Explicit approaches to modular abelian varieties

by William Arthur Stein

B.S. (Northern Arizona University) 1994

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 Hendrik Lenstra, Chair Professor Bjorn Poonen Professor Bin Yu Spring 2000 The dissertation of William Arthur Stein is approved:

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

University of California at Berkeley Spring 2000 Explicit approaches to modular abelian varieties

Copyright 2000 by William Arthur Stein

Abstract

Explicit approaches to modular abelian varieties by

William Arthur Stein

Doctor of Philosophy in Mathematics

University of California at Berkeley

Professor Hendrik Lenstra, Chair

I investigate the Birch and Swinnerton-Dyer conjecture, which ties together the constellation of invariants attached to an abelian variety. I attempt to verify this conjecture for certain specific modular abelian varieties of dimension greater than one. The key idea is to use Barry Mazur’s notion of visibility, coupled with explicit computations, to produce lower bounds on the Shafarevich-Tate group. I have not finished the proof of the conjecture in these examples; this would require computing explicit upper bounds on the order of this group.

I next describe how to compute in spaces of modular forms of weight at least two. I give an integrated package for computing, in many cases, the following invariants of a modular abelian variety: the modular degree, the rational part of the special value of the L-function, the order of the component group at primes of multiplicative reduction, the period lattice, upper and lower bounds on the torsion subgroup, and the real volume. Taken together, these algorithms are frequently sufficient to compute the odd part of the conjectural order of the Shafarevich-Tate group of an analytic rank 0 optimal quotient of J_{0}(N), with N square-free. I have not determined the exact structure of the component group, the order of the component group at primes whose square divides the level, or the exact order of the torsion subgroup in all cases. However, I do provide generalizations of some of the above algorithms to higher weight forms with nontrivial character.

Professor Hendrik Lenstra Dissertation Committee Chair

i

To my parents and my grandmother, Annette Maurer.

Contents

List of Figures v

List of Tables vi

List of Symbols vii

Preface 1

1 The Birch and Swinnerton-Dyer conjecture 2

1.1 The BSD conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 The ratio L(A, 1)/Ω_{A} . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Torsion subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.4 Upper bounds on #� (A) . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.5 Lower bounds on #� (A) . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.6 Motivation for considering abelian varieties . . . . . . . . . . . . . . 6

1.2 Existence of nontrivial visible elements of � (A) . . . . . . . . . . . . . . . 6 1.3 Description of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 Table 1.2: Shafarevich-Tate groups at prime level . . . . . . . . . . . 10 1.3.3 Tables 1.3–1.6: New visible Shafarevich-Tate groups . . . . . . . . . 11

1.4 Further visibility computations . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.1 Does � become visible at higher level? . . . . . . . . . . . . . . . . 12 1.4.2 Positive rank example . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Modular symbols 19 2.1 The definition of modular symbols . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Cuspidal modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Duality between modular symbols and modular forms . . . . . . . . . . . . 21 2.4 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 The ∗-involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4.3 The Atkin-Lehner involutions . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.1 Computing coset representatives . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Compatibility with modular forms . . . . . . . . . . . . . . . . . . . 27 2.6 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.1 Conversion between modular and Manin symbols . . . . . . . . . . . 28 2.6.2 Hecke operators on Manin symbols . . . . . . . . . . . . . . . . . . . 29 2.6.3 The cuspidal and boundary spaces in terms of Manin symbols …. 30 2.6.4 Computing the boundary map . . . . . . . . . . . . . . . . . . . . . 30

2.7 The complex torus attached to a modular form . . . . . . . . . . . . . . . . 32 2.7.1 The case when the weight is 2 . . . . . . . . . . . . . . . . . . . . . . 34

3 Applications of modular symbols 35 3.1 Computing the space of modular symbols . . . . . . . . . . . . . . . . . . . 35 3.2 Computing the Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Representing and enumerating Dirichlet characters . . . . . . . . . . . . . . 38 3.4 The dimension of S_{k}(N, ε) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 Decomposing the space of modular symbols . . . . . . . . . . . . . . . . . . 40

3.5.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5.2 Efficient computation of Hecke operators on the dual space . . . . . 42 3.5.3 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.5.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5.5 Sorting and labeling eigenforms . . . . . . . . . . . . . . . . . . . . . 44

3.6 Intersections and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.1 A strategy for computing congruences . . . . . . . . . . . . . . . . . 47 3.7 The rational period mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.8 The images of cuspidal points . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.8.1 Rational torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.8.2 Upper bound on torsion: Counting points mod p . . . . . . . . . . . 50

3.9 The modular degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.10 The rational part of L(A_{f}, j) . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10.1 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.10.2 Winding elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.10.3 Real and minus volumes . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.10.4 The theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.10.5 Bounding the denominator of the ratio . . . . . . . . . . . . . . . . . 55

3.11 The Manin constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.11.1 The primes that might divide c_{A} . . . . . . . . . . . . . . . . . . . . 57 3.11.2 Numerical evidence for the c_{A} = 1 conjecture . . . . . . . . . . . . . 58

3.12 Analytic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.12.1 Extended modular symbols . . . . . . . . . . . . . . . . . . . . . . . 59 3.12.2 Numerically computing period integrals . . . . . . . . . . . . . . . . 60 3.12.3 The W_{N}-trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.12.4 Computing the period mapping . . . . . . . . . . . . . . . . . . . . . 64 3.12.5 Computing special values . . . . . . . . . . . . . . . . . . . . . . . . 64 3.12.6 The real and minus volume associated to A_{f} . . . . . . . . . . . . . 65 3.12.7 The component groups c_{∞}^{+} and c_{∞}^{−} . . . . . . . . . . . . . . . . . . . 66 3.12.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4 Component groups of optimal quotients 70

4.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1.1 Néron models and component groups . . . . . . . . . . . . . . . . . . 70 4.1.2 Motivating problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.3 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Optimal quotients of Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 The closed fiber of the Néron model . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Rigid uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4.1 Raynaud’s uniformization . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4.2 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5.1 Description of the component group in terms of the monodromy pairing 76 4.6 Optimal quotients of J_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.1 Modular curves and semistability . . . . . . . . . . . . . . . . . . . . 80 4.6.2 Newforms and optimal quotients . . . . . . . . . . . . . . . . . . . . 80 4.6.3 Homology and the modular degree . . . . . . . . . . . . . . . . . . . 80 4.6.4 Rational points of the component group (Tamagawa numbers) … 81

4.7 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.7.1 Conjectures and questions . . . . . . . . . . . . . . . . . . . . . . . . 82 4.7.2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography 88

List of Figures

3.1 T-shirt design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

List of Tables

1.1 Odd invisible |� _{E}| > 1, all N ≤ 5500 (from Table 1 of [18]) . . . . . . . . . 12 1.2 Shafarevich-Tate groups at prime level. (The entries in the columns “mod

deg” and “#� _{an}” are only really the odd parts of “mod deg” and “#� _{an}”.) 15 1.3 New visible Shafarevich-Tate groups . . . . . . . . . . . . . . . . . . . . . . 16 1.4 Explanatory factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.5 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.6 Component groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Volumes associated to level one cusp forms. . . . . . . . . . . . . . . . . . . 68 3.2 CM elliptic curves of weight > 2. . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Volumes of higher dimensional abelian varieties. . . . . . . . . . . . . . . . . 69

4.1 Component groups at low level . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Big L(A, 1)/Ω_{A} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 Big component groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Component groups of quotients of J_{0}(N) . . . . . . . . . . . . . . . . . . . . 86 4.5 Component groups of quotients of J_{0}(p)^{−} . . . . . . . . . . . . . . . . . . . 87

List of Symbols

Symbol  Definition  Page
A^{∨}  dual to A  33
B_{k}(N, ε)  module of boundary modular symbols  21
c_{A}  Manin constant of A  57
m_{A}  modular degree  50
e_{i}  ith winding element X^{i−1}Y^{k−2−(i−1)}{0, ∞}  52
M_{k}(N, ε)  module of modular symbols  20
\overline{M}_{k}(N, ε)  module of extended modular symbols  59
M[I]  ∩_{a∈I} ker(a)
P (X, Y){α, β}  higher weight modular symbol  20
[P (X, Y), (u, v)]  higher weight Manin symbol  27
S_{k}(N, ε)  module of cuspidal modular symbols  21
T_{n}  nth Hecke operator  29
V_{k}  module of homogeneous polynomials of degree k  20
W_{d}  dth Atkin-Lehner involution  23
α_{t}, β_{t}  degeneracy maps  24
Θ_{f}  rational period mapping  47
σ, τ  σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} )  27
Φ_{f}  analytic period mapping  59
Φ_{A,p}  component group of A at p  71
Ω_{A}  real volume  66
⟨ , ⟩  integration pairing  22
∗  star involution  23

Acknowledgements

It is a pleasure to thank my thesis adviser, Hendrik Lenstra, for patiently guiding my intellectual development in innumerable ways. In addition, Robert Coleman, Barry Mazur, and Ken Ribet generously shared their countless ideas and unbounded enthusiasm with me. Robert helped me to understand p-adics, Barry taught me to visualize Shafarevich-Tate groups, and Ken explained congruences, component groups, and tutored me in the elusive art of mathematical exposition.

This thesis would not have been possible without the questions that arose out of exhilarating collaborations with Amod Agashe, Kevin Buzzard, Loïc Merel, David Moulton, Ed Schaefer, Helena Verrill, and Joe Wetherell. I would like to thank Ahmed Abbes, Matt Baker, Brian Conrad, János Csirik, Jordan Ellenberg, Edray Goins, David Jones, Ezra Miller, Bjorn Poonen, and Mak Trifkovic for asking and answering many stimulating questions.

I wish to thank Hendrik Lenstra and Ken Ribet for providing me with financial support during part of the writing of this thesis, Sarah M. Hallam for providing further support, the Cal@SiliconValley university fellowship for supporting me throughout my final year, and the Magma group for their computational support.

Finally, I would like to thank Bjorn Poonen for thoroughly reading this thesis and pointing out many ways in which it could be improved.

Preface

The object of numerical computation is theoretical advance.

–A. O. L. Atkin, see [5]

The definition of the spaces of modular forms as functions on the upper half plane satisfying a certain equation is very abstract. The definition of the Hecke operators even more so. Nevertheless, one wishes to carry out explicit investigations into these objects.

We are fortunate that we now have methods available that allow us to transform the vector space of cusp forms of given weight and level into a concrete object, which can be explicitly computed. We have the work of Atkin-Lehner, Birch, Swinnerton-Dyer, Manin, Merel, and many others to thank for this (see [6, 16, 45]). For example, the Eichler-Selberg trace formula, as extended in [30], can be used to compute characteristic polynomials of Hecke operators. One can compute Hecke operators using Brandt matrices and quaternion algebras [32, 52]; another closely related method involves the module of enhanced supersingular elliptic curves [47]. In the course of computing large tables of invariants of elliptic curves in [16], Cremona demonstrated the power of systematic computation using modular symbols.

Various methods often must be used in concert to obtain information about the package of invariants attached to a modular form. For example, computing orders of component groups of optimal quotients of J_{0}(N) involves computations on the module of supersingular elliptic curves combined with modular symbols techniques (see Chapter 4).

Chapter 1 is an attempt to systematically prove the Birch and Swinnerton-Dyer conjecture for a certain finite list of rank-0 quotients of J_{0}(N) that have nontrivial Shafarevich-Tate groups. The key idea is to use Barry Mazur’s notion of visibility, coupled with explicit computations, to produce lower bounds on the Shafarevich-Tate group. I have not finished the proof of the conjecture in these examples; this would require computing explicit upper bounds on the order of this group. However, I obtain explicit formulas and data that will be helpful in further investigations.

The following three chapters describe the algorithms used in Chapter 1, along with generalizations to eigenforms on Γ_{1}(N) of integral weight greater than two. I have used these algorithms to investigate the Artin Conjecture [12], Serre’s conjecture, and many other problems not described in this thesis. I have implemented most of the algorithms that are described in Chapters 2–4 in both Magma and C++; this implementation should be available in the standard release of Magma in versions 2.7 and greater.

William A. Stein University of California, Berkeley

Chapter 1

The Birch and Swinnerton-Dyer conjecture

Now that the Shimura-Taniyama conjecture has been proved, many experts consider the Birch and Swinnerton-Dyer conjecture (BSD conjecture) to be one of the main outstanding problems in the field (see [19, pg. 549] and [68, Intro.]). This conjecture ties together many of the arithmetic and analytic invariants of an elliptic curve. At present, there is no general class of elliptic curves for which the full BSD conjecture is known, though a slightly weakened form is known for a fairly broad class of complex multiplication elliptic curves of analytic rank 0 (see [55]), and several deep partial results have been obtained during the last twenty years (see, e.g., [27] and [33]).

Approaches to the BSD conjecture that rely on congruences between modular forms are likely to require a deeper understanding of the analogue of the BSD conjecture for higher-dimensional abelian varieties. As a first step, this chapter presents theorems and explicit computations of some of the arithmetic invariants of modular abelian varieties.

The reader is urged to also read A. Agashe’s 2000 Berkeley Ph.D. thesis which cover similar themes. The paper of Cremona and Mazur’s [18] paints a detailed experimental picture of the way in which congruences link Mordell-Weil and Shafarevich-Tate groups of elliptic curves.

1.1 The BSD conjecture

By [10] we now know that every elliptic curve over Q is a quotient of the curve X_{0}(N), whose complex points are the isomorphism classes of pairs consisting of a (generalized) elliptic curve and a cyclic subgroup of order N. Let J_{0}(N) denote the Jacobian of X_{0}(N); this is an abelian variety of dimension equal to the genus of X_{0}(N) whose points correspond to the degree 0 divisor classes on X_{0}(N). The survey article [21] is a good guide to the facts and literature about the family of abelian varieties J_{0}(N).

Following Mazur [41], we make the following definition.

Definition 1.1 (Optimal quotient). An optimal quotient of J_{0}(N) is a quotient A of J_{0}(N) by an abelian subvariety.

Consider an optimal quotient A such that L(A, 1) ≠ 0. By [35], A(Q) and � (A) are both finite. The BSD conjectureasserts that

\frac{L(A, 1)}{Ω_{A}} = \frac{#� (A) · ∏_{p|N} c_{p}}{#A(Q) · #A^{∨}(Q)}.

Here the Shafarevich-Tate group

� (A) := ker (H^{1}(Q, A) → ∏_{v} H^{1}(Q_{v}, A))

is a measure of the failure of the local-to-global principle; the Tamagawa numbers c_{p} are the orders of the groups of rational points of the component groups of A (see Chapter 4); the real number Ω_{A} is the measure of A(R) with respect to a basis of differentials having everywhere nonzero good reduction (see Section 3.12.6); and A^{∨} is the abelian variety dual to A (see [50, §9]). This chapter makes a small contribution to the long-term goal of verifying the above conjecture for many specific abelian varieties on a case-by-case basis. In a large list of examples, we compute the conjectured order of � (A), up to a power of 2, and then show that � (A) is at least as big as conjectured. We also discuss methods to obtain upper bounds on #� (A), but do not carry out any computations in this direction. This is the first step in a program to verify the above conjecture for an infinite family of quotients of J_{0}(N).

1.1.1 The ratio L(A, 1)/Ω_{A}

Extending classical work on elliptic curves, A. Agashe and the author proved the following theorem.

Theorem 1.2. Let m be the largest square dividing N. The ratio L(A, 1)/Ω_{A} is a rational number that can be explicitly computed, up to a unit (conjecturally 1) in Z[1/(2m)].

Proof. The proof uses modular symbols combined with an extension of the argument used by Mazur in [41] to bound the Manin constant. The modular symbols part of the proof for L-functions attached to newforms of weight k ≥ 2 is given in Section 3.10; it involves expressing the ratio L(A, 1)/Ω_{A} as the lattice index of two modules over the Hecke algebra. The bound on the Manin constant is given in Section 3.11.

The author has computed L(A, 1)/Ω_{A} for all simple optimal quotients of level N ≤ 1500; this table can be obtained from the author’s web page.

Remark 1.3. The method of proof should also give similar results for special values of twists of L(A, s), just as it does in the case dim A = 1 (see [16, Prop. 2.11.2]).

1.1.2 Torsion subgroup

We can compute upper and lower bounds on #A(Q)_{tor}, see Section 3.8; these frequently determine #A(Q)_{tor}.

These methods, combined with the method used to obtain Theorem 1.2, yield the following corollary, which supports the expected cancellation between torsion and c_{p} coming from the reduction map sending rational points to their image in the component group of A. The corollary also generalizes to higher weight forms, thus suggesting a geometric way to think about reducibility of modular Galois representations.

Corollary 1.4. Let n be the order of the image of (0) − (∞) in A(Q), and let m be the largest square dividing N. Then n · L(A, 1)/Ω_{A} ∈ Z[1/(2m)].

For the proof, see Corollary 3.48 in Chapter 3.

1.1.3 Tamagawa numbers

We prove the following theorem in Chapter 4.

Theorem 1.5. When p^{2} ∤ N, the number c_{p} can be explicitly computed (up to a power of 2).

We can compute the order c_{p} of the group of rational points of the component group, but not its structure as a group. When p^{2} | N it may be possible to compute c_{p} using the Drinfeld-Katz-Mazur model of X_{0}(N), but we have not yet done this. There are also good bounds on the primes that can divide c_{p} when p^{2} | N.

Systematic computations (see Section 4.7.1) using this formula suggest the following conjectural refinement of a result of Mazur [40].

Conjecture 1.6. Suppose N is prime and A is an optimal quotient of J_{0}(N) corresponding to a newform f. Then A(Q)_{tor} is generated by the image of (0) − (∞) and c_{p} = #A(Q)_{tor}. Furthermore, the product of the c_{p} over all simple optimal quotients corresponding to newforms equals the numerator of (N − 1)/12.

I have checked this conjecture for all N ≤ 997 and, up to a power of 2, for all N ≤ 2113. The first part is known when A is an elliptic curve (see [48]). Upon hearing of this conjecture, Mazur reportedly proved it when all “q-Eisenstein quotients” are simple. There are three promising approaches to finding a complete proof. One involves the explicit formula of Theorem 1.5; another is based on Ribet’s level lowering theorem (see [53]), and a third makes use of a simplicity result of Merel (see [46]).

