_{f∈S_{p}} \frac{⟨e, f⟩^{2}}{det_{V_{f}} t_{e}} .  (4.9)

If f′ is a normalized eigenform for all the Hecke operators (including U_{p} and U_{q}), then it is easy to see that t_{e}f′ = ⟨e, f′⟩f′ (look at the a_{1}’s of both sides).

If h ∈ S_{pq}, then h is an eigenform for all the Hecke operators. So det_{V_{h}} t_{e} = ⟨e, h⟩. Thus

if h ∈ S_{pq}, then

\frac{⟨e, h⟩}{det_{V_{h}} t_{e}} = 1.  (4.10)

But if g ∈ S_{q}, then g(z) and g(pz) are eigenvectors for all the Hecke operators T_{ℓ} for ℓ ≠ p, q and for U_{q}, but not for U_{p}. However U_{p} preserves the subspace V_{g}. Define (B_{p}g)(z) = g(pz). Then on Fourier expansions (see [AL70, §3]),

B_{p}(∑_{n≥1} a_{n}q^{n}) = ∑_{n≥1} a_{n}q^{np},
U_{p}(∑_{n≥1} a_{n}q^{n}) = ∑_{n≥1} a_{np}q^{n},
and T_{ℓ}(∑_{n≥1} a_{n}q^{n}) = ∑_{n≥1} a_{np}q^{n} + ∑_{n≥1} a_{n}q^{np}.

Since g is an eigenform for T_{p} with eigenvalue a_{p} = a_{p}(g), from the above formulas, it is easy to see that U_{p}(g) = a_{p}g − p · B_{p}g and U_{p}(B_{p}g) = g. Thus the characteristic polynomial of U_{p} on V_{g} is U_{p}^{2} −a_{p}U_{p} +p. By [CE98], the roots of this polynomial are distinct and so the action of U_{p} is diagonalizable. If α_{1} and α_{2} are the eigenvalues, then an easy check shows that g − α_{2} · B_{p}g and g − α_{1} · B_{p}g are eigenvectors with eigenvalues α_{1} and α_{2} respectively. Thus we can use this eigenbasis to compute det t_{e} on V_{g}. Since ⟨e, B_{p}g⟩ = ⟨e, g⟩/p, we get

det_{V_{g}} t_{e} = ⟨e, g − α_{1}B_{p}g⟩⟨e, g − α_{2}B_{p}g⟩ = ⟨e, g⟩^{2}(1 − α_{1}/p)(1 − α_{2}/p)

= ⟨e, g⟩^{2}(1 − (α_{1} + α_{2})/p + α_{1}α_{2}/p^{2}) = ⟨e, g⟩^{2}(1 − a_{p}/p + p/p^{2})
= ⟨e, g⟩^{2}(1 + p − a_{p})/p.

So if g ∈ S_{q}, then

\frac{⟨e, g⟩^{2}}{det_{V_{g}} t_{e}} = \frac{p}{1 + p − a_{p}(g)}.  (4.11)

Similarly, we can show that if f ∈ S_{p}, then

\frac{⟨e, g⟩^{2}}{det_{V_{f}} t_{e}} = \frac{q}{1 + q − a_{q}(f)}.  (4.12)

Putting (4.10), (4.11), and (4.12) in (4.9), we get

\frac{L_{J_{e}}(1)}{disc(Te × S_{e} → C)} = \frac{1}{∏_{f∈S_{p}}(1 + q − a_{q}(f)) · ∏_{g∈S_{q}}(1 + p − a_{p}(g))} .  (4.13)

Finally, from Lemma 3.2.10, we get

c_{∞}(J_{e}) = |\frac{(H/Ĥ_{e})^{+}}{(1 + c)(H/Ĥ_{e})}|.

Putting this and (4.13) in (4.6), we get Theorem 4.2.1.

4.3 Some other extensions

• In Sections 3.2 and 4.1.3 we proved formulas for L_{A}(1)/Ω_{A} for quotients A of the new quotient of J_{0}(N). In Section 4.2.2, we worked out the formula for the ratio of the special L-value to the real volume for the winding quotient of level a product of two distinct primes (which is not a quotient of the new quotient of J_{0}(N) in general). An inspection of the proofs shows that our methods will work whenever one can simultaneously diagonalize the action of the Hecke operators. By [CE98], one can do this diagonalization for any quotient of J_{0}(N) when N is cube-free.

• W. Stein suggested that one can generalize the formula in Theorem 4.1.1 to higher weight modular forms, in which case the abelian varieties get replaced by motives. Thus it may have applications to the conjectures of Deligne, Beilinson, Bloch and Kato (e.g., see [BK90]), which are analogs of the BSD conjecture for motives.

• So far, we considered only those quotients of J_{0}(N) whose special L-value was nonzero. There is also a BSD formula for quotients that have L-value equal to zero (given in Conjecture 1.1.2). Our Theorem 4.1.1, together with the ideas in [MT87], suggests a way to find formulas for the left-hand side of the BSD formula in this case.

• Each term in the BSD formula for quotients of J_{0}(N) is a Hecke module; so there might be a finer version of the formula that gives an equality of Fitting ideals (for example, this has been done for elliptic curves with complex multiplication in [Gro82]).

Chapter 5

Detecting invisible elements of the Shafarevich-Tate group

Mazur [Maz98] introduced the concept of visibility of elements of the Shafarevich-Tate group of optimal modular elliptic curves. We generalize the notion to arbitrary abelian varieties and find, based on calculations that assume the Birch and Swinnerton-Dyer conjecture, that there are elements of the Shafarevich-Tate group of certain abelian subvarieties of J_{0}(p) and J_{1}(p) that are not visible in J_{0}(p) and J_{1}(p) respectively. This chapter is basically a generalization of the results given in [Maz98] for quotients of J_{0}(N) that are elliptic curves to quotients of arbitrary dimension. This chapter is fairly independent of the rest of the thesis, except that we refer to the short Section 3.1 for some definitions, and that Proposition 5.3.3 depends on some earlier sections.

5.1 Definitions and the result

Let J be an abelian variety and B be an abelian subvariety of J, both defined over Q. The Galois cohomology group H^{1}(Q, B) is isomorphic to the group of principal homogeneous spaces, or torsors, of B [Lan91, III.4.2]. A B-torsor V is said to be visible in J if it is isomorphic over Q to a subvariety of J. The Tate-Shafarevich group of B, denoted � _{B}, consists of equivalence classes of principal homogeneous spaces of B that are locally trivial everywhere; it is conjectured to be finite. An element of the Shafarevich-Tate of B is said to be visible in J if the corresponding torsor is visible in J. If an element is not visible in J, we say that it is invisible (it will be clear from the context what the ambient abelian variety J is).

The notion of visibility arose when Mazur was looking for natural spaces where can one embed the principal homogeneous spaces of an elliptic curve (see [Maz98] for details). Adam Logan, based on Cremona’s tables, studied instances of non-trivial Shafarevich-Tate groups for elliptic curves E that were quotients of J_{0}(N) for N square-free and less than 3000. The order of the elements of the Shafarevich-Tate group of E that are visible in J_{0}(N) divides the modular degree of E, and thus by comparing the order of the Shafarevich-Tate group (as predicted by the Birch and Swinnerton-Dyer conjecture) with the modular degrees, they tried to detect elements that are not visible in J_{0}(N). The only instance of an invisible element they could convincingly detect was for the level N = 2849, which was not visible in J_{0}(N); but they could not test whether this element becomes visible in J_{1}(N) or not.

As before, if A is an abelian variety, then let Â denote the dual abelian variety of A. Recall the definitions of Section 3.1. Let p be a prime and J_{e} = J_{e}(p) throughout this chapter. The dual map Ĵ_{e} → { \atop J_{0}(p)} = J_{0}(p) is an injection (e.g., by [Maz98, Prop. 9]). Thus we can view Ĵ_{e} as a subvariety of J_{0}(p) and talk about the visibility of its torsors in J_{0}(p). We have a map J_{0}(p) →^{π^{∗}} J_{1}(p) obtained via Picard functoriality from the map π : X_{1}(p) → X_{0}(p). Its kernel is finite. Let Ĵ_{e}′ denote the image of Ĵ_{e} in J_{1}(p) under π^{∗}.