The formula that lies behind Theorem 1.5 probably has a natural analogue in weight greater than 2. One could then guess that it produces Tamagawa numbers of motifs attached to eigenforms of higher weight; however, we have no idea if this is really the case. These numbers appear in the conjectures of Bloch and Kato, which generalize the BSD conjecture to motifs (see [7]). Anyone wishing to try to compute them should be aware of Neil Dummigan’s paper [22], which gives some information about the Tamagawa numbers of motifs attached by Scholl in [57] to modular eigenforms.

1.1.4 Upper bounds on #� (A)

V. Kolyvagin (see [34]) and K. Kato (see, e.g., [58]) constructed Euler systems that were used to prove that � (A) is finite when L(A, 1) ≠ 0. To verify the full BSD conjecture for certain abelian varieties, we must make the Kolyvagin-Kato finiteness bound explicit. Kolyvagin’s bounds involve computations with Heegner points, and Kato’s involve a study of the Galois representations associated to A.

Kolyvagin’s bounds

In [33], Kolyvagin obtains explicit upper bounds for #� (A) for a certain (finite) list of elliptic curves A by computing the index in A(K) of the subgroup generated by the Heegner point, where K is a suitable imaginary quadratic extension. In [35], Kolyvagin and Logachev generalize Kolyvagin’s earlier results; in Section 1.6, “Unsolved problems”, they say that: “If one were to compute the height of a Heegner point y [...] considered in the present paper, then one would have succeeded in obtaining an upper bound for #� for this curve.” (By “curve” they mean abelian variety.) This suggests that explicit computations should yield upper bounds on the order of � (A), but that they had not yet figured out how to carry out such computations.

Kato’s bounds

Kato has constructed Euler systems coming from K_{2}-groups of modular curves. These can be used to prove the following theorem (see, e.g., [56, Cor. 3.5.19]).

Theorem 1.7 (Kato). Suppose E is an elliptic curve over Q without complex multiplication that E has conductor N, that E has good reduction at p, that p does not divide 2r_{E} ∏_{q|N} L_{q}(q^{−1})#E(Q_{q})_{tor}, and the Galois representation ρ_{E,p} : G_{Q} → Aut(E[p]) is surjective. Then

#� (E)_{p^{∞}} divides \frac{L(E, 1)}{Ω_{E}}.

Here L_{q}(x) is the local Euler factor at q and the constant r_{E} arises in the construction of Kato’s Euler system. Rubin suggests that computing r_{E} is not very difficult (private communication). Appropriate variants of Kato’s arguments give similar results for quotients of J_{0}(N) of arbitrary dimension, though these have not been written down.

1.1.5 Lower bounds on #� (A)

One approach to showing that � (A) is as at least as large as predicted by the BSD conjecture is suggested by Mazur’s notion of the visible part � (A)^{∘} of � (A) (see [18, 43]). Let A^{∨} ⊂ J_{0}(N) be the dual to A. The visible part of � (A^{∨}) is the kernel of the natural map � (A^{∨}) → � (J_{0}(N)). Mazur observed that if an element of order p in � (A^{∨}) is visible, then it is explained by a “jump in the rank of Mordell-Weil” in the sense that there is another abelian subvariety B ⊂ J_{0}(N) such that p | #(A^{∨} ∩ B) and the rank of B is positive.

Mazur’s observation can be turned around: if there is another abelian variety B of positive rank such that p | #(A^{∨} ∩ B), then, under mild hypotheses (see Theorem 1.8), there is an element of � (A^{∨}) of order p. From a computational point of view it is easy to understand the intersections A^{∨} ∩ B; see Section 3.6. From a theoretical point of view, nontrivial intersections “correspond” to congruences between modular forms. Thus the well-developed theory of congruences between modular forms can be used to obtain a lower bound on #� (A^{∨}).

Invisible elements of #� (A^{∨})

Numerical experiments suggest that as A^{∨} varies, � (A^{∨}) is often not visible inside of J_{0}(N). For example (see Table 1.2), the BSD conjecture predicts the existence of invisible elements of odd order in � (A^{∨}) for almost half of the 37 optimal quotients of prime level ≤ 2113.

Visibility at higher level

For every integer M (Ribet [54] tells us which M to choose), we can ask whether � (A^{∨}) maps to 0 under one of the natural maps A^{∨} → J_{0}(NM); that is, we can ask whether � (A^{∨}) “becomes visible at level NM.” We have been unable to prove in any particular case that � (A^{∨}) is not visible at level N, but becomes visible at some level NM. See Section 1.4.1 for some computations which strongly indicate that such examples exist.

Visibility in some Jacobian

Johan de Jong proved that if E is an elliptic curve over a number field K and c ∈ H^{1}(K, E) then there is a Jacobian J and an imbedding E ↪ J such that c maps to 0 under the natural map H^{1}(K, E) → H^{1}(K, J) (see Remark 3 in [18]); de Jong’s proof appears to generalize when E is replaced by an abelian variety, but time does not permit going into the details here.

1.1.6 Motivation for considering abelian varieties

If A is an elliptic curve, then explaining � (A) using only congruences between elliptic curves will probably fail. This is because pairs of non-isogenous elliptic curves with isomorphic p-torsion for large p are, according to E. Kani’s conjecture, extremely rare. It is crucial to understand what happens in all dimensions.

Within the range accessible by computer, abelian varieties exhibit more richly textured structure than elliptic curves. For example, there is a visible element of prime order 83341 in the Shafarevich-Tate group of an abelian variety of prime conductor 2333; in contrast, over all optimal elliptic curves of conductor up to 5500, it appears that the largest order of an element of a Shafarevich-Tate group is 7 (see the discussion in [18]).

1.2 Existence of nontrivial visible elements of � (A)

The reader who wants to see tables of Shafarevich-Tate groups can safely skip to the next section.

Without relying on any unverified conjectures, we will use the following theorem to exhibit abelian varieties A such that the visible part of � (A) is nonzero. In the following theorem we do not assume that J is the Jacobian of a curve.

1.2. EXISTENCE OF NONTRIVIAL VISIBLE ELEMENTS OF � (A) 7

Theorem 1.8. Let A and B be abelian subvarieties of an abelian variety J such that A∩B is finite and A(Q) is finite. Assume that B has purely toric reduction at each prime at which J has bad reduction. Let p be an odd prime at which J has good reduction, and assume that p does not divide the orders of any of the (geometric) component groups of A and B, or the orders of the torsion subgroups of (J/B)(Q) and B(Q). Suppose further that B[p] ⊂ A ∩ B. Then there exists an injection

B(Q)/pB(Q) ↪ � (A)^{∘}

of B(Q)/pB(Q) into the visible part of � (A).

Proof. Let C = J/A. The long exact sequence of Galois cohomology associated to the short exact sequence

0 → A → J → C → 0

begins

0 → A(Q) → J(Q) → C(Q) ⟶^{δ} H^{1}(Q, A) → · · · .

Because B[p] ⊂ A, the map B → C, obtained by composing the inclusion B ↪ J with J → C, factors through multiplication-by-p, giving the following commutative diagram:

\xymatrix{ & B \ar[d] \ar[r]^{p} & B \ar[d] \\ A \ar[r] & J \ar[r] & C.}

Because B(Q)[p] = 0 and B(Q) ∩ A(Q) = 0, we deduce the following commutative diagram with exact rows:

\xymatrix{ & 0 \ar[d] & K_{1} \ar[d] & K_{2} \ar[d] \\ 0 \ar[r] & B(Q) \ar[r]^{p} \ar[d] & B(Q) \ar[dr]^{π} \ar[r] \ar[d] & B(Q)/pB(Q) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & J(Q)/A(Q) \ar[r] \ar[d] & C(Q) \ar[r] & δ(C(Q)) \ar[r] & 0 \\  & K_{3},}

where K_{1} and K_{2} are the indicated kernels and K_{3} is the cokernel. We have the snake lemma exact sequence

0 → K_{1} → K_{2} → K_{3}.

Because B(Q)[p] = 0 and K_{2} is a p-torsion group, we have K_{1} = 0. The quotient J(Q)/B(Q) has no p-torsion because it is a subgroup of (J/B)(Q); also, A(Q) is a finite group of order coprime to p, so K_{3} = J(Q)/(A(Q) + B(Q)) has no p-torsion. Thus K_{2} = 0.

The above argument shows that B(Q)/pB(Q) is a subgroup of H^{1}(Q, A). However, H^{1}(Q, A) contains infinitely many elements of order p (see [59]), whereas � (A)[p] is a finite group, so we must work harder to deduce that B(Q)/pB(Q) lies in � (A)[p]. Fix x ∈ B(Q). We must show that π(x) lies in � (A)[p]; equivalently, that res_{v}(π(x)) = 0 for all places v of Q.

At the archimedean place v = ∞, the restriction res_{v}(π(x)) is killed by 2 and the odd prime p, hence res_{v}(π(x)) = 0.

Suppose that v is a place at which J has bad reduction. By hypothesis, B has purely toric reduction, so over the maximal unramified extension Q_{v}^{ur} of Q_{v} there is an isomorphism B ≅ G_{m}^{d}/Γ of Gal(\overline{Q}_{v}/Q_{v}^{ur})-modules, for some “lattice” Γ. For example, when dim B = 1, this is the Tate curve representation of B. Let n be the order of the component group of B at v; thus n equals the order of the cokernel of the valuation map Γ → Z^{d}. Choose a representative P = (x_{1}, …, x_{d}) ∈ G_{m}^{d} for the point x. Then nP can be adjusted by elements of Γ so that each of its components x_{i} ∈ G_{m} has valuation 0. By assumption, p is a prime at which J has good reduction, so p ≠ v; it follows that there is a point Q ∈ G_{m}^{d}(Q_{v}^{ur}) such that pQ = nP. Thus the cohomology class res_{v}(π(nx)) is unramified at v. By [51, Prop. I.3.8],

H^{1}(Q_{v}^{ur}/Q_{v}, A(Q_{v}^{ur})) = H^{1}(Q_{v}^{ur}/Q_{v}, Φ_{A,v}(\overline{F}_{v})),

where Φ_{A,v} is the component group of A at v. Since the component group Φ_{A,v}(\overline{F}_{v}) has order n, it follows that

res_{v}(π(nx)) = n res_{v}(π(x)) = 0.

Since the order p of res_{v}(π(x)) is coprime to n, we conclude that res_{v}(π(x)) = 0.

Next suppose that J has good reduction at v and that v is odd, in the sense that the residue characteristic of v is odd. To simplify notation in this paragraph, since v is a nonarchimedean place of Q, we will also let v denote the odd prime number which is the residue characteristic of v. Let A, J, C, be the Néron models of A, J, and C, respectively (for more on Néron models, see Chapter 4). Let A, J, C, also denote the sheaves on the étale-site over Spec(Z_{v}) determined by the group schemes A, J, and C, respectively. Since v is odd, 1 = e < v − 1, so we may apply [8, Thm. 7.5.4] to conclude that the sequence of group schemes

0 → A → J → C → 0

is exact; in particular, it is exact as a sequence of sheaves on the étale site (see the proof of [8, Thm. 7.5.4]). Thus it is exact on the stalks, so by [49, 2.9(d)] the sequence

0 → A(Z_{v}^{ur}) → J (Z_{v}^{ur}) → C(Z_{v}^{ur}) → 0

is exact; by the Néron mapping property the sequence

0 → A(Q_{v}^{ur}) → J(Q_{v}^{ur}) → C(Q_{v}^{ur}) → 0

is also exact. Thus res_{v}(π(x)) in unramified, so it arises by inflation from an element of H^{1}(Q_{v}^{ur}/Q_{v}, A). By [51, Prop. I.3.8],

H^{1}(Q_{v}^{ur}/Q_{v}, A) ≅ H^{1}(Q_{v}^{ur}/Q_{v}, Φ_{A,v}),

where Φ_{A,v} is the component group of A at v. Since A has good reduction, Φ_{A,v} = 0, hence res_{v}(π(x)) = 0.

If J has bad reduction at v = 2, then we already dealt with 2 above. Consider the case when J has good reduction at 2. Because the absolute ramification index e of Z_{2} is 1, which is not less than v − 1 = 1, we can not apply [8, Thm. 7.5.4]. However, we can modify our situation by an isogeny of degree a power of 2, then apply a different theorem as follows. The 2-primary subgroup Ψ of A ∩ B is rational as a subgroup over Q, in the sense that the conjugates of any point in Ψ are also contained in Ψ. The abelian varieties J̃ = J/Ψ, Ã = A/Ψ, and B̃ = B/Ψ also satisfy the hypothesis of the theorem we are proving. By [8, Prop. 7.5.3(a)], the corresponding sequence of Néron models

0 → Ã → J̃ → C̃ → 0

is exact, so the sequence

0 → Ã(Q_{v}^{ur}) → J̃(Q_{v}^{ur}) → C̃(Q_{v}^{ur}) → 0

is exact. Thus the image of res_{v}(π(x)) in H^{1}(Q_{v}, Ã) is unramified. It equals 0, again by [51, Prop. 3.8], since the component group of Ã at v has order a power of 2 (in fact it is trivial, since Ã has good reduction at 2), whereas π(x) has odd prime order p. Thus res_{v}(π(x)) = 0, since the kernel of H^{1}(Q_{v}, A) → H^{1}(Q_{v}, Ã) is a finite group of 2-power order.

1.3 Description of tables

In this section we describe our tables of optimal quotients of J_{0}(N), which have nontrivial Shafarevich-Tate group. The tables, which can be found on pages 15–18, were computed using a combination of Hecke [64], LiDIA, NTL, Pari, and most successfully Magma [9]. The component group computations at non-prime level rely on Kohel’s quaternion algebra package, which was also written in Magma.

We compute the conjectural order of the Shafarevich-Tate group of an abelian variety A, and then make assertions about the Shafarevich-Tate group of A^{∨}. This is justified because the order of � (A^{∨}) equal the order of � (A), since both are finite and the Cassells-Tate pairing sets up a nondegenerate duality between them.

1.3.1 Notation

Each optimal quotient A of J_{0}(N) is denoted by a label, such as 389E, which consists of a level N and a letter indicating the isogeny class. In the labeling, N is a positive integer and the isogeny class is given by a letter: the first isogeny class is labeled A, the second is labeled B, the third labeled C, and so on. Thus 389E is the fifth isogeny class of optimal quotient of J_{0}(389), corresponding to a Galois-conjugacy class of newforms. The isogeny classes that we consider are in bijection with the Galois-conjugacy classes of newforms in S_{2}(Γ_{0}(N)). The classes of newforms are ordered as described in Section 3.5.5.

WARNING: The odd part of a rational number x is x/2^{v}, where v = ord_{2}(x). In the tables, only the odd parts of the arithmetic invariants of A are given.

1.3.2 Table 1.2: Shafarevich-Tate groups at prime level

Table 1.2 was constructed as follows. Using the results of Section 3.10, we computed the odd part of the conjectural order #� _{an}(A) of the Shafarevich-Tate group of every optimal quotient of J_{0}(p) that corresponds to a single Galois conjugacy-class of eigenforms and has analytic rank 0, for p a prime with p ≤ 2161. We also computed a few sporadic examples of prime level p with p > 2161. The sporadic examples appear at the bottom of the table below a horizontal line.

Notation

The columns of the table contain the following information. The abelian varieties A for which #� _{an}(A) is greater than 1 are laid out in the first column of Table 1.2. The second column contains the dimensions of the abelian varieties in the first column. The third column contains the odd part (i.e., largest odd divisor) of the order of the Shafarevich-Tate group, as predicted by the BSD conjecture. Column four contains the odd parts of the modular degrees of the abelian varieties in the first column.

The fifth column contains an optimal quotient B of J_{0}(p) of positive analytic rank, such that the ℓ-torsion of B^{∨} is contained in A^{∨}, when one exists, where ℓ is a divisor of #� _{an}(A). We computed this intersection using the algorithm described in Section 3.6. Such a B is called an explanatory factor. When no such abelian varieties exists, we write “NONE” in the fifth column. The sixth column contains the dimensions of the abelian varieties in the fifth column, and the seventh column contains the odd parts of the modular degrees of the abelian varieties in the fifth column.

Ranks of the explanatory factors

That the explanatory factors have positive analytic rank follows from our modular symbols computation of L(B, 1)/Ω_{B}. This is supported by the table in [11], except in the case 2333A, where there is a mistake in [11] (see below).

The explanatory factor 389A is the first elliptic curve of rank 2. The table in [11] suggests that the explanatory factor 1061B is the first 2-dimensional abelian variety (attached to a newform) whose Mordell-Weil group when tensored with the field of fractions F of the corresponding ring of Fourier coefficients, is of dimension 2 over F. Similarly 1567B appears to be the first 3-dimensional one of rank 2, and 2333A is the first 4-dimensional one of rank 2. It thus seems very likely that the ranks of each explanatory factor is exactly 2, though we have not proved this.

Discussion of the data

There are 23 examples in which � (A) is visible and 18 in which � (A) is invisible. The largest visible � (A) found occurs at level 2333 and has order at least 83341^{2} (83341 is prime). The largest invisible � (A) occurs in a 112-dimensional abelian variety at level 2111 and has order at least 211^{2}.

The example 1283C demonstrates that #� _{an}(A) can divide the modular degree, yet be invisible. In this case 5 divides #� _{an}(A). Since 5 divides the modular degree, it follows that there must be other non-isogenous subvarieties of J_{0}(1283) that intersect 1283C in a subgroup of order divisible by 5. In this case, the only such subvariety is 1283A, which has dimension 2 and whose 5-torsion is contained in 1283C. However 1283A has analytic (hence algebraic) rank 0, so #� _{an}(A) cannot be visible.

The cases 1483D, 1567D, 2029C, and 2593B are interesting because all of � , even though it has two nontrivial p-primary components in each of these cases, is made visible in a single B. In the case 1913E only the 5-primary component of � is visible in 1913A, but still both the 5-primary and 61-primary components of � are visible in 1913C.

Examples 1091C and 1429B were first found in [1] and 1913B in [18].

Errata to Brumer’s paper

Contrary to our computations, [11] suggests that 2333A has rank 0. However, the author pointed the discrepancy out to Brumer who replied:

I looked in vain for information about θ-relations on 2333 and have concluded that I never ran the interval from 2300 to 2500 or else had lost all info by the time I wrote up the paper. The punchline: 4 relations for 2333 and 2 relations for 2381 (also missing from the table).