Based on calculations concerning the order of � _{J_{e}} as predicted by the Birch and Swinnerton-Dyer conjecture, we find (Theorems 5.3.4 and 5.3.5) that, for p = 1091, there is an element of � Ĵ_{e} that is not visible in J_{0}(p) and whose image in � Ĵ_{e}′ is not visible in J_{1}(p). This result was mentioned in [Aga99]. After [Maz98] and [Aga99] appeared, other instances of invisibility have been found; e.g., see [CM99] and [AS99b]. The existence of visible elements is closely related to congruences between modular forms of analytic rank zero and modular forms of analytic rank greater than zero (see [AS99b] for details).

5.2 The strategy to detect invisible elements

As in the introduction, let J be an abelian variety and B be an abelian subvariety of J, both defined over Q. The definition of visibility that we gave is one possible extension of the definition given by Mazur. Here is another definition: we say that a B-torsor V is visible as a torsor in J, if there is a subvariety V′ of J such that the group law of J gives an action of B on V′ (both B and V′ are contained in J), and there is an isomorphism of B-torsors ι : V ⟶^{≅} V′ (i.e., an isomorphism of varieties over Q that respects the B-action).

In [CM99] and [AS99b], the definition used is yet another one: visible elements are defined as the the elements in the kernel of the map � _{B} → � _{J}. We show that this definition is the same as the notion of visibility as a torsor.

Proposition 5.2.1. Let J be an abelian variety and B be an abelian subvariety of J, both defined over Q. Let V be a B-torsor. Then V is visible as a torsor in J if and only if the cocycle class corresponding to V is in the kernel of the map H^{1}(Q, B) → H^{1}(Q, J).

Proof. It is convenient to use the notion of sheaf torsors (see [Mil80, § III.4]). If B is an abelian variety over Q, let ST (B) denote the equivalence classes of sheaf torsors of B. If V is a sheaf-torsor, pick P ∈ V (\overline{Q}); then we get a cocycle given by σ ↦ σ(P) − P ∈ B(\overline{Q}), where σ ∈ Gal(\overline{Q}/Q). One can show that this gives an element of H^{1}(Q, B) that is independent of the choice of the point P above. Thus we get a canonical map ST (B) → H^{1}(Q, B). By Theorems 1.7, 3.9, 2.10, and 4.6 in Chapter III of [Mil80], this map is an isomorphism.

If V is a B-sheaf torsor, we define the pushout V ×^{B} J as the sheaf whose section over a Q-algebra R of finite type is the set of orbits of V (R) × J(R) under the action of B(R), where B(R) acts on V (R) in the usual way, but on J(R) the action is by the inverse of the group law on J(R). Also V (R) × J(R) has an action of J(R) on the second component, which is compatible with the B(R) action, and thus we have an action of J(R) on (V ×^{B} J)(R). Hence V ×^{B} J is a J-torsor.

The map H^{1}(Q, B) → H^{1}(Q, J) induces a map ST (B) → ST (J). We first claim that the image of (the sheaf torsor corresponding to) V under this induced map is the pushout V ×^{B} J.

Proof of the claim. Pick P ∈ V (\overline{Q}). Let σ ∈ Gal(\overline{Q}/Q). Just for this proof, we write the torsor action as composition. The cocycle in H^{1}(Q, B) corresponding to V maps σ to a_{σ}, where a_{σ} is the unique element of B(\overline{Q}) such that σ(P) = a_{σ}(P). Now consider the point (P, 0) in V (\overline{Q}) × J(\overline{Q}) and let Q be its image in (V ×^{B} J)(\overline{Q}). Then an easy check shows that σ(Q) = a_{σ}(Q) where, now, a_{σ} is considered an element of J(\overline{Q}). So the cocycle in H^{1}(Q, J) corresponding to V ×^{B} J maps σ to a_{σ} ∈ J(\overline{Q}). But this is exactly the image of V under the map H^{1}(Q, B) → H^{1}(Q, J). Since σ was an arbitrary element of Gal(\overline{Q}/Q), that proves the claim.

Now we are ready to prove the Proposition. Suppose V is visible as a torsor in J. Let ι : V → J be as in the definition of visibility as a torsor in J. Then consider the map of sheaf torsors j : V → V ×^{B} J induced by the map on sections V (R) → V (R) × J(R) given by v ↦ (v, −ι(v)). If v_{1}, v_{2} ∈ V (R), then they are translates by an element of B(R); but then −ι(v_{1}) and −ι(v_{2}) are also translates by the same element of B(R). Hence the images of v_{1} and v_{2} under j are the same; i.e., the image of the map V (R) → (V ×^{B} J)(R) is a point. This point is also Galois invariant (since the map j is defined over Q). Hence this gives us a point of V ×^{B} J over Q. But that makes V ×^{B} J the trivial torsor. Hence by the claim above, the cocycle class corresponding to V in H^{1}(Q, B) maps to the trivial element of H^{1}(Q, J).

Conversely, suppose the cocycle class corresponding to V is in the kernel of the map H^{1}(Q, B) → H^{1}(Q, J). By the claim above, this means that there is an isomorphism ϕ : V ×^{B} J ⟶^{≅} J over Q. Now let R be any Q-algebra of finite type and consider the map ψ : V (R) → (V ×^{B} J)(R) induced by the map V (R) → V (R) × J(R) given by v ↦ (v, 0). An easy check shows that the composite V (R) →^{ψ} (V ×^{B} J)(R) →^{ϕ} J(R) is an injection and that the action of B(R) is preserved. By Yoneda’s lemma, we have a monomorphism, i.e., a closed immersion V → J, and the action of B is preserved. This shows that V is visible as a torsor in J.

If an element is visible as a torsor then it is clear that it is also visible (in the sense we defined it). We do not know if the converse is true. However we have a result which says that the converse is true up to an automorphism of B; we state it next. From now on, whenever we use the term “visible”, we mean the definition we gave in Section 5.1.

Recall that J is an abelian variety and B is an abelian subvariety of J, both defined over Q. Consider the following condition on the pair (J, B):

(∗) if J ∼ B × C is an isogeny over \overline{Q}, where C is another abelian variety, then no simple factor (over \overline{Q}) of B is isogenous (over \overline{Q}) to a simple factor (over \overline{Q}) of C.

The following lemma was stated without proof in [Aga99].

Lemma 5.2.2. Let B be an abelian subvariety of J such that the pair (J, B) satisfies (∗). Let V be a B-torsor that is visible in J; so V can be considered as an element of H^{1}(Q, B).

Consider the natural map ĩ : H^{1}(Q, B) → H^{1}(Q, J) obtained from the embedding i of B in J. Then there exists an automorphism ϕ of B (defined over Q) such that V maps to 0

under the composite H^{1}(Q, B) \underrightarrow{ϕ̃} H^{1}(Q, B) \underrightarrow{ĩ} H^{1}(Q, J), where ϕ̃ is the automorphism of H^{1}(Q, B) induced by ϕ.

Proof. Suppose V is a B-torsor visible in J and let V′ be the subvariety of J isomorphic to V over Q (given by the definition of visibility). Since V is a B-torsor, we have B ≅ V ≅ V′ over \overline{Q}. Consider the composite map B ⟶^{≅} V′ → J/B defined over \overline{Q}. Up to translation, it is a homomorphism of abelian varieties. Its image has to be a point because otherwise (∗) would be violated. Hence the image of V′ → J/B is also a point. Thus V′ is a translate of B (over \overline{Q}) and hence has an action of B by translation. In the following, let σ denote an arbitrary element of Gal(\overline{Q}/Q). As a cocycle in H^{1}(Q, B), V′ is given by σ ↦ σ(Q) − Q, where Q is any fixed element of V′(\overline{Q}), and the subtraction is the usual subtraction in J. But then V′ maps to 0 in H^{1}(Q, J) under ĩ since Q ∈ V′(\overline{Q}) ⊆ J(\overline{Q}). Next, let P be an element of V (\overline{Q}). Then the element of H^{1}(Q, B) corresponding to V is the class of the cocycle σ ↦ σ(P) −_{V} P, where we use subscripts under the group action symbol to distinguish actions of B on different torsors. Let ι : V → V′ be the isomorphism between V and V′ (over Q). Then the element of H^{1}(Q, B) corresponding to V′ is given by σ ↦ σ(ι(P)) −_{V′} ι(P). Consider the map ϕ : B → B given by a ↦ ι(P +_{V} a) −_{V′} ι(P). The map ϕ is defined over Q and is a homomorphism of abelian varieties since it takes the identity element of B to itself. It takes the torsor V to V′ and thus ĩ(ϕ̃(V)) = ĩ(V′) = 0. It is an automorphism since it has an inverse given by a ↦ ι^{−1}(ι(P) +_{V′} a) −_{V} P.