1.3.3 Tables 1.3–1.6: New visible Shafarevich-Tate groups

Let n denote the largest odd square dividing the numerator of L(A, 1)/Ω_{A}. Table 1.3 lists those A such that for some p | n there exists a quotient B of J_{0}(N), corresponding to a newform and having positive analytic rank, such that (A^{∨} ∩ B^{∨})[p] ≠ 0. Our search was systematic up to level 1001, but there might be omitted examples between levels 1001 and 1028. Table 1.4 contains further arithmetic information about each explanatory factor. Table 1.6 gives the quantities involved in the formula of Chapter 4 for Tamagawa numbers, for each of the abelian varieties A in Table 1.3.

Notation

Most of the notation is the same as in Table 1.2. However the additional columns w_{q} and c_{p} contain the signs of the Atkin-Lehner involutions and the Tamagawa numbers, respectively. These are given in order, from smallest to largest prime divisor of N.

In each case B has dimension 1. When 4 | N, we write “a” for c_{2} to remind us that we did not compute c_{2} because the reduction at 2 is additive. Again only odd parts of the invariants are given. Section 4.7.2 contains a discussion of the notation used in the headings of Table 1.6.

Remarks on the data

The explanatory factors B of level ≤ 1028 are exactly the set of rank 2 elliptic curves of level ≤ 1028.

Table 1.1: Odd invisible |� _{E}| > 1, all N ≤ 5500 (from Table 1 of [18])

E  \sqrt{|� _{E}|}  m_{E}  F  m_{F} Remarks
2849A  3  2^{5} · 5 · 61  NONE −
3364C  7  2^{6} · 3^{2} · 5^{2} · 7 none  −
4229A  3  2^{3} · 3 · 7 · 13 none  −
4343B  3  2^{4} · 1583  NONE −
4914N  3  2^{4} · 3^{5}  none  − E has rational 3-torsion
5054C  3  2^{3} · 3^{3} · 11 none  −
5073D  3  2^{5} · 3 · 5 · 7 · 23 none  −
5389A  3  2^{2} · 2333  NONE −

1.4 Further visibility computations

1.4.1 Does � become visible at higher level?

This section is concerned with whether or not the examples of invisible elements of Shafarevich-Tate groups of elliptic curves of level N that are given in [18] become visible in abelian surfaces inside appropriate J_{0}(Np). We analyze each of the cases in Table 1 of [18]. For the reader’s convenience, the part of this table which concerns us is reproduced as Table 1.1. The most interesting examples we give are 2849A and 5389A. As in [18], the assertions below assume the truth of the BSD conjecture.

How we found the explanatory curves

We use a naive heuristic observation to find possible explanatory curves of higher level, even though their conductors are out of the range of Cremona’s tables. Note that we have not proved that these factors are actually explanatory in any cases, and expect that in some cases they are not.

First we recall some of the notation from Table 1 of [18], which is partially reproduced below. The “NONE” label in the row corresponding to an elliptic curve E indicates that there are elements in � (E) whose order does not divide the modular degree of E, and hence they must be invisible. The label “none” indicates only that no suitable explanatory elliptic curves could be found, so � (E) is not visible in an abelian surface inside J_{0}(N); it could, however, be visible in the full abelian variety J_{0}(N).

Studying the Weierstrass equations corresponding to the curves in [18] reveals that the elliptic curves labeled “NONE” have unusually large height, as compared to their conductors. However, the explanatory factors often have unusually small height. Motivated by this purely heuristic observation, we make a table of all elliptic curves of the form

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6},

with a_{1}, a_{2}, a_{3} ∈ {−1, 0, 1}, |a_{4}|, |a_{6}| < 1000, and conductor bounded by 50000. The bound on the conductor is required only so that the table will fit within computer storage. This table took a few days to generate.

2849A

Barry Mazur and Adam Logan found the first known example of an invisible Shafarevich-Tate group. This was � (E), where E is the elliptic curve 2849A, which has minimal Weierstrass equation

E : y^{2} + xy + y = x^{3} + x^{2} − 53484x − 4843180.

Consulting our table of curves of small height, we find an elliptic curve F of conductor 8547 = 2849 · 3 such that f_{E} ≡ f_{F} (mod 3), where f_{E} and f_{F} are the newforms attached to E and F, respectively. This is a congruence for all eigenvalues a_{p} attached to E and F. The elliptic curve F is defined by the equation

F : y^{2} + xy + y = x^{3} + x^{2} − 154x − 478.

Cremona’s program mwrank reveals that the Mordell-Weil group of F has rank 2. Thus maybe � (E) becomes visible at level 8547. Unfortunately, visibility of � (E) is not implied by Theorem 1.8 because the geometric component group of F at 3 has order divisible by 3.

4343B

Consider the elliptic curve E labeled 4343B. According to Table 1 of [18], � (E) has order 9, but the modular degree prevents � (E) from being visible in J_{0}(4343). At level 21715 = 5 · 4343 there is an elliptic curve F of rank 1 that is congruent to E. Its equation is

F : y^{2} − xy − y = x^{3} − x^{2} + 78x − 256.

5389A

The last curve labeled “NONE” in the table is curve 5389A, which has minimal Weierstrass equation

y^{2} + xy + y = x^{3} − 35590x − 2587197.

The main theorem of [54] implies that there exists a newform that is congruent modulo 3 to the newform corresponding to 5389A and of level 3 · 5389. This is because (−2)^{2} = (3 + 1)^{2} (mod 3). However, our table of curves of small height does not contain any curve of conductor 3 · 5389. Next we observe that (−2)^{2} ≡ (7 + 1)^{2} (mod 3), so using Ribet’s theorem we can instead augment the level by 7. Our table of small-height curves contains just one (up to isogeny) elliptic curve of conductor 37723, and luckily the corresponding newform is congruent modulo 3 to the newform corresponding to 5389A (away from primes dividing the level)! The Weierstrass equation of this curve is

F : y^{2} − y = x^{3} + x^{2} + 34x − 248.

According to Cremona’s program mwrank, the rank of F is 2.

3364C, 4229A, 5073D

Perhaps � (E) is already visible in some of the cases in which the curve is labeled “none”, because the method fails in most of these cases. Each of the curves 3364C, 4229A, and 5073D is labeled “none”. In none of these 3 cases are we able to find an explanatory factor at higher level, within the range of our table of elliptic curves of small height.

4194N, 5054C

The curve 4914N is labeled “none” and we find the remark “E has rational 3-torsion”. There is a congruent curve F of conductor 24570 given by the equation

F : y^{2} − xy = x^{3} − x^{2} − 15x − 75,

and F (Q) = {0}. The curve 5054C is labeled “none” and its Shafarevich-Tate group contains invisible elements of order 3. We find a congruent curve of level 25270 and rank 1. The equation of the congruent curve is

F : y^{2} − xy = x^{3} + x^{2} − 178x + 882.

1.4.2 Positive rank example

The abelian varieties with nontrivial � (A) that one finds in both ours and Cremona’s tables all have rank 0. In this section we outline a computation which sugggests, but does not prove, that there is a positive-rank abelian subvariety A of J_{0}(5077) such that � (A) possesses a nontrivial visible element of order 31.

According to [16], the smallest conductor elliptic curve E of rank 3 is found in J = J_{0}(5077). The number 5077 is prime, and the isogeny decomposition of J is^{1}

J ∼ A × B × E,

where each of A, B, and E are abelian subvarieties of J associated to newforms, which have dimensions 205, 216, and 1, respectively. Using Remark 3.38 or [69], we find that the modular degree of E is 1984 = 2^{6} · 31. The sign of the Atkin-Lehner involution on E is the same as its sign on A, so E[31] ⊂ A. We have E(Q) ≅ Z × Z × Z, and the component group of E is trivial. The numerator of (5077 − 1)/12 is 3^{2} · 47, so [40] implies that none of the abelian varieties above have 31-torsion. It might be possible to find an analogue of Theorem 1.8 that applies when A has positive rank, and deduce in this case that � (A) contains visible elements of order 31.

1This decomposition was found in about one minute using the Mestre-Oesterlé method of graphs (see [47]).

1.4. FURTHER VISIBILITY COMPUTATIONS 15 Table 1.2: Shafarevich-Tate groups at prime level. (The entries in the columns “mod deg” and “#� _{an}” are only really the odd parts of “mod deg” and “#� _{an}”.)

A  dim #� _{an}(A)  mod deg(A)  B  dim mod deg(B)
389E 20  5^{2}  5  389A  1  5
433D 16  7^{2}  3 · 7 · 37  433A  1  7
563E 31  13^{2}  13  563A  1  13
571D  2  3^{2}  3^{2} · 127  571B  1  3
709C 30  11^{2}  11  709A  1  11
997H 42  3^{4}  3^{2}  997B  1  3
1061D 46  151^{2}  61 · 151 · 179  1061B  2  151
1091C 62  7^{2}  1  NONE
1171D 53  11^{2}  3^{4} · 11  1171A  1  11
1283C 62  5^{2}  5 · 41 · 59  NONE
1429B 64  5^{2}  1  NONE
1481C 71  13^{2}  5^{2} · 2833  NONE
1483D 67  3^{2} · 5^{2}  3 · 5  1483A  1  3 · 5
1531D 73  3^{2}  3  1531A  1  3
1559B 90  11^{2}  1  NONE
1567D 69 7^{2} · 41^{2}  7 · 41  1567B  3  7 · 41
1613D 75  5^{2}  5 · 19  1613A  1  5
1621C 70  17^{2}  17  1621A  1  17
1627C 73  3^{4}  3^{2}  1627A  1  3^{2}
1693C 72  1301^{2}  1301  1693A  3  1301
1811D 98  31^{2}  1  NONE
1847B 98  3^{6}  1  NONE
1871C 98  19^{2}  14699  NONE
1877B 86  7^{2}  1  NONE
1907D 90  7^{2}  3 · 5 · 7 · 11  1907A  1  7
1913B 1  3^{2}  3 · 103  1913A  1  3 · 5^{2}
1913E 84 5^{4} · 61^{2}  5^{2} · 61 · 103  1913A,C 1, 2 3 · 5^{2}, 5^{2} · 61
1933C 83  3^{2} · 7^{2}  3 · 7  1933A  1  3 · 7
1997C 93  17^{2}  1  NONE
2027C 94  29^{2}  29  2027A  1  29
2029C 90 5^{2} · 269^{2}  5 · 269  2029A  2  5 · 269
2039F 99  3^{4} · 5^{2} 19 · 29 · 7759 · 3214201 NONE
2063C 106  13^{2}  61 · 139  NONE
2089J 91  11^{2}  3 · 5 · 11 · 19 · 73 · 139 2089B  1  11
2099B 106  3^{2}  1  NONE
2111B 112  211^{2}  1  NONE
2113B 91  7^{2}  1  NONE
2161C 98  23^{2}  1  NONE
2333C 101 83341^{2}  83341  2333A  4  83341
2339C 114  3^{8}  6791  NONE
2411B 123  11^{2}  1  NONE
2593B 109 67^{2} · 2213^{2}  67 · 2213  2593A  4  67 · 2213

16 CHAPTER 1. THE BIRCH AND SWINNERTON-DYER CONJECTURE

Table 1.3: New visible Shafarevich-Tate groups

A  dim #� _{an} w_{q}  c_{p} #A(Q) \frac{#A(Q)·L(A,1)}{Ω_{A}}  mod deg(A)  B
389E 20  5^{2}  −  97  97  5^{2}  5  389A
433D 16  7^{2}  −  3^{2}  3^{2}  7^{2}  3 · 7 · 37  433A
446F  8  11^{2}  +− 1, 3  3  11^{2}  11 · 359353  446B
563E 31 13^{2}  −  281  281  13^{2}  13  563A
571D 2  3^{2}  −  1  1  3^{2}  3^{2} · 127  571B
655D 13  3^{4}  +− 1, 1  1  3^{4}  3^{2} · 19 · 515741 655A
664F  8  5^{2}  −+ a, 1  1  5^{2}  5  664A
681B  1  3^{2}  +− 1, 1  1  3^{2}  3 · 5^{3}  681C
707G 15 13^{2}  +− 1, 1  1  13^{2}  13 · 800077  707A
709C 30 11^{2}  −  59  59  11^{2}  11  709A
718F  7  7^{2}  +− 1, 1  1  7^{2}  7 · 151 · 35573  718B
794G 14 11^{2}  +− 3, 1  3  11^{2}  3 · 7 · 11 · 47 · 35447 794A
817E 15  7^{2}  +− 1, 5  5  7^{2}  7 · 79  817A
916G 9  11^{2}  −+ a, 1  1  11^{2}  3^{9} · 11 · 17 · 239 916C
944O 6  7^{2}  +− a, 1  1  7^{2}  7  944E
997H 42  3^{4}  −  83  83  3^{4}  3^{2}  997BC
1001L 7  7^{2} + − + 1, 1, 1  1  7^{2}  7 · 19 · 47 · 2273 1001C
1028E 14 11^{2}  −+ a, 1  3  3^{4} · 11^{2}  3^{13} · 11  1028A

1.4. FURTHER VISIBILITY COMPUTATIONS 17

Table 1.4: Explanatory factors

B  rank w_{q}  c_{p} #A(Q) mod deg(A)  Comments
389A  2  −  1  1  5  first curve of rank 2
433A  2  −  1  1  7
446B  2  +− 1, 1  1  11  called 446D in [16]
563A  2  −  1  1  13
571B  2  −  1  1  3
655A  2  +− 1, 1  1  3^{2}
664A  2  −+ 1, 1  1  5
681C  2  +− 1, 1  1  3
707A  2  +− 1, 1  1  13
709A  2  −  1  1  11
718B  2  +− 1, 1  1  7
794A  2  +− 1, 1  1  11
817A  2  +− 1, 1  1  7
916C  2  −+ 3, 1  1  3 · 11
944E  2  +− 1, 1  1  7
997B  2  −  1  1  3
997C  2  −  1  1  3
1001C 2 + − + 1, 3, 1  1  3^{2} · 7
1028A 2  −+ 3, 1  1  3 · 11  intersects 1028E mod 11

Table 1.5: Factorizations

\begin{matrix}446 = 2 · 223 & 655 = 5 · 131 & 664 = 2^{3} · 83 & 681 = 3 · 227 \\ 707 = 7 · 101 & 718 = 2 · 359 & 794 = 2 · 397 & 817 = 19 · 43 \\ 916 = 2^{2} · 229 & 944 = 2^{4} · 59 & 1001 = 7 · 11 · 13 & 1028 = 2^{2} · 257 \end{matrix}

18 CHAPTER 1. THE BIRCH AND SWINNERTON-DYER CONJECTURE

Table 1.6: Component groups

A  dim p w_{q} #Φ_{X,p}  m_{X,p}  #Φ_{A,p}(\overline{F}_{p})
389E 20 389 −  97  5 · 97  97
433D 16 433 −  3^{2}  3^{3} · 7 · 37  3^{2}
446F  8 223 −  3  3 · 11 · 359353  3
2 +  3  3 · 11  3 · 359353
563E 31 563 − 281  13 · 281  281
571D 2 571 −  1  3^{2} · 127  1
655D 13 131 −  1  3^{2} · 19 · 515741  1
5 +  1  3^{2}  19 · 515741
664F  8 83 +  1  5  1
681B  1 227 −  1  3 · 5^{3}  1
3 +  1  3 · 5^{2}  5
707G 15 101 −  1  13 · 800077  1
7 +  1  13  800077
709C 30 709 −  59  11 · 59  59
718F  7 359 −  1  7 · 151 · 35573  1
2 +  1  7  151 · 35573
794G 14 397 −  3  3^{2} · 7 · 11 · 47 · 35447  3
2 +  3  3 · 11  3^{2} · 7 · 47 · 35447
817E 15 43 −  5  5 · 7 · 79  5
19 +  1  7  79
916G 9 229 +  1  3^{9} · 11 · 17 · 239  1
944O 6 59 −  1  7  1
997H 42 997 −  83  3^{2} · 83  83
1001L 7 13 +  1  7 · 19 · 47 · 2273  1
11 −  1  7 · 19 · 47 · 2273  1
7 +  1  7 · 19 · 47  2273
1028E 14 257 +  1  3^{13} · 11  1

Chapter 2

Modular symbols

Modular symbols permeate this thesis. In their simplest incarnation, modular symbols provide a finite presentation for the homology group H_{1}(X_{0}(N), Z) of the Riemann surface X_{0}(N). This presentation is equipped with such a rich structure that from it we can deduce the action of the Hecke operators; this is already sufficient information for us to compute a basis for the space S_{2}(Γ_{0}(N), C) of cusp forms.

We recall the definition of spaces of modular symbols in Sections 2.1–2.2. Then in Section 2.3, we review the duality between modular symbols and modular forms. In Section 2.4, we see that modular symbols are furnished with analogues of each of the standard operators that one finds on spaces of modular forms, and in Section 2.5 we see that the same is true of the degeneracy maps. Section 2.6 describes Manin symbols, which supply a convenient finite presentation for the space of modular symbols. Finally, Section 2.7 introduces the complex torus attached to a newform, which appears in various guises throughout this thesis.

Before continuing, we offer an apology. We will only consider modular symbols that are already equipped with a fixed Dirichlet character. Though fixing a character complicates the formulas, the resulting increase in efficiency is of extreme value in computational applications. Fixing a character allows us to compute in just the part of the space of modular symbols for Γ_{1}(N) that interests us. We apologize for any inconvenience this may cause the less efficiency minded reader.

Acknowledgment. This chapter and the next were greatly influenced by the publications of Cremona [15, 16] and Merel [45], along with the foundational contributions of Manin [38], Mazur [42, 39], and Shokurov [63]. Cremona’s book [16] provides a motivated roadmap that guides the reader who wishes to compute with modular symbols in the familiar context of elliptic curves, and Merel’s article provides an accessible overview of the action of Hecke operators on higher weight modular symbols, and the connection between modular symbols and related cohomology theories.

2.1 The definition of modular symbols

Fix a positive integer N, an integer k ≥ 2, and a continuous homomorphism

ε : (Z/NZ)^{∗} → C^{∗}

such that ε(−1) = (−1)^{k}. We call N the level, k the weight, and ε the Dirichlet character.