In [CM99], semi-stable elliptic curves were considered, and hence the only possible automorphisms of B were multiplication by ±1 and so all the definitions of visibility coincided (see [CM99, Remark 2]).

Let J be an abelian variety that is self-dual, i.e., an abelian variety J together with an isomorphism Ĵ ⟶^{≅} J, using which we implicitly identify Ĵ with J. Let A be an optimal quotient of J such that the dual map Â → J is injective. If (J, Â) satisfy (∗), then the composite Â → J → A is an isogeny; call it f. Define n_{A} as the exponent of kerf (this definition is consistent with Definition 2.2.5). In this situation, the following proposition gives a way of detecting invisible elements of � Â.

Proposition 5.2.3. Let J be an abelian variety that is self-dual and let A be an optimal quotient such that the dual map Â → J is injective, and such that and (J, Â) satisfy (∗). Then all elements of � Â that are visible in J are killed by multiplication by n_{A}.

Proof. Recall that f was the composite map Â → J → A, which is an isogeny. There is an isogeny g : A → Â such that g ∘ f = multiplication by n_{A} (e.g., see [Mil86a, §8]; we called g an isogeny conjugate to f in Definition 2.3.3). So we have maps

Â ⟶ J ⟶ A ⟶^{g} Â

such that the composite is multiplication by n_{A}. Suppose V is an element of � Â that is visible in J. Apply Lemma 5.2.2 with B = Â and let ϕ be the automorphism of Â as given by the lemma. Consider the maps

Â ⟶^{ϕ} Â ⟶ J ⟶ A ⟶ Â ⟶^{ϕ^{−1}} Â.

The composite is again multiplication by n_{A}. This induces maps

� Â \underrightarrow{ϕ̃} � Â ⟶ � _{J} ⟶ � _{A} ⟶ � Â \underrightarrow{ϕ̃^{−1}} � Â,

where the composite is again multiplication by n_{A}. Consider V as an element of the first group � Â in this sequence. Then by Lemma 5.2.2, its image in � _{J} is trivial; hence it is killed under the composite; i.e., it is killed by multiplication by n_{A}.

Thus every element of � Â that has order prime to n_{A} is not visible in J. We will use this technique in the next section.

5.3 Discovery of invisible elements

We continue to use the notation of Sections 5.1 and 5.2. Let J_{0}(p)′ denote the image of J_{0}(p) in J_{1}(p).

Lemma 5.3.1. Let J denote J_{0}(p) or J_{1}(p), and B denote an abelian subvariety of J_{0}(p) or J_{0}(p)′ respectively. Then the pair (J, B) satisfies (∗).

Proof. First the case of J = J_{0}(p): this follows because in a decomposition of J_{0}(p) up to isogeny, no two simple factors can be isogenous over Q by the multiplicity one theorem and not even over \overline{Q} because p is squarefree (using [Rib75, Prop. 3.1]). Next the case J = J_{1}(p): No simple factor of B can be isogenous to another simple factor of J_{0}(p)′ (by the same argument above). Suppose B′ is a simple factor of B that is isogenous to a simple factor of J_{1}(p)/J_{0}(p)′. Now J_{1}(p)/J_{0}(p)′ has everywhere good reduction over some extension of Q (this follows from [DR73, §5, Ex. 3.7(i)]), hence so does B′. But J_{0}(p) has purely multiplicative reduction at p by [DR73, §5, Thm. 6.9], so it cannot have a factor with good reduction even after a base extension. This contradiction finishes the proof of the lemma.

In what follows, p = 1091 and for ease of notation, we frequently denote J_{0}(p) simply by J_{0}.

Proposition 5.3.2. For p = 1091, the elements of � Ĵ_{e} that are visible in J_{0}(p) are killed by multiplication by 2.

Proof. For p = 1091, we have J_{e} = J_{0}^{−}, where J_{0}^{−} = J_{0}/(1 + W_{p})J_{0} and W_{p} is the Atkin-Lehner involution (this was checked by a calculation and also follows from [Bru95, §8]). By combining the quotient map J_{0} → J_{e} and its dual map, and identifying Ĵ_{0} with J_{0} using the usual canonical isomorphism (e.g., see [Mil86b, Thm 6.6]), we get the maps Ĵ_{e} → J_{0} → J_{e}; call the composite f. By [Maz98, Prop. 8], f is an isogeny; let K denote its kernel. The group K is the intersection of Ĵ_{e} and (1 + W_{p})J_{0}. On the former group, W_{p} acts as −1, and on the latter group, W_{p} acts as +1. We conclude that K is killed by multiplication by 2. The proposition now follows from Lemma 5.3.1 (with J = J_{0} and B = Ĵ_{e}) and Proposition 5.2.3.

Proposition 5.3.3. Assume the Birch and Swinnerton-Dyer formula (Conjecture 3.2.1) for the prime p = 1091. Then for p = 1091, there is an element of order 7 in � Ĵ_{e}.

Proof. We will use notation from Sections 1.2 and 3.2. Combining the BSD formula (Conjecture 3.2.1) with Theorem 3.2.2, we get

|� _{J_{e}}| · c_{p}(J_{e}) · c_{Je} · n = |J_{e}(Q)_{tor}| · |Ĵ_{e}(Q)_{tor}| · |\frac{H^{+}}{Ĥ_{e}^{+} + H_{e}^{+}} | · |\frac{H_{e}^{+}}{ℑe} |  (5.1)

By a calculation (see Table 3.1) we find that, for p = 1091, 7 divides the factor |H_{e}^{+}/ℑe| in the equation given above. Thus 7 divides the right-hand side of equation (5.1). We will check that it does not divide any factor of the left-hand side other than |� _{J_{e}}|.

First consider c_{p}(J_{e}). We apply [BLR90, Prop. 7.5.3] to the exact sequence

0 → (1 + W_{p})J_{0} → J_{0} → J_{0}/(1 + W_{p})J_{0} → 0

to conclude that a power of 2 kills the cokernel of the map of Néron models J_{0} → J_{e}. Hence we have that away from 2, c_{p}(J_{e}) divides the number of connected components in the special fiber at p of J_{0}, which is n by [Maz77, Thm. A.1]. But in our case, n = num((1091 − 1)/12) = 545, so 7 does not divide c_{p}(J_{e}). Next, by Theorem 2.2.3, the only prime that can divide c_{Je} is 2, so certainly 7 does not divide c_{Je}. Finally 7 does not divide n = 545.

So looking at equation (5.1), one concludes that 7 divides |� _{J_{e}}|. Next, by the Cassels-Tate pairing, � Ĵ_{e} is finite, and |� Ĵ_{e}| = |� _{J_{e}}|. Hence 7 divides |� Ĵ_{e}|. Thus � Ĵ_{e} has a non-trivial element of order 7.

Theorem 5.3.4. Assume the Birch and Swinnerton-Dyer formula (Conjecture 1.2.1) for the prime p = 1091. Then for p = 1091, � Ĵ_{e} has an element that is not visible in J_{0}(p).

Proof. The element of order 7 of � Ĵ_{e} from Proposition 5.3.3 is not visible in J_{0}(p) by Proposition 5.3.2.

Theorem 5.3.5. For p = 1091, the image of the element of order 7 of � Ĵ_{e} (from Proposition 5.3.3) in � Ĵ_{e}′ is not visible in J_{1}(p).