Consider the quotient of the abelian group generated by all formal symbols {α, β}, with α, β ∈ P^{1}(Q) = Q ∪ {∞}, by the following relations:

{α, β} + {β, γ} + {γ, α} = 0,

for all α, β, γ ∈ P^{1}(Q). Let M be the torsion-free quotient of this group by its torsion subgroup. Because M is torsion free, {α, α} = 0 and {α, β} = −{β, α}.

Remark 2.1. One is motivated to consider these relations by viewing {α, β} as the homology class of an appropriate path from α to β in the upper half plane.

Let V_{k−2} be the Z-submodule of Z[X, Y] made up of all homogeneous polynomials of degree k − 2, and set M_{k} := V_{k−2} ⊗ M. For g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(Q) and P ∈ V_{k−2}, let

gP (X, Y) = P (det(g)g^{−1} (\begin{matrix}X \\ Y \end{matrix})) = P ((\begin{matrix}d & −b \\ −c & a \end{matrix}) (\begin{matrix}X \\ Y \end{matrix}))
= P (dX − bY, −cX + aY).

This defines a left action of GL_{2}(Q) on V_{k−2}; it is a left action because

(gh)P (v) = P (det(gh)(gh)^{−1}v) = P (det(h)h^{−1} det(g)g^{−1}v)
= gP (det(h)h^{−1}v) = g(hP (v)).

Combining this action with the action of GL_{2}(Q) on P^{1}(Q) by linear fractional transformations gives a left action of GL_{2}(Q) on M_{k}:

g(P ⊗ {α, β}) = g(P) ⊗ {g(α), g(β)}.

Finally, for g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N), let ε(g) := ε(\overline{a}), where \overline{a} ∈ Z/NZ is the reduction modulo N of a.

Let

Z[ε] := Z[ε(a) : a ∈ Z/NZ]

be the subring of C generated by the values of the character ε.

Definition 2.2 (Modular symbols). The space of modular symbols M_{k}(N, ε) of level N, weight k and character ε is the largest torsion-free quotient of M_{k} ⊗ Z[ε] by the Z[ε]- submodule generated by the relations gx − ε(g)x for all x ∈ M_{k} and all g ∈ Γ_{0}(N).

Denote by P {α, β} the image of P ⊗ {α, β} in M_{k}(N, ε). For any Z[ε]-algebra R, let

M_{k}(N, ε; R) := M_{k}(N, ε) ⊗_{Z[ε]} R.

See Section 3.1 for an algorithm which can be used to compute M_{k}(N, ε; Q(ε)).

2.2 Cuspidal modular symbols

Let B be the free abelian group generated by the symbols {α} for all α ∈ P^{1}(Q). There is a left action of GL_{2}(Q) on B given by g{α} = {g(α)}. Let B_{k} := V_{k−2} ⊗ B, and let GL_{2}(Q) act on B_{k} by g(P {α}) = (gP){g(α)}.

Definition 2.3 (Boundary modular symbols). The space B_{k}(N, ε) of boundary modular symbols is the largest torsion-free quotient of B_{k} ⊗ Z[ε] by the relations gx = ε(g)x for all g ∈ Γ_{0}(N) and x ∈ B_{k}.

Denote by P {α} the image of P ⊗ {α} in B_{k}(N, ε). The boundary map

δ : M_{k}(N, ε) → B_{k}(N, ε)

is defined by

δ(P {α, β}) = P {β} − P {α}.

Definition 2.4 (Cuspidal modular symbols). The space S_{k}(N, ε) of cuspidal modular symbols is the kernel of δ.

The three spaces defined above fit together in the following exact sequence:

0 → S_{k}(N, ε) → M_{k}(N, ε) ⟶^{δ} B_{k}(N, ε).

2.3 Duality between modular symbols and modular forms

For any positive integer k, any C-valued function f on the complex upper half plane

𝔥 := {z ∈ C : im(z) > 0},

and any matrix γ ∈ GL_{2}(Q), define a function f|[γ]_{k} on 𝔥 by

(f|[γ]_{k})(z) = det(γ)^{k−1} \frac{f(γz)}{(cz + d)^{k}} .

Definition 2.5 (Cusp forms). Let S_{k}(N, ε) be the complex vector space of holomorphic functions f(z) on 𝔥 that satisfy the equation

f|[γ]_{k} = ε(γ)f

for all γ ∈ Γ_{0}(N), and such that f is holomorphic and vanishes at all cusps, in the sense of [21, pg. 42].

Definition 2.6 (Antiholomorphic cusp forms). Let \overline{S}_{k}(N, ε) be the space of antiholomorphic cusp forms; the definition is as above, except

\frac{f(γz)}{(c\overline{z} + d)^{k}} = \overline{ε}(γ)f(z)

for all γ ∈ Γ_{0}(N).

There is a canonical isomorphism of real vector spaces between S_{k}(N, ε) and \overline{S}_{k}(N, ε) that associates to f the antiholomorphic cusp form defined by the function z ↦ \overline{f(z)}.

Theorem 2.7 (Merel). There is a pairing

⟨ , ⟩ : (S_{k}(N, ε) ⊕ \overline{S}_{k}(N, ε)) × M_{k}(N, ε; C) → C

given by

⟨f ⊕ g, P \{α, β\}⟩ = ∫_{α}^{β} f(z)P (z, 1)dz + ∫_{α}^{β} g(z)P (\overline{z}, 1)d\overline{z},

where the path from α to β is, except for the endpoints, contained in 𝔥. The pairing is perfect when restricted to S_{k}(N, ε; C).

Proof. Take the ε part of each side of [45, Thm. 3].

2.4 Linear operators

2.4.1 Hecke operators

For each positive integer n and each space V of modular symbols or modular forms, there is a Hecke operator T_{n}, which acts as a linear endomorphism of V. For the definition of T_{n} on modular symbols, see [45, §2]. Alternatively, because we consider only modular symbols with character, the following recipe completely determines the Hecke operators. First, when n = p is prime, we have

T_{p}(x) = [(\begin{matrix}p & 0 \\ 0 & 1 \end{matrix}) + ∑_{r mod p} (\begin{matrix}1 & r \\ 0 & p \end{matrix})] x,

where the first matrix is omitted if p | N. If m and n are coprime, then T_{mn} = T_{m}T_{n}. Finally, if p is a prime, r ≥ 2 is an integer, ε is the Dirichlet character of associated to V, and k is the weight of V, then

T_{p^{r}} = T_{p}T_{p^{r−1}} − ε(p)p^{k−1}T_{p^{r−2}}.

Definition 2.8. The Hecke algebra associated to V is the subring

T = T_{V} = Z[… T_{n} …]

of End(V) generated by all Hecke operators T_{n}, with n = 1, 2, 3, . . ..

Proposition 2.9. The pairing of Theorem 2.7 respects the action of the Hecke operators, in the sense that ⟨fT, x⟩ = ⟨f, T x⟩ for all T ∈ T, x ∈ M_{k}(N, ε), and f ∈ S_{k}(N, ε) ⊕ \overline{S}_{k}(N, ε).

Proof. See [45, Prop. 10].

2.4.2 The ∗-involution

The matrix j = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) defines an involution ∗ of M_{k}(N, ε) given by x ↦ x^{∗} = j(x). Explicitly,

(P (X, Y){α, β})^{∗} = P (X, −Y){−α, −β}.

Because the space of modular symbols is constructed as a quotient, it is not obvious that the ∗-involution is well defined.

Proposition 2.10. The ∗-involution is well defined.

Proof. Recall that M_{k}(N, ε) is the largest torsion-free quotient of the free Z[ε]-module generated by symbols x = P {α, β} by the submodule generated by relations γx − ε(γ)x for all γ ∈ Γ_{0}(N). In order to check that the operator ∗ is well defined, it suffices to check, for any x ∈ M_{k}, that ∗(γx − ε(γ)x) is of the form γ′y − ε(γ′)y, for some y in M_{k}. Note that if γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N), then jγj^{−1} = ( \begin{matrix}a & −b \\ −c & d \end{matrix} ) is also in Γ_{0}(N) and ε(jγj^{−1}) = ε(γ). We have

j(γx − ε(γ)x) = jγx − jε(γ)x
= jγj^{−1}jx − ε(γ)jx
= (jγj^{−1})(jx) − ε(jγj^{−1})(jx).

If f is a modular form, let f^{∗} be the holomorphic function \overline{f(−\overline{z})}, where the bar denotes complex conjugation. The Fourier coefficients of f^{∗} are the complex conjugates of those of f; though f^{∗} is again a holomorphic modular form, its character is \overline{ε} instead of ε. The pairing of Theorem 2.7 is the restriction of a pairing on the full spaces without character, and we have the following proposition.

Proposition 2.11. We have

⟨f^{∗}, x^{∗}⟩ = \overline{⟨f, x⟩}.

Definition 2.12 (Plus-one quotient). The plus-one quotient M_{k}(N, ε)_{+} is the largest torsion-free quotient of M_{k}(N, ε) by the relations x^{∗} − x = 0 for all x ∈ M_{k}(N, ε). Similarly, the minus-one quotient is the quotient of M_{k}(N, ε) by all relations x^{∗} + x = 0, for x ∈ M_{k}(N, ε).

WARNING 2.13. We were forced to make a choice in our definition of the operator ∗. Fortunately, it agrees with that of [16, §2.1.3], but not with the choice made in [45, §1.6].

2.4.3 The Atkin-Lehner involutions

In this section we assume that k is even and ε^{2} = 1. The assumption on ε is necessary only so that the involution we are about to define preserves M_{k}(N, ε). More generally, it is possible to define a map which sends M_{k}(N, ε) to M_{k}(N, \overline{ε}).

To each divisor d of N such that (d, N/d) = 1 there is an Atkin-Lehner involution W_{d} of M_{k}(N, ε), which is defined as follows. Using the Euclidean algorithm, choose integers x, y, z, w such that

dxw − (N/d)yz = 1.

Next let g = ( \begin{matrix}dx & y \\ Nz & dw \end{matrix} ) and define

W_{d}(x) := \frac{1}{d^{\frac{k−2}{2}}} · g(x).

For example, when d = N we have g = ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ). The factor of d^{\frac{k−2}{2}} is necessary to normalize W_{d} so that it is an involution.

On modular forms there is an Atkin-Lehner involution, also denoted W_{d}, which acts by W_{d}(f) = f|[W_{d}]_{k}. These two like-named involutions are compatible with the integration pairing:

⟨W_{d}(f), x⟩ = ⟨f, W_{d}(x)⟩.

2.5 Degeneracy maps

In this section, we describe natural maps between spaces of modular symbols of different levels.

Fix a positive integer N and a Dirichlet character ε : (Z/NZ)^{∗} → C^{∗}. Let M be a positive divisor of N that is divisible by the conductor of ε, in the sense that ε factors through (Z/MZ)^{∗} via the natural map (Z/NZ)^{∗} → (Z/MZ)^{∗} composed with some uniquely defined character ε′ : (Z/MZ)^{∗} → C^{∗}. For any positive divisor t of N/M, let T = ( \begin{matrix}1 & 0 \\ 0 & t \end{matrix}) and fix a choice D_{t} = {T γ_{i} : i = 1, …, n} of coset representatives for Γ_{0}(N)\T Γ_{0}(M).

WARNING 2.14. There is a mistake in [45, §2.6]: The quotient “Γ_{1}(N)\Γ_{1}(M)T ” should be replaced by “Γ_{1}(N)\T Γ_{1}(M)”.

Proposition 2.15. For each divisor t of N/M there are well-defined linear maps

α_{t} : M_{k}(N, ε) → M_{k}(M, ε′),  α_{t}(x) = (tT^{−1})x = (\begin{matrix}t & 0 \\ 0 & 1 \end{matrix}) x
β_{t} : M_{k}(M, ε′) → M_{k}(N, ε),  β_{t}(x) = ∑_{T γ_{i}∈D_{t}} ε′(γ_{i})^{−1}T γ_{i}x.

Furthermore, α_{t} ∘ β_{t} is multiplication by t^{k−2} · [Γ_{0}(M) : Γ_{0}(N)].

Proof. To show that α_{t} is well defined, we must show that for each x ∈ M_{k}(N, ε) and γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N), that we have

α_{t}(γx − ε(γ)x) = 0 ∈ M_{k}(M, ε′).

We have

α_{t}(γx) = (\begin{matrix}t & 0 \\ 0 & 1 \end{matrix}) γx = (\begin{matrix}a & tb \\ c/t & d \end{matrix}) (\begin{matrix}t & 0 \\ 0 & 1 \end{matrix}) x = ε′(a) (\begin{matrix}t & 0 \\ 0 & 1 \end{matrix}) x,

so

α_{t}(γx − ε(γ)x) = ε′(a)α_{t}(x) − ε(γ)α_{t}(x) = 0.

We next verify that β_{t} is well defined. Suppose that x ∈ M_{k}(M, ε′) and γ ∈ Γ_{0}(M); then ε′(γ)^{−1}γx = x, so

β_{t}(x) = ∑_{T γ_{i}∈D_{t}} ε′(γ_{i})^{−1}T γ_{i}ε′(γ)^{−1}γx

= ∑_{T γ_{i}γ∈D_{t}} ε′(γ_{i}γ)^{−1}T γ_{i}γx.

This computation shows that the definition of β_{t} does not depend on the choice D_{t} of coset representatives. To finish the proof that β_{t} is well defined we must show that, for γ ∈ Γ_{0}(M), we have β_{t}(γx) = ε′(γ)β_{t}(x) so that β_{t} respects the relations that define M_{k}(M, ε). Using that β_{t} does not depend on the choice of coset representative, we find that for γ ∈ Γ_{0}(M),

β_{t}(γx) = ∑_{T γ_{i}∈D_{t}} ε′(γ_{i})^{−1}T γ_{i}γx

= ∑_{T γ_{i}γ^{−1}∈D_{t}} ε′(γ_{i}γ^{−1})^{−1}T γ_{i}γ^{−1}γx

= ε′(γ)β_{t}(x).

To compute α_{t} ∘ β_{t}, we use that #D_{t} = [Γ_{0}(N) : Γ_{0}(M)]:

α_{t}(β_{t}(x)) = α_{t} (∑_{T γ_{i}} ε′(γ_{i})^{−1}T γ_{i}x)

= ∑_{T γ_{i}} ε′(γ_{i})^{−1}(tT^{−1})T γ_{i}x

= t^{k−2} ∑_{T γ_{i}} ε′(γ_{i})^{−1}γ_{i}x

= t^{k−2} ∑_{T γ_{i}} x

= t^{k−2} · [Γ_{0}(N) : Γ_{0}(M)] · x.

The scalar factor of t^{k−2} appears instead of t, because t is acting on x as an element of GL_{2}(Q) not as an an element of Q.

Definition 2.16 (New and old modular symbols). The subspace M_{k}(N, ε)^{new} of new modular symbols is the intersection of the kernels of the α_{t} as t runs through all positive divisors of N/M and M runs through positive divisors of M strictly less than N and divisible by the conductor of ε. The subspace M_{k}(N, ε)^{old} of old modular symbols is the subspace generated by the images of the β_{t} where t runs through all positive divisors of N/M and M runs through positive divisors of M strictly less than N and divisible by the conductor of ε.

WARNING: The new and old subspaces need not be disjoint, as the following example illustrates! This is contrary to the statement on page 80 of [45].

Example 2.17. We justify the above warning. Consider, for example, the case N = 6, k = 2, and trivial character. The spaces M_{2}(2) and M_{2}(3) are each of dimension 1, and each is generated by the modular symbol {∞, 0}. The space M_{2}(6) is of dimension 3, and is generated by the 3 modular symbols {∞, 0}, {−1/4, 0}, and {−1/2, −1/3}. The space generated by the 2 images of M_{2}(2) under the 2 degeneracy maps has dimension 2, and likewise for M_{2}(3). Together these images generate M_{2}(6), so M_{2}(6) is equal to its old subspace. However, the new subspace is nontrivial because the two degeneracy maps M_{2}(6) → M_{2}(2) are equal, as are the two degeneracy maps M_{2}(6) → M_{2}(3). In particular, the intersection of the kernels of the degeneracy maps has dimension at least 1 (in fact, it equals 1).

Computationally, it appears that something similar to this happens if and only if the weight is 2, the character is trivial, and the level is composite. This behavior is probably related to the nonexistence of a characteristic 0 Eisenstein series of weight 2 and level 1.

The following tempting argument is incorrect; the error lies in the fact that an element of the old subspace is a linear combination of β_{t}(y)’s for various y’s and t’s: “If x is in both the new and old subspace, then x = β_{t}(y) for some modular symbol y of lower level. This implies x = 0 because

0 = α_{t}(x) = α_{t}(β_{t}(y)) = t^{k−2} · [Γ_{0}(N) : Γ_{0}(M)] · y.”

Remark 2.18. The map β_{t}∘α_{t} cannot in general be multiplication by a scalar since M_{k}(M, ε′) usually has smaller dimension than M_{k}(N, ε).

2.5.1 Computing coset representatives

Definition 2.19 (Projective line mod N). Let N be a positive integer. Then the projective line P^{1}(N) is the set of pairs (a, b), with a, b ∈ Z/NZ and gcd(a, b, N) = 1, modulo the eqivalence relation which identifies (a, b) and (a′, b′) if and only if ab′ ≡ ba′ (mod N).

Let M be a positive divisor of N and t a divisor of N/M. The following random algorithm computes a set D_{t} of representatives for the orbit space Γ_{0}(M)\T Γ_{0}(N). There are deterministic algorithms for computing D_{t}, but all of the ones the author has found are vastly less efficient than the following random algorithm.

Algorithm 2.20. Let Γ_{0}(N/t, t) denote the subgroup of SL_{2}(Z) consisting of matrices that are upper triangular modulo N/t and lower triangular modulo t. Observe that two right cosets of Γ_{0}(N/t, t) in SL_{2}(Z), represented by ( \begin{matrix}a & b \\ c & d \end{matrix} ) and ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ), are equivalent if and only if (a, b) = (a′, b′) as points of P^{1}(t) and (c, d) = (c′, d′) as points of P^{1}(N/t). Using the following algorithm, we compute right coset representatives for Γ_{0}(N/t, t) inside Γ_{0}(M).

1. Compute the number [Γ_{0}(M) : Γ_{0}(N)] of cosets.

2. Compute a random element x ∈ Γ_{0}(M).

3. If x is not equivalent to anything generated so far, add it to the list.

4. Repeat steps (2) and (3) until the list is as long as the bound of step (1).

There is a natural bijection between Γ_{0}(N)\T Γ_{0}(M) and Γ_{0}(N/t, t)\Γ_{0}(M), under which T γ corresponds to γ. Thus we obtain coset representatives for Γ_{0}(N)\T Γ_{0}(M) by left multiplying each coset representative of Γ_{0}(N/t, t)\Γ_{0}(M) by T.

2.5.2 Compatibility with modular forms

The degeneracy maps defined above are compatible with the corresponding degeneracy maps α̃_{t} and β̃_{t} on modular forms. This is because the degeneracy maps on modular forms are defined by summing over the same coset representatives D_{t}. Thus we have the following compatibilities.

⟨α̃_{t}(f), x⟩ = ⟨f, α_{t}(x)⟩,
⟨β̃_{t}(f), x⟩ = ⟨f, β_{t}(x)⟩.

If p is prime to N, then T_{p}α_{t} = α_{t}T_{p} and T_{p}β_{t} = β_{t}T_{p}.

2.6 Manin symbols

From the definition given in Section 2.1, it is not obvious that M_{k}(N, ε) is of finite rank. The Manin symbols provide a finite presentation of M_{k}(N, ε) that is vastly more useful from a computational point of view.

Definition 2.21 (Manin symbols). The Manin symbols are the set of pairs

[P (X, Y), (u, v)]

where P (X, Y) ∈ V_{k−2} and 0 ≤ u, v < N with gcd(u, v, N) = 1.

Define a right action of GL_{2}(Q) on the free Z[ε]-module M generated by the Manin symbols as follows. The element g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) acts by

[P, (u, v)]g = [g^{−1}P (X, Y), (u, v)g] = [P (aX + bY, cX + dY), (au + cv, bu + dv)].

Let σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} ). Let M_{k}(N, ε)′ be the largest torsion-free quotient of M by the relations

x + xσ = 0,
x + xτ + xτ^{2} = 0,
ε(λ)[P, (u, v)] − [P, (λu, λv)] = 0.

Theorem 2.22. There is a natural isomorphism φ : M_{k}(N, ε)′ ⟶ M_{k}(N, ε) given by

[X^{i}Y^{2−k−i}, (u, v)] ↦ g(X^{i}Y^{k−2−i}{0, ∞})

where g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) is any matrix such that (u, v) ≡ (c, d) (mod N).

Proof. In [45, §1.2, §1.7] it is proved that φ ⊗_{Z[ε]} C is an isomorphism, so φ is injective and well defined. The discussion in Section 2.6.1 below (“Manin’s trick”) shows that every element in M_{k}(N, ε) is a Z[ε]-linear combination of elements in the image, so φ is surjective as well.

2.6.1 Conversion between modular and Manin symbols

For some purposes it is better to work with modular symbols, and for others it is better to work with Manin symbols. For example, there are descriptions of the Atkin-Lehner involution in terms of both Manin and modular symbols; it appears more efficient to compute this involution using modular symbols. On the other hand, practically Hecke operators can be computed more efficiently using Manin symbols. It is thus essential to be able to convert between these two representations. The conversion from Manin to modular symbols is straightforward, and follows immediately from Theorem 2.22. The conversion back is accomplished using the method used to prove Theorem 2.22; it is known as “Manin’s trick”, and involves continued fractions.

Given a Manin symbol [X^{i}Y^{k−2−i}, (u, v)], we write down a corresponding modular symbol as follows. Choose ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) such that (c, d) ≡ (u, v) (mod N). This is possible by Lemma 1.38 of [62, pg. 20]; in practice, finding ( \begin{matrix}a & b \\ c & d \end{matrix} ) is not completely trivial, but can be accomplished using the extended Euclidean algorithm. Then

[X^{i}Y^{k−2−i}, (u, v)] ⟷ ( \begin{matrix}a & b \\ c & d \end{matrix} ) (X^{i}Y^{k−2−i}\{0, ∞\})

= (dX − bY)^{i}(−cX + aY)^{2−k−i} \{\frac{b}{d}, \frac{a}{c}\} .

In the other direction, suppose that we are given a modular symbol P (X, Y){α, β} and wish to represent it as a sum of Manin symbols. Because

P {a/b, c/d} = P {a/b, 0} + P {0, c/d},

it suffices to write P {0, a/b} in terms of Manin symbols. Let

0 = \frac{p_{−2}}{q_{−2}} = \frac{0}{1}, \frac{p_{−1}}{q_{−1}} = \frac{1}{0}, \frac{p_{0}}{1} = \frac{p_{0}}{q_{0}} , \frac{p_{1}}{q_{1}} , \frac{p_{2}}{q_{2}} , …, \frac{p_{r}}{q_{r}} = \frac{a}{b}

denote the continued fraction convergents of the rational number a/b. Then

p_{j}q_{j−1} − p_{j−1}q_{j} = (−1)^{j−1} for − 1 ≤ j ≤ r.

If we let g_{j} = (\begin{matrix}(−1)^{j−1}p_{j} & p_{j−1} \\ (−1)^{j−1}q_{j} & q_{j−1} \end{matrix}), then g_{j} ∈ SL_{2}(Z) and

P (X, Y)\{0, a/b\} = P (X, Y) ∑_{j=−1}^{r} \{\frac{p_{j−1}}{q_{j−1}} , \frac{p_{j}}{q_{j}} \}

= ∑_{j=−1}^{r} g_{j}((g_{j}^{−1}P (X, Y)){0, ∞})

= ∑_{j=−1}^{r}[g_{j}^{−1}P (X, Y), ((−1)^{j−1}q_{j}, q_{j−1})].

Note that in the jth summand, g_{j}^{−1}P (X, Y), replaces P (X, Y). Since g_{j} ∈ SL_{2}(Z) and P (X, Y) has integer coefficients, the polynomial g_{j}^{−1}P (X, Y) also has integer coefficients, so no denominators are introduced.

The continued fraction expansion [c_{1}, c_{2}, …, c_{n}] of the rational number a/b can be computed using the Euclidean algorithm. The first term, c_{1}, is the “quotient”: a = bc_{1} + r, with 0 ≤ r < b. Let a′ = b, b′ = r and compute c_{2} as a′ = b′c_{2} + r′, etc., terminating when the remainder is 0. For example, the expansion of 5/13 is [0, 2, 1, 1, 2]. The numbers

d_{i} = c_{1} + \frac{1}{c_{2} + \frac{1}{c_{3}+···}}

will then be the (finite) convergents. For example if a/b = 5/13, then the convergents are

0/1, 1/0, d_{1} = 0, d_{2} = \frac{1}{2}, d_{3} = \frac{1}{3}, d_{4} = \frac{2}{5}, d_{5} = \frac{5}{13}.

2.6.2 Hecke operators on Manin symbols

Thoerem 2 of [45] gives a description of the Hecke operators T_{n} directly on the space of Manin symbols. This avoids the expense of first converting a Manin symbol to a modular symbol, computing T_{n} on the modular symbol, and then converting back. For the reader’s convenience, we very briefly recall Merel’s theorem here, along with an enhancement due to Cremona.

As in [16, §2.4], define R_{p} as follows. When p = 2,

R_{2} := \{(\begin{matrix}1 & 0 \\ 0 & 2 \end{matrix}) , (\begin{matrix}2 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}2 & 1 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 1 & 2 \end{matrix})\} .

When p is odd, R_{p} is the set of 2 × 2 integer matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) with determinant p, and either

1. a > |b| > 0, d > |c| > 0, and bc < 0; or

2. b = 0, and |c| < d/2; or

3. c = 0, and |b| < a/2.

Proposition 2.23. For [P (X, Y), (u, v)] ∈ M_{k}(N, ε) and p a prime, we have

T_{p}([P (X, Y), (u, v)]) = ∑_{g∈R_{p}}[P (X, Y), (u, v)].g

= ∑_{( \begin{matrix}a & b \\ c & d \end{matrix} )∈R_{p}}[P (aX + bY, cX + dY), (au + cv, bu + dv)],

where the sum is restricted to matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) such that gcd(au + cv, bu + dv, N) = 1.

Proof. For the case k = 2 and an algorithm to compute R_{p}, see [16, §2.4]. The general case follows from [45, Theorem 2] applied to the set S of [45, §3] by observing that when p is an odd prime S′_{p} is empty.

2.6.3 The cuspidal and boundary spaces in terms of Manin symbols

This section is a review of Merel’s explicit description of the boundary map in terms of Manin symbols for Γ = Γ_{1}(N) (see [45, §1.4]). In the next section, we describe a very efficient way to compute the boundary map.

Let R be the equivalence relation on Γ\Q^{2} which identifies the element [Γ ( \begin{matrix}λu \\ λv \end{matrix} )] with sign(λ)^{k}[Γ ( \begin{matrix}u \\ v \end{matrix})], for any λ ∈ Q^{∗}. Denote by B_{k}(Γ) the finite dimensional Q-vector space with basis the equivalence classes (Γ\Q^{2})/R. The dimension of this space is #(Γ\P^{1}(Q)).

Proposition 2.24. The map

μ : B_{k}(Γ) → B_{k}(Γ), P \{\frac{u}{v}\} ↦ P (u, v) [Γ (\begin{matrix}u \\ v \end{matrix})]

is well defined and injective. Here u and v are assumed coprime.

Thus the kernel of δ : S_{k}(Γ) → B_{k}(Γ) is the same as the kernel of μ ∘ δ.

Proposition 2.25. Let P ∈ V_{k−2} and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z). We have

μ ∘ δ([P, (c, d)]) = P (1, 0)[Γ ( \begin{matrix}a \\ c \end{matrix})] − P (0, 1)[Γ ( \begin{matrix}b \\ d \end{matrix} )].

2.6.4 Computing the boundary map

In this section we describe how to compute the map δ : M_{k}(N, ε) → B_{k}(N, ε) given in the previous section. The algorithm presented here generalizes the one in [16, §2.2]. To compute the image of [P, (c, d)], with g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z), we must compute the class of [( \begin{matrix}a \\ c \end{matrix})] and of [( \begin{matrix}b \\ d \end{matrix} )]. Instead of finding a canonical form for cusps, we use a quick test for equivalence modulo scalars. In the following algorithm, by the ith standard cusp we mean the ith basis vector for a basis of B_{k}(N, ε). The basis is constructed as the algorithm is called successively. We first give the algorithm, then prove the facts used by the algorithm in testing equivalence.

Algorithm 2.26. Given a cusp [( \begin{matrix}u \\ v \end{matrix})] this algorithm computes an integer i and a scalar α such that [( \begin{matrix}u \\ v \end{matrix})] is equivalent to α times the ith standard cusp. First, using Proposition 2.27 and Algorithm 2.28, check whether or not [( \begin{matrix}u \\ v \end{matrix})] is equivalent, modulo scalars, to any cusp found so far. If so, return the index of the representative and the scalar. If not, record ( \begin{matrix}u \\ v \end{matrix}) in the representative list. Then, using Proposition 2.30, check whether or not ( \begin{matrix}u \\ v \end{matrix}) is forced to equal zero by the relations. If it does not equal zero, return its position in the list and the scalar 1. If it equals zero, return the scalar 0 and the position 1; keep ( \begin{matrix}u \\ v \end{matrix}) in the list, and record that it is zero.

In the case considered in Cremona’s book [16], the relations between cusps involve only the trivial character, so they do not force any cusp classes to vanish. Cremona gives the following two criteria for equivalence.

Proposition 2.27 (Cremona). Let ( \begin{matrix}u_{i} \\ v_{i} \end{matrix}), i = 1, 2 be written so that gcd(u_{i}, v_{i}) = 1.

1. There exists g ∈ Γ_{0}(N) such that g ( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ) = ( \begin{matrix}u_{2} \\ v_{2} \end{matrix} ) if and only if

s_{1}v_{2} ≡ s_{2}v_{1} (mod gcd(v_{1}v_{2}, N)), where s_{j} satisfies u_{j}s_{j} ≡ 1 (mod v_{j}).

2. There exists g ∈ Γ_{1}(N) such that g ( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ) = ( \begin{matrix}u_{2} \\ v_{2} \end{matrix} ) if and only if

v_{2} ≡ v_{1} (mod N) and u_{2} ≡ u_{1} (mod gcd(v_{1}, N)).

Proof. The first is Proposition 2.2.3 of [16], and the second is Lemma 3.2 of [15].

Algorithm 2.28. Suppose ( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ) and ( \begin{matrix}u_{2} \\ v_{2} \end{matrix} ) are equivalent modulo Γ_{0}(N). This algorithm computes a matrix g ∈ Γ_{0}(N) such that g ( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ) = ( \begin{matrix}u_{2} \\ v_{2} \end{matrix} ). Let s_{1}, s_{2}, r_{1}, r_{2} be solutions to s_{1}u_{1} − r_{1}v_{1} = 1 and s_{2}u_{2} − r_{2}v_{2} = 1. Find integers x_{0} and y_{0} such that x_{0}v_{1}v_{2} + y_{0}N = 1.

Let x = −x_{0}(s_{1}v_{2} − s_{2}v_{1})/(v_{1}v_{2}, N) and s′_{1} = s_{1} + xv_{1}. Then g = (\begin{matrix}u_{2} & r_{2} \\ v_{2} & s_{2} \end{matrix}) · (\begin{matrix}u_{1} & r_{1} \\ v_{1} & s′_{1} \end{matrix})^{−1}

sends ( \begin{matrix}u_{1} \\ v_{1} \end{matrix} ) to ( \begin{matrix}u_{2} \\ v_{2} \end{matrix} ).

Proof. This follows from the proof of Proposition 2.27 in [16].

To see how the ε relations, for nontrivial ε, make the situation more complicated, observe that it is possible that ε(α) ≠ ε(β) but

ε(α) [(\begin{matrix}u \\ v \end{matrix})] = [γ_{α} (\begin{matrix}u \\ v \end{matrix})] = [γ_{β} (\begin{matrix}u \\ v \end{matrix})] = ε(β) [(\begin{matrix}u \\ v \end{matrix})];

One way out of this difficulty is to construct the cusp classes for Γ_{1}(N), then quotient out by the additional ε relations using Gaussian elimination. This is far too inefficient to be useful in practice because the number of Γ_{1}(N) cusp classes can be unreasonably large. Instead, we give a quick test to determine whether or not a cusp vanishes modulo the ε-relations.

Lemma 2.29. Suppose α and α′ are integers such that gcd(α, α′, N) = 1. Then there exist integers β and β′, congruent to α and α′ modulo N, respectively, such that gcd(β, β′) = 1.

Proof. By [62, 1.38] the map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective. By the Euclidean algorithm, there exist integers x, y and z such that xα + yα′ + zN = 1. Consider the matrix ( \begin{matrix}y & −x \\ α & α′ \end{matrix} ) ∈ SL_{2}(Z/NZ) and take β, β′ to be the bottom row of a lift of this matrix to SL_{2}(Z).

Proposition 2.30. Let N be a positive integer and ε a Dirichlet character of modulus N. Suppose ( \begin{matrix}u \\ v \end{matrix}) is a cusp with u and v coprime. Then ( \begin{matrix}u \\ v \end{matrix}) vanishes modulo the relations

[γ ( \begin{matrix}u \\ v \end{matrix})] = ε(γ) [( \begin{matrix}u \\ v \end{matrix})] , all γ ∈ Γ_{0}(N)

if and only if there exists α ∈ (Z/NZ)^{∗}, with ε(α) ≠ 1, such that

v ≡ αv (mod N),
u ≡ αu (mod gcd(v, N)).

Proof. First suppose such an α exists. By Lemma 2.29 there exists β, β′ ∈ Z lifting α, α^{−1} such that gcd(β, β′) = 1. The cusp ( \begin{matrix}βu \\ β′v \end{matrix} ) has coprime coordinates so, by Proposition 2.27

and our congruence conditions on α, the cusps ( \begin{matrix}βu \\ β′v \end{matrix} ) and ( \begin{matrix}u \\ v \end{matrix}) are equivalent by an element

of Γ_{1}(N). This implies that [( \begin{matrix}βu \\ β′v \end{matrix} )] = [( \begin{matrix}u \\ v \end{matrix})]. Since [( \begin{matrix}βu \\ β′v \end{matrix} )] = ε(α) [( \begin{matrix}u \\ v \end{matrix})], our assumption that ε(α) ≠ 1 forces [( \begin{matrix}u \\ v \end{matrix})] = 0.

Conversely, suppose [( \begin{matrix}u \\ v \end{matrix})] = 0. Because all relations are two-term relations, and the Γ_{1}(N)-relations identify Γ_{1}(N)-orbits, there must exists α and β with

[γ_{α} (\begin{matrix}u \\ v \end{matrix})] = [γ_{β} (\begin{matrix}u \\ v \end{matrix})] and ε(α) ≠ ε(β).

Indeed, if this did not occur, then we could mod out by the ε relations by writing each [γ_{α} ( \begin{matrix}u \\ v \end{matrix})] in terms of [( \begin{matrix}u \\ v \end{matrix})], and there would be no further relations left to kill [( \begin{matrix}u \\ v \end{matrix})]. Next observe that

[γ_{β^{−1}α} (\begin{matrix}u \\ v \end{matrix})] = [γ_{β^{−1}}γ_{α} (\begin{matrix}u \\ v \end{matrix})] = ε(β^{−1}) [γ_{α} (\begin{matrix}u \\ v \end{matrix})] = ε(β^{−1}) [γ_{β} (\begin{matrix}u \\ v \end{matrix})] = [(\begin{matrix}u \\ v \end{matrix})] .

Applying Proposition 2.27 and noting that ε(β^{−1}α) ≠ 1 shows that β^{−1}α satisfies the properties of the “α” in the statement of the proposition we are proving.

We enumerate the possible α appearing in Proposition 2.30 as follows. Let g = (v, N) and list the α = v · \frac{N}{g} · a + 1, for a = 0, …, g − 1, such that ε(α) ≠ 0.