Proof. Let J_{1} = J_{1}(p). Consider the series of maps

J_{0} ⟶^{π^{∗}} J_{1} ⟶^{≅} J_{1} ⟶^{π_{∗}} J_{0},

where the map π_{∗} is obtained from π : X_{1}(p) → X_{0}(p) via the Albanese functoriality. The composite is just multiplication by deg(π)= (p − 1)/2 = 545. Let ψ denote the quotient map J_{0} → J_{e} and let ψ̂ denote its dual. Then the composite

Ĵ_{e} \underrightarrow{ψ̂} J_{0} ⟶^{π^{∗}} J_{1} ⟶^{≅} J_{1} ⟶^{π_{∗}} J_{0} ⟶^{ψ} J_{e}

is an isogeny and the only primes that can divide the order of this isogeny are the prime 2 and the primes that divide 545, i.e., 5 and 109. Hence the element of order 7 in � Ĵ_{e} does not get killed in � _{J_{1}} and so there is a nontrivial element of order 7 in � Ĵ_{e}′. Call it V and suppose it is visible in J_{1}. Then by Lemma 5.2.2 and Lemma 5.3.1(with J = J_{1} and B = Ĵ_{e}′), there is an automorphism ϕ of Ĵ_{e}′ such that V is killed under the composite

� Ĵ′_{e} \underrightarrow{ϕ̃} � Ĵ_{e}′ → � _{J_{1}}. Now in the composite

Ĵ_{e} ⟶^{π^{∗}} Ĵ_{e}′ ↪ J_{1} ⟶^{π_{∗}} J_{0} ⟶^{ψ} J_{e},  (5.2)

which is an isogeny, the first map Ĵ_{e} →^{π^{∗}} Ĵ_{e}′ is also an isogeny. So the rest of (5.2), i.e., the composite

Ĵ_{e}′ ↪ J_{1} ⟶^{π_{∗}} J_{0} ⟶^{ψ} J_{e}

is also an isogeny and the only primes that can divide its degree are 2, 5 and 109. Attaching the automorphism ϕ, we find that the composite

Ĵ_{e}′ ⟶^{ϕ} Ĵ_{e}′ ↪ J_{1} ⟶^{π_{∗}} J_{0} ⟶^{ψ} J_{e}

is also an isogeny; call it f. The only primes that can divide the degree of f are 2, 5 and 109. There is an isogeny g : J_{e} → Ĵ_{e}′ such that g ∘ f is multiplication by the degree of the isogeny f (e.g., see [Mil86a, §8]). Since V is killed under the sequence of maps

� Ĵ′_{e} \underrightarrow{ϕ̃} � Ĵ_{e}′ → � _{J_{1}}, it is killed under the map induced by g ∘ f, i.e., it is killed by multiplication by an integer that is not divisible by 7. But V has order 7 in � Ĵ′_{e} ; this contradiction shows that V is an element of � Ĵ′_{e} that is not visible in J_{1}.

A similar result of the existence of an invisible element was found for p = 1429, where 5 divides |H_{e}^{+}/ℑe| (see Table 3.1). Note that as a byproduct, we have that the Shafarevich-Tate groups of J_{0}(1091) and J_{1}(1091) are non-trivial (assuming the Birch and Swinnerton-Dyer conjecture).

Question 5.3.6. Mazur [Maz99] showed that if E is an elliptic curve over Q, then every element of � _{E} of order 3 is visible in an abelian surface contained in the Jacobian of the modular curve of some (unknown) level. So the natural question is whether the element of � _{J_{e}(1091)} that is not visible in J_{0}(1091) is visible in J_{0}(M) for some integer M that is a multiple of 1091. W. Stein found that J_{e}(1091) shared a congruence mod 7 with an elliptic curve of level 2·1091 and rank 1, and using this one hopes to show that the element under consideration becomes visible in J_{0}(2·1091) (by generalizing work of Mazur: see [Maz98] and [AS99b]).