Working in the plus one or minus one quotient. Let s be a sign, either +1 or −1. To compute S_{k}(N, ε)_{s} it is necessary to replace B_{k}(N, ε) by its quotient modulo the additional relations [( \begin{matrix}−u \\ v \end{matrix})] = s [( \begin{matrix}u \\ v \end{matrix})] for all cusps ( \begin{matrix}u \\ v \end{matrix}). Algorithm 2.26 can be modified to deal with this situation as follows. Given a cusp x = ( \begin{matrix}u \\ v \end{matrix}), proceed as in Algorithm 2.26 and check if either ( \begin{matrix}u \\ v \end{matrix}) or ( \begin{matrix}−u \\ v \end{matrix}) is equivalent (modulo scalars) to any cusp seen so far. If not, use the following trick to determine whether the ε and s-relations kill the class of ( \begin{matrix}u \\ v \end{matrix}): use the unmodified Algorithm 2.26 to compute the scalars α_{1}, α_{2} and standard indices i_{1}, i_{2} associated to ( \begin{matrix}u \\ v \end{matrix}) and ( \begin{matrix}−u \\ v \end{matrix}), respectively. The s-relation kills the class of ( \begin{matrix}u \\ v \end{matrix}) if and only if i_{1} = i_{2} but α_{1} ≠ sα_{2}.

2.7 The complex torus attached to a modular form

Fix integers N ≥ 1, k ≥ 2, and let ε be a mod N Dirichlet character. For the rest of this section assume that ε^{2} = 1.

We construct a lattice in Hom(S_{k}(N, ε), C) that is invariant under complex conjugation and under the action of the Hecke operators. The quotient of Hom(S_{k}(N, ε), C) by this lattice is a complex torus J_{k}(N, ε), which is equipped with an action of the Hecke operators and of complex conjugation.

The reader may wish to compare our construction with a closely related construction of Shimura [60]. Shimura observes that the Petersson pairing gives his torus the structure of an abelian variety over C. Note that his torus is, a priori, different than our torus. We do not know if our torus has the structure of abelian variety over C.

When k = 2, the torus J_{2}(N, ε) is the set of complex points of an abelian variety, which is actually defined over Q; when k > 2, the study of these complex tori is of interest in trying to understand the conjectures of Bloch and Kato (see [7]) on motifs attached to modular forms.

Let S = S_{k}(N, ε) (respectively, S = S_{k}(N, ε)) be the space of cuspidal modular symbols (respectively, cusp forms) of weight k, level N, and character ε. The Hecke algebra T acts in a way compatible with the integration pairing ⟨ , ⟩ : S × S → C. This pairing induces a T-module homomorphism Φ : S → S^{∗} = Hom_{C}(S, C), called the period mapping. Because ε^{2} = 1, the ∗-involution preserves S.

Proposition 2.31. The period mapping Φ is injective and Φ(S) is a lattice in S^{∗}.

Proof. By Theorem 2.7,

S ⊗_{R} C ≅ Hom_{C}(S ⊕ \overline{S}, C).

Because ε^{2} = 1, we have S = S_{k}(N, ε; R) ⊗_{R} C. Set S_{R} := S_{k}(N, ε; R) and likewise define \overline{S}_{R}. We have

Hom_{C}(S ⊕ \overline{S}, C) = Hom_{R}(S_{R} ⊕ \overline{S}_{R}, R) ⊗_{R} C.

Let S_{R} = S_{k}(N, ε; R) and S_{R}^{+} be the subspace fixed under ∗. By Proposition 2.11 we have maps

S_{R}^{+} → Hom_{R}(S_{R} ⊕ \overline{S}_{R}, R) → Hom_{R}(S_{R}, R)

and

S_{R}^{−} → Hom_{R}(S_{R} ⊕ \overline{S}_{R}, iR) → Hom_{R}(S_{R}, iR).

The map S_{R}^{+} → Hom_{R}(S_{R}, R) is an isomorphism: the point is that if ⟨•, x⟩, for x ∈ S_{R}^{+}, vanishes on S_{R} then it vanishes on the whole of S ⊕ \overline{S}. Likewise, the map S_{R}^{−} → Hom_{R}(S_{R}, iR) is an isomorphism. Thus

S ⊗ R = S_{R} ≅ Hom_{R}(S_{R}, R) ⊕ Hom_{R}(S_{R}, iR) ≅ Hom_{C}(S, C).

Finally, we observe that S is by definition torsion free, which completes the proof.

The torus J_{k}(N, ε) fits into an exact sequence

0 ⟶ S ⟶^{Φ} Hom_{C}(S, C) ⟶ J_{k}(N, ε) ⟶ 0.

Let f ∈ S be a newform and S_{f} the complex vector space spanned by the Galois conjugates of f. The period map Φ_{f} associated to f is the map S → Hom_{C}(S_{f}, C) obtained by composing Φ with restriction to S_{f}. Set

A_{f} := Hom_{C}(S_{f}, C)/Φ_{f}(S).

We associate to f a subtorus of J as follows. Let I_{f} = Ann_{T}(f) be the annihilator of f in the Hecke algebra, and set

A_{f}^{∨} := Hom_{C}(S, C)[I_{f}]/Φ(S[I_{f}])

where Hom_{C}(S, C)[I_{f}] = ∩_{t∈I_{f}} ker(t).

The following diagram summarizes the tori just defined; its columns are exact but its rows need not be.

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ S[I_{f}] \ar[r] \ar[d] & S \ar[r] \ar[d] & Φ_{f}(S) \ar[d] \\ Hom_{C}(S, C)[I_{f}] \ar[r] \ar[d] & Hom_{C}(S, C) \ar[r] \ar[d] & Hom_{C}(S[I_{f}], C) \ar[d] \\ A_{f}^{∨} \ar[r] \ar[d] & J_{k}(N, ε) \ar[r] \ar[d] & A_{f} \ar[d] \\ 0 & 0 & 0}  (2.1)

2.7.1 The case when the weight is 2

When k = 2 and ε = 1 the above is just Shimura’s classical association of an abelian variety to a modular form; see [62, Thm. 7.14] and [61]. In this case A_{f} and A_{f}^{∨} are abelian varieties that are defined over Q. Furthermore A_{f} is an optimal quotient of J, in the sense that the kernel of the map J → A_{f} is connected. For a summary of the main results in this situation, see Section 4.6.

Chapter 3

Applications of modular symbols

In the previous chapter we introduced several spaces of modular symbols, and observations such as “Manin’s trick” suggested that we could compute with them. The duality between modular symbols and modular forms hints that modular symbols might be useful in computing information about modular forms. In the present chapter, we gather together the fruits of our investigation into this connection.

Sections 3.1–3.5 of this chapter give a method to compute the irreducible components of the spaces M_{k}(N, ε) of modular symbols. In Section 3.6 we observe that computing intersections of certain abelian varieties can be reduced to linear algebra over Z by viewing the abelian varieties as complex tori and considering the appropriate diagrams. In Sections 3.7, we continue this trend by pointing out that many invariants of the complex torus attached to a modular form can be computed without computing any approximate period lattices. In Section 3.8, we discuss well-known methods for computing both an upper and lower bound on the order of the torsion subgroup of certain abelian varieties. Section 3.9 presents an algorithm for computing the modular degree of the complex torus associated to a newform.

In Section 3.10 we aim squarely at the problem of gathering data related to the Birch and Swinnerton-Dyer conjecture and its generalizations, where we give a formula for the rational numbers |L(A_{f}, j)/Ω_{j}| attached to a newform. In Section 3.11 we compare the ratio computed in the previous section to the one considered in the Birch and Swinnerton-Dyer conjecture; the two numbers differ by a Manin constant, which we bound. Finally, in Section 3.12 we give algorithms for approximating the period lattice and related numerical quantities associated to a newform of arbitrary weight.

3.1 Computing the space of modular symbols

Definition 3.1. Let W be a subspace of a finite-dimensional vector space V. To compute the quotient V/W means to give a matrix representing the projection V → V/W, with respect to some basis for V and some basis B for V/W, along with a lift to V of each element of B.

In other words, to compute V/W means to create a reduction function that assigns to each element of V its canonical representative on the “free basis” B.

Let N be a positive integer, fix a mod N Dirichlet character ε, let K := Q[ε] be the smallest extension containing the values of ε, and let O := Z[ε].

Algorithm 3.2. Given a positive integer N, a Dirichlet character ε, and an integer k ≥ 2 this algorithm computes M_{k}(N, ε; K). We compute the quotient presentation given in Theorem 2.22 in three steps.

1. Let V_{1} be the finite-dimensional K-vector space generated by the Manin symbols [X^{i}Y^{k−2−i}, (u, v)] for i = 0, …, k − 2 and 0 ≤ u, v < N with gcd(u, v, N) = 1. Let W_{1} be the subspace of V_{1} generated by all differences

[X^{i}Y^{k−2−i}, (λu, λv)] − ε(λ)[X^{i}Y^{k−2−i}, (u, v)].

Because all relations are two-term, it is easy to compute V_{2} := V_{1}/W_{1}. In computing this quotient, we do not have to explicitly compute the large matrix representing the linear map V_{1} → V_{2}, as it can be replaced by a suitable “reduction procedure” involving arithmetic in Z/NZ.

2. Let σ act on the set of Manin symbols as in Section 2.6; thus

[X^{i}Y^{k−2−i}, (u, v)]σ = (−1)^{i}[Y^{i}X^{k−2−i}, (v, −u)].

Let W_{2} be the subspace of V_{2} generated by the sums x + xσ for x ∈ V_{2}. Because all relations are two-term relations, it is easy to compute V_{3} := V_{2}/W_{2}.

3. Let τ act on Manin symbols as in Section 2.6; thus

[X^{i}Y^{k−2−i}, (u, v)]τ = [(−Y)^{i}(X − Y)^{k−2−i}, (v, −u − v)].

Note that the symbol on the right can be written as a sum of generating Manin symbols. Let W_{3} be the subspace of V_{3} generated by the sums x + xτ + xτ^{2} where x varies over the images of a basis of V_{2} (not just a basis for V_{3}!). Using some form of Gaussian elimination, we compute V_{3}/W_{3}. Finally, M_{k}(N, ε; K) ≈ V_{3}/W_{3}.

Proof. For λ ∈ (Z/NZ)^{∗}, denote by ⟨λ⟩ the right action of λ on Manin symbols; thus

[X^{i}Y^{k−2−i}, (u, v)]⟨λ⟩ = [X^{i}Y^{k−2−i}, (λu, λv)].

By Theorem 2.22 the space M_{k}(N, ε; K) is isomorphic to the quotient of the vector spaces V_{1} of Step 1 modulo the relations x + xσ = 0, x + xτ + xτ^{2} = 0, and x⟨λ⟩ = λx as x varies over all Manin symbols and λ varies over (Z/NZ)^{∗}.

As motivation, we note that a naive computation of V_{1} modulo the σ, τ, and ⟨λ⟩ relations using Gaussian elimination is far too inefficient. This is why we compute the quotient in three steps. The complexity of Steps 1 and Steps 2 are negligible. The difficulty occurs in Step 3; at least the relations of this step occur in a space of dimension much smaller than that of V_{1}.

To see that the algorithm is correct, it is necessary only to observe that σ and τ both commute with all diamond-bracket operators ⟨λ⟩; this is an immediate consequence of the above formulas. We remark that in Step 3 it is in general necessary to compute the quotient by all relations x + xτ + xτ^{2} with x the image of a basis vector for V_{2} instead of just x in V_{3} because σ and τ do not commute.

Remark 3.3. In implementing the above algorithm, one should take special care in Steps 1 and 2 because the relations can together force certain of the Manin symbols to equal 0. For example, there might be relations of the form x_{1} + x_{2} = 0 and x_{1} − x_{2} = 0 which together force x_{1} = x_{2} = 0. Remark 3.4. To compute the plus-one quotient M_{k}(N, ε; K)_{+}, it is necessary to modify Step 2 of Algorithm 3.2 by including in W_{2} the differences x − xI where I = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), and

[X^{i}Y^{k−2−i}, (u, v)]I = (−1)^{i}[X^{i}Y^{k−2−i}, (−u, v)].

Likewise, to compute the minus-one quotient we include the sums x + xI. Note, as in the remarks in the proof of Algorithm 3.2, we can not add in the I relations in Step 1 because I and σ do not commute.

Algorithm 3.5. Given a positive integer N, a Dirichlet character ε, and an integer k ≥ 2, this algorithm computes the O-modules M_{k}(N, ε) and S_{k}(N, ε). (We assume as given algorithms for performing standard operations on O-modules.)

1. Using Algorithm 3.2 compute the K-vector space V := M_{k}(N, ε; K).

2. Compute the O-lattice L in V generated by the classes of the finitely many symbols [X^{i}Y^{k−2−i}, (u, v)] for i = 0, …, k − 2 and 0 ≤ u, v < N with gcd(u, v, N) = 1. It is only necessary to take one symbol in each ε-equivalence class, so there are (k − 2 + 1) · #P^{1}(Z/NZ) generating symbols. This computes M_{k}(N, ε).

3. To compute the submodule S_{k}(N, ε) of L, we use the algorithm of Section 2.6.4 to compute the boundary map δ : M_{k}(N, ε; K) → B_{k}(N, ε; K). Then S_{k}(N, ε) is the kernel of δ restricted to the lattice L.

As a check, using the formulas of Section 3.4, we compute the dimension of the space S_{k}(N, ε) of cusp forms and compare with the dimension of S_{k}(N, ε; K) computed in Algorithm 3.5. The latter dimension must equal twice the former one.

3.2 Computing the Hecke algebra

In this section we give an upper bound on the number of Hecke operators needed to generate the Hecke algebra as a Z-module. The bound on Hecke operators is an application of [66], which was described to the author by Ribet and Agashe when k = 2 and the level is prime. There are much better bounds on the number of Hecke operators needed to generate the Hecke algebra as a ring, but we do not investigate them here.

Let Γ be a subgroup of SL_{2}(Z) that contains Γ_{1}(N) for some N. Let S_{k}(Γ; C) be the space of weight-k cuspforms for Γ, and let T ⊂ End(S_{k}(Γ; C)) be the corresponding Hecke algebra. We now give a bound r such that the Hecke operators T_{n}, with n ≤ r, generate T as a Z-module.

For any ring R ⊂ C, let S_{k}(Γ; R) denotes the space of cuspforms for Γ with Fourier coefficients in R. Since S_{k}(Γ; C) = S_{k}(Γ; Z) ⊗_{Z} C, it makes sense to define

S_{k}(Γ; R) := S_{k}(Γ; Z) ⊗_{Z} R

for any ring R. The following proposition is well known.

Proposition 3.6. For any ring R, the pairing

T_{R} ⊗_{R} S_{k}(N; R) → R

that sends (T, f) to a_{1}(T f) is a perfect pairing, where T_{R} = T⊗_{Z} R. Furthermore, we have (T_{n}, f) = a_{n}(f), where T_{n} is the nth Hecke operator.

Let

μ = [SL_{2}(Z) : Γ],

and denote by ⌈x⌉ the smallest integer ≥ x.

Theorem 3.7 (Sturm). Let λ be a prime ideal in the ring O of integers in some number field. If f ∈ S_{k}(Γ; O) satisfies a_{n}(f) ≡ 0 (mod λ) for n ≤ ⌈\frac{k}{12}μ⌉, then f ≡ 0 (mod λ).

Proof. Theorem 1 of [66].

Proposition 3.8. If f ∈ S_{k}(Γ) satisfies a_{n}(f) = 0 for n ≤ r = ⌈\frac{k}{12}μ⌉, then f = 0.

Proof. We must show that the composite map S_{k}(Γ) ↪ C[[q]] → C[[q]]/(q^{r+1}) is injective. Because C is a flat Z-module and S_{k}(Γ; Z) ⊗ C = S_{k}(Γ), it suffices to show that the map F : S_{k}(Γ; Z) → Z[[q]]/(q^{r+1}) is injective. Suppose F (f) = 0, and let p be a prime number. Then a_{n}(f) = 0 for n ≤ r, hence plainly a_{n}(f) ≡ 0 (mod p) for any such n. Theorem 3.7 implies that f ≡ 0 (mod p). Duplicating this argument shows that the coefficients of f are divisible by all primes p, so they are 0.

Theorem 3.9. As a Z-module, T is generated by T_{1}, …, T_{r}, where r = ⌈\frac{k}{12}μ⌉.

Proof. Let Z be the submodule of T generated by T_{1}, T_{2}, …, T_{r}. Consider the exact sequence of additive abelian groups 0 → Z ⟶^{i} T → T/Z → 0. Let p be a prime and tensor this sequence with F_{p} to obtain the exact sequence

Z ⊗ F_{p} ⟶^{\overline{i}} T ⊗ F_{p} → (T/Z) ⊗ F_{p} → 0.

Put R = F_{p} in Proposition 3.6, and suppose that f ∈ S_{k}(N, F_{p}) pairs to 0 with each of T_{1}, …, T_{r}. Then by Proposition 3.6, a_{m}(f) = a_{1}(T_{m}f) = 0 in F_{p} for each m, 1 ≤ m ≤ r. Theorem 3.7 then asserts that f = 0. Thus the pairing, when restricted to the image of Z ⊗F_{p} in T⊗F_{p}, is also perfect. Thus dim_{F_{p}} \overline{i}(Z ⊗F_{p}) = dim_{F_{p}} S_{k}(N, F_{p}) = dim_{F_{p}} T⊗F_{p}, so (T/Z) ⊗ F_{p} = 0; repeating this argument for all p shows that T/Z = 0.

3.3 Representing and enumerating Dirichlet characters

Recall that a Dirichlet character is a homomorphism ε : (Z/NZ)^{∗} → C^{∗}.

The following lemma is well known.

Lemma 3.10. If p is an odd prime, then (Z/p^{n}Z)^{∗} is a cyclic group. The group (Z/2^{n}Z)^{∗} is generated by −1 and 5.

We use the following representation of Dirichlet characters. Factor N as a product of prime powers: N = ∏_{i=1}^{r} p_{i}^{e_{i}} with p_{i} < p_{i+1} and each e_{i} > 0; then (Z/NZ)^{∗} ≅ ∏_{i=1}^{r}(Z/p_{i}^{e_{i}}Z)^{∗}. If p_{i} is odd then the lemma implies that (Z/p_{i}^{e_{i}}Z)^{∗} is cyclic. If p_{1} = 2, then (Z/p_{1}^{e_{1}}Z)^{∗} is a product ⟨−1⟩ × ⟨5⟩ of two cyclic groups, both possibly trivial. For each i, we let a_{i} ∈ (Z/p_{i}^{e_{i}}Z)^{∗} be the smallest generator of the ith factor (Z/p_{i}^{e_{i}}Z)^{∗}. If p_{1} = 2, let a_{1} and a_{2} correspond to the two factors ⟨−1⟩ and ⟨5⟩, respectively; then a_{3} corresponds to p_{2}, etc. Here a_{i} is smallest in the sense that the minimal lift ã_{i} ∈ Z_{>0} is smallest. Let n be the exponent of (Z/NZ)^{∗}, and let ζ = e^{2πi/n} ∈ C^{∗}. To give ε is the same as giving the images of each generator of a_{i} as a power of ζ. We thus represent ε as a vector of elements of C^{∗} with respect to a canonically chosen, but unnatural, basis.

Alternatively, the vector representing a character ε can be equivalently viewed as a vector in (Z/nZ)^{r}, where again n is the exponent of (Z/NZ)^{∗}. Such a vector represents a character if and only if the ith component of the vector has additive order dividing φ(p_{i}^{e_{i}}). If p_{1} = 2, then there are r + 1 entries instead of r entries, and the condition is suitably modified. If a vector v = [d_{1}, …, d_{r}] represents a character ε, then each of the Galois conjugate characters is represented by [md_{1}, …, md_{r}] where m varies over elements of (Z/nZ)^{∗}.

When performing actual machine computations, we work in the smallest field that contains all of the values of ε. Thus if d = gcd(d_{1}, …, d_{r}, n), then we work in the subfield Q(ζ^{d}), which is cheaper than working in Q(ζ).

It is sometimes important to work in characteristic ℓ. Then the notation is as above, except ζ is replaced by a primitive mth root of unity, where m is the prime-to-ℓ part of n. Note that the primitive nth roots of unity in characteristic ℓ need not be conjugate; for example, both 2 and 3 are square roots of −1 in F_{5}, but they are not conjugate. Thus we must specify ζ as part of the notation when giving a mod ℓ Dirichlet character.

Example 3.11. Suppose N = p is an odd prime. The group of mod p Dirichlet characters (in characteristic 0) is isomorphic to Z/(p − 1)Z, and two characters a and b are Galois conjugate if and only if there is an element x ∈ (Z/(p−1)Z)^{∗} such that xa = b. A character is determined up to Galois conjugacy by its order, so the set of classes of mod p Dirichlet characters are in bijection with the set of divisors d of p − 1 = #(Z/pZ)^{∗}.

Let p be an odd prime. The quadratic mod p character is denoted [(p − 1)/2]. The quadratic mod 2p character is denoted by [0, 0, (p − 1)/2]; the quadratic mod 4p character is denoted [(p − 1)/2, 0, (p − 1)/2]. If n ≥ 3, then the exponent of (Z/2^{n}Z)^{∗} is 2^{n−2}, so the nontrivial mod 2^{n} character that factors through (Z/4Z)^{∗} is denoted [2^{n−3}, 0].

Definition 3.12. The conductor of a character ε : (Z/NZ)^{∗} → C^{∗} is the smallest divisor M of N such that ε factors through the natural reduction map (Z/NZ)^{∗} → (Z/MZ)^{∗}.

For simplicity, we assume that N is odd. To compute the conductor of ε, let v be the vector in (Z/nZ)^{r} that represents ε, as above. Since both (Z/p_{i}^{e_{i}}Z)^{∗} and (Z/p_{i}^{d}Z)^{∗} are cyclic and the reduction map is surjective, we find that p_{i}^{d}, with d ≤ e_{i}, divides the conductor of ε if and only if the ith component of v has additive order dividing φ(p_{i}^{d}). We can thus compute the power of p_{i} dividing the conductor of ε by computing the smallest d such that p_{i}^{d} ≡ p_{i}^{d−1} modulo the order of the ith component of v.

3.4 The dimension of S_{k}(N, ε)

An explicit formula for the dimension of S_{k}(N, ε) is given in [13], without proof. For the reader’s convenience, we reproduce it here.

Theorem 3.13 (Cohen-Oesterlé). Let k ≥ 2 be an integer and ε : (Z/NZ)^{∗} → C^{∗} be a Dirichlet character such that ε(−1) = (−1)^{k}. Then

dim S_{k}(N, ε) = δ + \frac{k − 1}{12} · N · ∏_{p|N} (1 + \frac{1}{p}) − \frac{1}{2} · ∏_{p|N} λ(r_{p}, s_{p}, p)
+ γ_{k} ∑_{{x∈(Z/NZ)^{∗}: x^{2}+1=0}} ε(x)  +  μ_{k} ∑_{{x∈(Z/NZ)^{∗}: x^{2}+x+1=0}} ε(x).

Let f be the conductor of ε, i.e., the smallest M such that ε factors through (Z/MZ)^{∗}. If p | N, then r_{p} (resp. s_{p}) denotes the exponent of p in the prime factorization of N (resp. f). Furthermore,

λ(r_{p}, s_{p}, p) := \begin{cases}p^{r′} + p^{r′−1} & if 2s_{p} ≤ r_{p} = 2r′ \\ 2p^{r′} & if 2s_{p} ≤ r_{p} = 2r′ + 1 \\ 2p^{r_{p}−s_{p}} & if 2s_{p} > r_{p} \end{cases}

γ_{k} := \begin{cases}0 & if k is odd \\ −\frac{1}{4} & if k ≡ 2 (mod 4) \\ \frac{1}{4} & if k ≡ 0 (mod 4) \end{cases}

μ_{k} := \begin{cases}0 & if k ≡ 1 (mod 3) \\ −\frac{1}{3} & if k ≡ 2 (mod 3) \\ \frac{1}{3} & if k ≡ 0 (mod 3) \end{cases}

δ := \begin{cases}1 & if k = 2 and ε is trivial \\ 0 & otherwise \end{cases}

3.5 Decomposing the space of modular symbols

Consider the space S_{k}(N, ε) of cuspidal modular symbols of level N and character ε over K = Q(ε). In this section we describe how to decompose the new part of S_{k}(N, ε) as a direct sum of T-modules corresponding to the Galois conjugacy classes of newforms with character ε. As an application, we can compute the q-expansions of the normalized cuspidal newforms of level N and character ε. Using the theory of Atkin-Lehner [4] as extended by Li [37], it is then possible to construct a basis for the space S_{k}(N, ε; C) of cusp forms.

The algorithm is, for the most part, a straightforward generalization of the method used by Cremona [16] to enumerate the Q-rational weight-two newforms corresponding to modular elliptic curves. Nevertheless, we present several tricks learned in the course of doing computations, which speed up the algorithm. One useful trick that Cremona also made use of is to work in the space dual to modular symbols as described in the next section.

3.5.1 Duality

Let K = Q[ε], and let S_{k}(N, ε; K)^{⊥} denote Hom_{K}(S_{k}(N, ε; K), K) equipped with its natural right T-action: for φ ∈ S_{k}(N, ε; K)^{⊥},

(φT)(x) = φ(T x).

The natural pairing

⟨ , ⟩ : S_{k}(N, ε; K)^{⊥} × S_{k}(N, ε; K) → K  (3.1)

given by ⟨φ, x⟩ = φ(x) satisfies ⟨φT, x⟩ = ⟨φ, T x⟩.

Viewing the elements T ∈ T as sitting inside End(S_{k}(N, ε; K)), the transpose map T ↦ T^{t} allows us to view S_{k}(N, ε; K)^{⊥} as a left T-module.

Proposition 3.14. Let V ⊂ S_{k}(N, ε; K)^{new} be an irreducible new T-submodule and set I = Ann_{T} V. Then the characteristic polynomial of each T_{p} on S_{k}(N, ε; K)^{⊥}[I] is the same as the characteristic polynomial of T_{p} on V.

Proof. We may assume for the purposes of proving the proposition that K = \overline{Q}. There is a basis of simultaneous T-eigenvectors for S_{k}(N, ε; K)^{new}. With respect to this basis, T acts via diagonal matrices. The systems of eigenvalues coming from the old subspace are distinct from the systems of eigenvalues on the new space. Thus the dimension of S_{k}(N, ε; K)^{⊥}[I] is the same as the dimension of V, instead of being too large. The proposition now follows by noting that the characteristic polynomial of a matrix is the same as the characteristic polynomial of its transpose.

The degeneracy maps α_{t} and β_{t} of Section 2.5 give rise to maps α_{t}^{⊥} and β_{t}^{⊥} between the dual spaces and having the dual properties to those of α_{t} and β_{t}. In particular, they commute with the Hecke operators T_{p} for p prime to N. The new and old subspace of S_{k}(N, ε; K)^{⊥} are defined as in Definition 2.16.

Algorithm 3.15. This algorithm computes a decomposition of S_{k}(N, ε; K)^{⊥ new} into irreducible submodules V.

Using Algorithm 3.2 compute S_{k}(N, ε; K). Then compute the maps β_{t} using Algorithm 2.20 and intersect the transposes of their kernels in order to obtain S_{k}(N, ε)^{⊥ new}. Compute the boundary map δ : S_{k}(N, ε; K) → B_{k}(N, ε; K) using Algorithm 2.26. We cut out the cuspidal submodule S_{k}(N, ε; K)^{⊥ new} using the Hecke operators, Algorithm 3.17, and Proposition 3.14. Set p = 2 and perform the following steps.

1. Using Algorithm 3.17, compute a matrix A representing the Hecke operator T_{p} on S_{k}(N, ε; K)^{⊥ new}.

2. Compute and factor the characteristic polynomial F of A.

3. For each irreducible factor f of F compute V_{f} = ker(f(A)). Then, compute the +1 and −1 eigen-subspaces V_{f}^{+} and V_{f}^{−} for the star involution. Let W denote one of these two eigen-subspaces, and use the following criteria to determine whether or not W is irreducible:

(a) If p is greater than the Sturm bound (see Theorem 3.9) then W must be irreducible.

(b) If the characteristic polynomial of some element T ∈ T acting on W is irreducible, then W is irreducible.

4. If W is irreducible, record W and consider the next factor of the characteristic polynomial in step 3. Otherwise, replace p by the next prime larger than p and replace S_{k}(N, ε; K)^{⊥ new} by W, then repeat the above sequence of steps, beginning with step 1.

3.5.2 Efficient computation of Hecke operators on the dual space

In this section we give a method for computing the action of the Hecke operators T_{p} ∈ T on an invariant subspace V ⊂ S_{k}(N, ε; K)^{⊥}. A naive way to compute the right action of T_{p} on V is to compute a matrix representing T_{p} on S_{k}(N, ε; K), transpose to obtain T_{p} on S_{k}(N, ε; K)^{⊥}, and then restrict to V using Gaussian elimination. To compute T_{p} on S_{k}(N, ε; K), observe that S_{k}(N, ε; K) has a basis e_{1},…, e_{n}, where each e_{i} is a Manin symbol [P, (c, d)], and that the action of T_{p} on [P, (c, d)] can be computed using Section 2.6.2.

In practice, d = dim V will often be much less than n; we now describe how to compute T_{p} on V in d/n of the time it takes using the above naive method. This is a substantial savings when d is small. Transposing the injection V ↪ S_{k}(N, ε; K)^{⊥}, we obtain a surjection S_{k}(N, ε; K) → V^{⊥}. There exists a subset e_{i_{1}},…, e_{i_{d}} of the e_{i} whose image forms a basis for V^{⊥}. With some care, it is then possible to compute T_{p} on V^{⊥} by computing T_{p} on each of e_{i_{1}},…, e_{i_{d}}.

In the rest of this section, we describe in terms of matrices a definite way to carry out this computation. Let V be an n × m matrix whose rows generate an n-dimensional subspace of an m-dimensional space of row vectors. Let T be an m×m-matrix and suppose that V has rank n and that V T is contained in the row space of V. Let E be an m × n matrix with the property that the n × n matrix V E is invertible, with inverse D.

Proposition 3.16. V T = V T EDV.

Proof. Observe that

V (EDV) = (V ED)V = IV = V.

Thus right multiplication by EDV

v ↦ vEDV

induces the identity map on the row space of V. Since V T is contained in the row space of V, we have

(V T)EDV = V T,

as claimed.

We have not computed T, but we can compute T on each basis element e_{1}, …, e_{d} of the ambient space–unfortunately, d is extremely large. Our problem: quickly compute the action of T^{t} on the invariant subspace spanned by the rows of V. Can this be done without having to compute T on all e_{i}? Yes, the following algorithm shows how using a subset of only n = dim V of the e_{i}.

Algorithm 3.17. Let T be any linear transformation which leaves V invariant and for which we can compute T (e_{i}) for i = 1, …, d. This algorithm computes the matrix representing the action of T on V while computing T (e_{i}) for only dim V of the i.

Choose any m × n matrix E whose columns are sparse linear combinations of the e_{i} and such that V E is invertible. For this we find a set of positions so that elements of the space spanned by the columns of V are determined by the entries in these spots. This is accomplished by row reducing, and setting E equal to the pivot columns. Using Gaussian elimination, compute the inverse D of the n × n matrix V E. The matrix representing the action of T with respect to V is then

V (T E)D = V (T E)(V E)^{−1}.

Proof. Let A be any matrix so that V A is the n × n identity matrix. By the proposition we have

V T A = (V T EDV)A = V T ED(V A) = V T ED = V (T E)D.

To see that V T A represents T, observe that by the proposition,

V T AV = (V T EDV)AV = (V T EDVA)V
= (V T ED)(VA)V = (V T ED)V = V T

so that V T A gives the correct linear combination of the rows of V.

3.5.3 Eigenvectors

Once a T-simple subspace of S^{∗} has been identified, the following algorithm, which was suggested to the author by H. Lenstra, produces an eigenvector defined over an extension of the base field.

Algorithm 3.18. Let A be an n × n matrix over an arbitrary field K and suppose that the characteristic polynomial f(x) = x^{n} + ⋯ + a_{1}x + a_{0} of A is irreducible. Let α be a root of f(x) in an algebraic closure \overline{K} of K. Factor f(x) over K(α) as f(x) = (x − α)g(x). Then for any element v ∈ K^{n} the vector g(A)v is either 0 or it is an eigenvector of A with eigenvalue α. The vector g(A)v can be computed by finding Av, A(Av), A(A(Av)), and then using that

g(x) = x^{n−1} + c_{n−2}x^{n−2} + ⋯ + c_{1}x + c_{0},

where the coefficients c_{i} are determined by the recurrence

c_{0} = −a_{0}/α, c_{i} = (c_{i−1} − a_{i})/α.

We will prove below that g(A)v ≠ 0 for all vectors v not in a proper subspace of K^{n}. Thus with high probability, a “randomly chosen” v will have the property that g(A)v ≠ 0. Alternatively, if v_{1}, … v_{n} form a basis for K^{n}, then g(A)v_{i} must be nonzero for some i.

Proof. By the Cayley-Hamilton theorem [36, XIV.3] we have that f(A) = 0. Consequently, for any v ∈ K^{n}, we have (A − α)g(A)v = 0 so that Ag(A)v = αv. Since f is irreducible it is the polynomial of least degree satisfied by A and so g(A) ≠ 0. Therefore g(A)v ≠ 0 for all v not in the proper closed subset ker(g(A)).

3.5.4 Eigenvalues

In this section we give an algorithm for computing the q-expansion of one of the newforms corresponding to a factor of S_{k}(N, ε; K)^{new}. This is a generalization of the algorithm described in [16, §2.9].

Algorithm 3.19. Given a factor V ⊂ S_{k}(N, ε; K)^{⊥ new} as computed by Algorithm 3.15 this algorithm computes the q-expansion of one of the corresponding Galois conjugate newforms.

1. Using Algorithm 3.17 compute the action of the ∗-involution (Section 2.4) on V. Then compute the +1 eigenspace V^{+} ⊂ V.

2. Find an element T ∈ T such that the characteristic polynomial of the matrix A of T acting on V^{+} is irreducible. Such a T must exist by the primitive element theorem [36, V.4]. (Note: It is not always the case that T can be taken to equal some Hecke operator T_{n}. The first example with k = 2 and ε = 1 occurs at level N = 512.)

3. Using Algorithm 3.5.3 compute an eigenvector e for A over an extension of K.

4. Because e is an eigenvector and the pairing given in Equation 3.1 respects the Hecke action, we have that for any Hecke operator T_{n} and element w ∈ S_{k}(N, ε; K), that

a_{n}⟨e, w⟩ = ⟨eT_{n}, w⟩ = ⟨e, T_{n}w⟩.

Choose w so that ⟨e, w⟩ ≠ 0. Then

a_{n} = \frac{⟨e, T_{n}w⟩}{⟨e, w⟩}.

The a_{n} can now be computed by computing ⟨e, w⟩ once and for all, and then computing ⟨e, T_{n}w⟩ for each n. It is best to choose w in such a way that T_{n}w can be computed quickly.

The beauty of this algorithm is that when w is a Manin symbol [P (X, Y), (c, d)] the computation of T_{p}w = ∑_{x∈R_{p}} wx is very quick, requiring us to only sum over the Heilbronn matrices of determinant p once.

In practice we compute only the eigenvalues a_{p} using the above algorithm, then use the following recurrences to obtain the a_{n}:

a_{nm} = a_{n}a_{m}  if (n, m) = 1, and
a_{p^{r}} = a_{p^{r−1}}a_{p} − ε(p)p^{k−2}a_{p^{r−2}}.

3.5.5 Sorting and labeling eigenforms

Systematically ordering the factors is essential, so that we can later refer to them. In Section 3.5.4 we saw how to associate to each new factor a sequence a_{n} of Hecke eigenvalues. These can be used to sort the factors.

Except in the case of weight 2 and trivial character, we use the following ordering. To each eigenvector associate the following sequence of integers

tr(a_{1}), tr(a_{2}), tr(a_{3}), tr(a_{4}), tr(a_{5}), tr(a_{6}), …

where the trace is from K_{f} = Q(… a_{n} …) down to Q. Sort the eigenforms by ordering the sequences in dictionary order with minus coming before plus. Since we included tr(a_{1}), this ordering gathers together factors of the same dimension. Furthermore, the sequence of traces determines the Galois conjugacy class of f, because the g = ∑_{n≥1} tr(a_{n})q^{n} is the trace of f, hence g lies in the C-vector space spanned by the Galois conjugates of f.