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List of Symbols

Symbol  Definition  Page
L_{A}(s)  L-function of an abelian variety  1
G^{+}  2
c_{∞}(A)  2
Ω_{A}  real volume  2
Â  dual abelian variety  2
R_{A}  regulator  2
� _{A}  Shafarevich-Tate group  2
A  Néron model  2
A^{0}  2
c_{p}(A)  order of arithmetic component group  2
|G|  order of a group  2
=^{?}  conjectured equality  2
N  level  2
X_{0}(N)  modular curve  2
J_{0}(N)  Jacobian of X_{0}(N)  2
T  Hecke algebra  2
T_{ℓ}  Hecke operator  2
U_{p}  Hecke operator  2
Γ_{0}(N)  congruence subgroup  3
S_{2}(Γ_{0}(N), R)  cusp forms with coefficients in R  3
M[I]  I-torsion in M  3
ω_{f}  3
S_{A}  4
c_{A}  generalized Manin constant  4
disc  discriminant of a pairing  4
degϕ_{E}  modular degree  6
c_{E}  Manin constant  6
r_{E}  congruence number  6
J  Jacobian of X_{0}(N)  7
B_{R}  Néron model  7
S_{2}(R)  cusp forms with coefficients in R  7
c_{A}  generalized Manin constant  8

Symbol  Definition  Page
M_{0}(N)  coarse moduli scheme  8
M_{0}(N)^{0}  smooth part of M_{0}(N)  8
ϕ_{A}  modular parametrization  8
q-exp  q-expansion map  8
C_{A}  9
J^{new} new quotient of J_{0}(N) 9
ϕ_{2}  9
ϕ_{1}  9
n_{A}  modular exponent  9
# G  order of a group  10
m  10
S  Spec(Z[\frac{1}{m}])  10
Ω  relative dualizing sheaf  10
q-exp  q-expansion map  10
q-exp  q-expansion map  10
C  11
W (I)  11
r_{A}  congruence number  11
m_{A}  congruence exponent  11
ϕ′  conjugate isogeny  12
e  winding element  15
I_{e}  winding ideal  15
J_{e}(N)  winding quotient  15
J_{e}  winding quotient of prime level p  15
H  homology group  16
H_{e} = H[I_{e}]  16
Î_{e}  16
Ĥ_{e}  16
ℑ  Eisenstein ideal  16
n  order of (0) − (∞)  16
c  complex conjugation involution  18
S_{e}  19
[x] Manin symbol 22
F (x)  23
σ_{1}(i)  23
r  Sturm bound  24
Φ_{p}(A)  component group  29
L(f, s)  L-function of a modular form  32
I_{f}  32
A_{f}  Shimura quotient  32
[L : M]  lattice index  33
f_{i}  conjugate of f  33

Symbol  Definition  Page
Φ  33
n order of the cuspidal subgroup 33
S_{f}  35
J_{e}  winding quotient of level pq  36
a_{m}(f)  mth Fourier coefficient of f  37
ℑ  37
S_{M}  37
J an abelian variety 42
π  43
π^{∗}  43
Ĵ_{e}′  43
(∗)  44
J_{0}  J_{0}(p)  46

Index

analytic rank 3 arithmetic component group 29

BSD formula 2

component group 29 congruence exponent 11 congruence number 6, 11 congruence primes 6 congruence primes of J_{e} 29 conjugate isogeny 12 cusp form 3 cuspidal subgroup 33

discriminant of a pairing 4

Eisenstein ideal 16 exponent of a finite group 9

Fourier coefficient of a cusp form 3 Fourier expansion of a cusp form 3

generalized Manin constant 8

Hecke algebra 2 Hecke operator 2

invisible element 42

Jacobian 2

lattice index 33 lattice 33 level 2 L-function 1

Manin constant 6 Manin symbol 22 modular abelian variety 3 modular curve 2 modular degree 6 modular exponent 9 modular symbol 22

principal homogeneous space 42 pushout 43

q-exp 8, 10

real volume 2 regulator 2

self-dual abelian variety 45 Shafarevich-Tate group 2 sheaf torsor 43 special L-value 3 strong modular elliptic curve 5

torsor 42

visible as a torsor 43 visible element 42

winding element 15

winding quotient of level N 15