When k = 2 and the character is trivial we use a different and somewhat complicated ordering because it extends the notation for elliptic curves that was introduced in the second edition of [16] and has since become standard. Sort the factors of S_{k}(N, ε)^{new} as follows. First by dimension, with smallest dimension first. Within each dimension, sort in binary order, by the signs of the Atkin-Lehner involutions with − corresponding to 0 and + to 1. For example, if there are three Atkin-Lehner involutions then the sign patterns are sorted as follows:

+ + +, − + +, + − +, − − +, + + −, − + −, + − −, − − −.

Finally, let p be the smallest prime not dividing N. Within each of the Atkin-Lehner classes, sort by the magnitudes of the K_{f}/Q-trace of a_{p} breaking ties by letting the positive trace be first. If there are still any ties, repeat the final step with the next smallest prime not dividing N, etc. (Note: It’s not clear to the author that ties will always eventually be broken, though in his computation they always have been.)

3.6 Intersections and congruences

Consider a complex torus J = V/Λ, and let A = V_{A}/Λ_{A} and B = V_{B}/Λ_{B} be subtori whose intersection A ∩ B is finite.

Proposition 3.20. There is a natural isomorphism of groups

A ∩ B ≅ (\frac{Λ}{Λ_{A} + Λ_{B}} )_{tor .}

Proof. There is an exact sequence

0 → A ∩ B → A ⊕ B → J.

Consider the diagram

\xymatrix{ & Λ_{A} ⊕ Λ_{B} \ar[d] \ar[r] & Λ \ar[r] \ar[d] & Λ/(Λ_{A} + Λ_{B}) \ar[d] \\  & V_{A} ⊕ V_{B} \ar[d] \ar[r] & V \ar[r] \ar[d] & V/(V_{A} + V_{B}) \ar[d] \\ A ∩ B \ar[r] & A ⊕ B \ar[r] & J \ar[r] & J/(A + B).}

Figure 3.1: T-shirt design

The snake lemma gives an exact sequence

0 → A ∩ B → Λ/(Λ_{A} + Λ_{B}) → V/(V_{A} + V_{B}).

Since V/(V_{A} + V_{B}) is a C-vector space, the torsion part of Λ/(Λ_{A} + Λ_{B}) must map to 0. No non-torsion in Λ/(Λ_{A} + Λ_{B}) could map to 0, because if it did then A ∩ B would not be finite. The lemma follows.

The following formula for the intersection of n subtori is obtained in a similar way.

Proposition 3.21. For i = 1, …, n let A_{i} = V_{i}/Λ_{i} be a subtorus of J = V/Λ, and assume that each pairwise intersection A_{i} ∩ A_{j} is finite. Then

A_{1} ∩ ⋯ ∩ A_{n} ≅ (\frac{Λ ⊕ ⋯ ⊕ Λ}{f(Λ_{1} ⊕ ⋯ ⊕ Λ_{n})}) ,

where f(x_{1}, …, x_{n}) = (x_{1} − x_{2}, x_{2} − x_{3}, x_{3} − x_{4}, …, x_{n−1} − x_{n}).

Remark 3.22. Using this proposition the author constructed the T-shirt design in Figure 3.1.

Example 3.23. L. Kilford of London, England has recently discovered an example at prime level 503 in which “multiplicity one” fails. One verification of his example uses the above proposition. Let E_{1}, E_{2}, and E_{3} be the three elliptic curves of conductor 503, and for each i = 1, 2, 3, let 𝔪_{i} be the maximal ideal of T ⊂ End(J_{0}(503)) generated by 2 and all T_{p} − a_{p}(E_{i}), with p prime. Each of the Galois representations E_{i}[2] is irreducible, and one can check that 𝔪_{1} = 𝔪_{2} = 𝔪_{3}. If multiplicity one holds, then E_{1}[2] = E_{2}[2] = E_{3}[2] inside of J_{0}(503). However, this is not the case, as a modular symbols computation in the integral homology H_{1}(X_{0}(N), Z) reveals that E_{1} ∩ E_{2} = {0}.

3.6.1 A strategy for computing congruences

Let N be a positive integer, k ≥ 2 an integer, and ε a mod N Dirichlet character. Suppose f and g are newforms in S_{k}(N, ε; \overline{Q}). The following proposition gives rise to an algorithm for computing most congruences between infinite Fourier expansions.

The advantage of the algorithm is that it only involves finite exact computations and does not rely on the computation of q-expansions. A disadvantage is that congruences between q-expansions need not be reflected by the corresponding modular symbols, so the proposition need not give all congruences. This is illustrated in Example 3.23.

The author first learned about this strategy from the section entitled “First strategy: Computing m-congruences of period lattices” in [18].

Proposition 3.24. Suppose f and g are newforms in S_{k}(N, ε; \overline{Q}). Let I_{f} and I_{g} be the corresponding annihilators in the Hecke algebra T. Let Λ = S_{k}(N, ε; O), and set Λ_{f} = Λ[I_{f}] and Λ_{g} = Λ[I_{g}]. If p | # (\frac{Λ}{Λ_{f} +Λ_{g}} )_{tor} then there is a prime ℘ of residue characteristic p such that f ≡ g (mod ℘).

Proof. Consider the exact sequence

0 → Λ_{f} ⊕ Λ_{g} → Λ → Λ/(Λ_{f} + Λ_{g}) → 0

where the first map is (a, b) ↦ a − b. Upon tensoring this sequence with F_{p} we obtain:

Z → (Λ_{f} ⊗ F_{p}) ⊕ (Λ_{g} ⊗ F_{p}) → Λ ⊗ F_{p} → (Λ/(Λ_{f} + Λ_{g})) ⊗ F_{p} → 0,

where Z = Tor^{1}(Λ/(Λ_{f} + Λ_{g}), F_{p}). Denote by im(Λ_{f}) the image of Λ_{f} ⊗ F_{p} in Λ ⊗ F_{p} and likewise for Λ_{g}. Our assumption that p divides the torsion part of Λ/(Λ_{f} + Λ_{g}) implies that Z is nonzero, so im(Λ_{f}) and im(Λ_{g}) have nonzero intersection inside the F_{p}-vector space Λ⊗F_{p}. The Hecke algebra T acts on im(Λ_{f}) through its action on f, that is, through the quotient T/I_{f}; similarly, T acts on im(Λ_{g}) through T/I_{g}. Thus T acts on the nonzero T ⊗ F_{p}-module im(Λ_{f}) ∩ im(Λ_{g}) through T/(I_{f} + I_{g} + p). This implies that I_{f} + I_{g} + p is not the unit ideal, which is equivalent to the assertion of the proposition.

3.7 The rational period mapping

Consider a triple (N, k, ε), and let K = Q[ε]. Let I be an ideal in the Hecke algebra T associated to (N, k, ε). The rational period mapping associated to I is a map from the space M_{k}(N, ε; K) of modular symbols to a finite dimensional K-vector space. It is a computable analogue of the classical integration pairing, and is of great value in extracting the rational parts of analytic invariants; e.g., of special values of L-functions. In the next section we use it to compute the image of cuspidal points on J(N, k, ε).

Definition 3.25. Let D := Hom_{K}(M_{k}(N, ε; K), K)[I]; the rational period mapping is the natural quotient map

Θ_{I} : M_{k}(N, ε; K) → \frac{M_{k}(N, ε; K)}{⋂ \{ker(φ) : φ ∈ D\}}.

If f ∈ S_{k}(N, ε) is a newform, set Θ_{f} := Θ_{I_{f}} where I_{f} is the annihilator of f in the Hecke algebra.

Algorithm 3.26. This algorithm computes Θ_{I}. Choose a basis for W = M_{k}(N, ε; K) and use it to view W as a space of column vectors equipped with a left action of T. View W^{∗} = Hom_{K}(M_{k}(N, ε; K), K) as the space of row vectors of length equal to dim M_{k}(N, ε; K); thus W^{∗} is dual to W via the natural pairing between row and column vectors. The Hecke operators act on W^{∗} on the right. Compute a basis φ_{1}, …, φ_{n} for the K-vector space W^{∗}[I]. Then the rational period mapping with respect to this basis is φ_{1} × ⋯ × φ_{n}; it is given by the matrix whose rows are φ_{1}, …, φ_{n}.

Proof. The kernels of φ_{1} × ⋯ × φ_{n} and Θ_{I} are the same.

Example 3.27. Let I be the annihilator of the newform f = q − 2q^{2} + ⋯ ∈ M_{2}(37, 1; Q) corresponding to the elliptic curve 37k2A. There is a basis for W = M_{2}(37, 1; Q) such that

T_{2} = (\begin{matrix}−1 & 1 & 1 & −1 & 0 \\ 1 & −1 & 1 & 0 & 0 \\ 0 & 0 & −2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 3 \end{matrix})

The characteristic polynomial of T_{2} is x^{2}(x+2)^{2}(x−3). Thus W [I] = ker(T_{2} +2) is spanned by the column vectors (1, −1, 0, 1/2, 0)^{t} and (0, 0, 1, −1/2, 0)^{t}, and W^{∗}[I] = ker(T_{2}^{t} + 2) is spanned by the row vectors (1, 0, −1, 0, 0) and (0, 1, −1, 0, 0). The rational period mapping is Θ_{I}((a, b, c, d, e)^{t}) = (a − c, b − c).

Lemma 3.28.

dim M_{k}(N, ε; K)[I] = dim Hom_{K}(M_{k}(N, ε; K), K)[I].

Proof. Let W = M_{k}(N, ε; K) and W^{∗} be its dual. Let a_{1}, …, a_{n} be a set of generators for I. Choose a basis for W that is compatible with the following filtration:

0 ⊂ (ker(a_{1}) ∩ ⋯ ∩ ker(a_{n})) ⊂ (ker(a_{1}) ∩ ⋯ ∩ ker(a_{n−1})) ⊂ ⋯ ⊂ ker(a_{1}) ⊂ W.

The rank of a matrix equals the rank of its transpose, so the dimension of ker(a_{1}) is the same as the dimension of ker(a_{1}^{t}), that is, dim W [(a_{1})] = dim W^{∗}[(a_{1})]. Since T is commutative, a_{2} leaves ker(a_{1}) invariant; because of how we chose our basis for W, the transpose of a_{2}|_{ker(a_{1})} is a_{2}^{t}|_{ker(a_{1}^{t})}. Thus again, dim(ker(a_{2}|_{ker(a_{1})})) equals dim(ker(a_{2}^{t}|_{ker(a_{1}^{t})})). Proceeding inductively, we prove the lemma.

Corollary 3.29. Suppose M_{k}(N, ε; K)[I] ⊂ S_{k}(N, ε; K), and let P : M_{k}(N, ε; K) → Hom_{C}(S_{k}(N, ε; C)[I], C) be the classical period map induced by the integration pairing. Then ker(P) = ker(Θ_{I}).

Proof. Since P (M_{k}(N, ε; O)) is known to be a finite-covolume O-lattice in the complex vector space Hom_{C}(S_{k}(N, ε; C)[I], C), the K-dimension of im(P) equals 2·dim_{C} S_{k}(N, ε; C)[I], which in turn equals dim_{K} M_{k}(N, ε; K)[I]. Thus by Lemma 3.28 the images im(P) and im(Θ_{I}) have the same dimension, hence ker(P) and ker(Θ_{I}) also have the same dimension. It thus suffices to prove the inclusion ker(Θ_{I}) ⊂ ker(P). Suppose Θ_{I}(x) = 0; then φ(x) = 0 for all x ∈ W^{∗}[I], where W = M_{k}(N, ε; K). Thus φ(x) = 0 for all φ ∈ (W ⊗ C)^{∗}[I]. Since the integration pairing that defines P respects the action of T, the composition of P with any linear functional lies in (W ⊗ C)^{∗}[I]. Thus P (x) = 0, as required.

3.8 The images of cuspidal points

Consider a triple (N, k, ε), and let K = Q[ε]. Recall that integration defines a period mapping

P : M_{k}(N, ε; K) → Hom_{C}(S_{k}(N, ε; C), C).

A cuspidal point of

J = J(N, k, ε) := \frac{Hom_{C}(S_{k}(N, ε; C), C)}{P (S_{k}(N, ε; O))}

is a point that is in the image under P of M_{k}(N, ε; O). It is of great interest to compute the structure of the cuspidal subgroup of J and of the quotients of J. For example, when k = 2 and ε = 1, the torus J can be identified with J_{0}(N)(C). In this case, Manin proved (see [38]) that the cuspidal point {0, ∞} is a torsion point in J_{0}(N)(Q), so its order gives a lower bound on J_{0}(N)(Q)_{tor}.

Algorithm 3.30 (Cuspidal subgroup). Let I be an ideal in the Hecke algebra T. This algorithm computes the cuspidal subgroup of the quotient A_{I} of J. Using Algorithm 3.5 compute M_{k}(N, ε; O) and S_{k}(N, ε; O). Using Algorithm 3.26, compute the rational period mapping Θ_{I}. Then the cuspidal subgroup is the subgroup of Θ_{I}(S_{k}(N, ε; O)) generated by the elements Θ_{I}(x) for x ∈ M_{k}(N, ε; O). In particular, the point of A_{I}(C) corresponding to X^{i}Y^{k−2−i}{α, β} is the image of Θ_{I}(X^{i}Y^{k−2−i}{α, β}) in the quotient of Θ_{I}(M_{k}(N, ε; O)) by Θ_{I}(S_{k}(N, ε; O)).

Example 3.31. This example continues Example 3.27. The basis chosen is also a basis for M_{2}(37, 1; Z), so by computing the boundary map, or the integer kernel of T_{2}(T_{2} + 2), we find that S_{2}(37, 1; Z) is spanned by (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), and (0, 0, 0, 1, 0). Thus Θ_{I}(S_{2}(37, 1; Z)) is generated by (1, 0) and (0, 1). The modular symbols {0, ∞} is represented by (0, 0, 0, 0, −1), so the image of the cusp (0) − (∞) ∈ J_{0}(37) is 0 in 37k2A.

The rational period mapping associated to 37k2B (with respect to some basis) is

Θ_{I}((a, b, c, d, e)^{t}) = (a − c − 2d + \frac{2}{3}e, b + c + 2d − \frac{2}{3}e).

Thus Θ_{I}(S_{2}(37, 1; Z)) is generated by (1, 0) and (0, 1). The image of \{0, ∞\} is is \frac{2}{3}(1, −1), so the image of (0) − (∞) in 37k2B has order 3.

3.8.1 Rational torsion

Let f be a newform of weight 2, and suppose ε = 1. Manin proved that (0) − (∞) defines an element of J_{0}(N)(Q)_{tor}. Thus the order of the image of (0) − (∞) provides a lower bounds on #A_{f}(Q)_{tor}. In general, many other points in the cuspidal subgroup can be rational. Determining which would give a better lower bound on the rational subgroup; the author has not yet carried out such computations (see, however, [65]).

3.8.2 Upper bound on torsion: Counting points mod p

Let f be a newform of weight 2, and suppose ε = 1. The Hecke algebra T acts through a quotient \overline{T} on the subspace of S_{2}(Γ_{0}(N)) spanned by the Galois conjugates of f. Let χ_{p}(X) be the characteristic polynomial of the image of T_{p} in \overline{T}. Suppose p ∤ N and let N_{p} = #A_{f}(F_{p}) be the number of points on the mod p reduction of the abelian variety A_{f}.

Proposition 3.32. For each prime p not dividing N,

N_{p} = χ_{p}(p + 1).

Proof. This is probably well-known, but we give a proof (which was suggested to the author by Matt Baker). It follows from the Eichler-Shimura theorem that the following relation holds in the endomorphism ring of A_{f}/F_{p}:

T_{p} = Frob + Ver = Frob +p/ Frob .

Let ℓ ≠ p be a prime. If the characteristic polynomial of Frob on an ℓ-adic Tate module of A_{f}/F_{p} is F (t), and the characteristic polynomial of T_{p} on differentials H^{0}(A_{f}/F_{p}, Ω) is f(t), then we have f(t) = x^{−d}F (x), where t = x + (p/x) and d = dim A_{f}. In other words, the relation above gives an easy conversion between f and F. Since it’s a general fact that #A_{f}(F_{p}) = F (1), we have #A_{f}(F_{p}) = f(p + 1).

The following theorem is proved using formal groups.

Theorem 3.33. Let A be an abelian variety over Q, with good reduction outside N. Suppose p ∤ N. Then the kernel of the reduction map A(Q)_{tor} → A(F_{p}) is killed by p. If p > 2 then the kernel is trivial.

By taking gcd’s we obtain an upper bound on #A(Q)_{tor}. This upper bound is not in general sharp; in fact, it is unchanged if A is replaced by any isogenous abelian variety. For example, X_{0}(11) and X_{1}(11) are isogenous, but have different torsion subgroups.

3.9 The modular degree

Let f be a newform of level N, weight k ≥ 2 and character ε such that ε^{2} = 1. In this section we define and give an algorithm to compute the modular degree of the torus A_{f} attached to f.

Definition 3.34. The modular map is the map θ_{f} : A_{f}^{∨} → A_{f} that is induced by the bottom row of Diagram 2.1 on page 34. The modular degree m_{f} of f (or of A_{f}) is the degree of this map. If f has weight two, then θ_{f} is a polarization so by [50, Thm. 13.3] its degree is a perfect square; in this case we instead define the modular degree m_{f} to be the positive square root of the degree of θ_{f}.

Remark 3.35. When E/Q be a modular elliptic curve of conductor N that is an optimal quotient of J_{0}(N), then m_{f} is the usual modular degree, which is the least degree of a map X_{0}(N) → E.

Remark 3.36. When k ≠ 2, the degree of θ_{f} need not be a perfect square. For example, there is a one-dimensional quotient A_{f} associated to the unique rational newform

f = q + 2q^{2} − 8q^{3} + 4q^{4} + 5q^{5} − 16q^{6} − 4q^{7} + ⋯ ∈ S_{4}(10)

such that the kernel of θ_{f} is isomorphic to Z/10Z.

Next, for a newform f let θ′_{f} be the part of # ker(θ_{f}) that is coprime to the level. There is a newform in f ∈ S_{4}(Γ_{0}(77)) such that θ′_{f} is not a perfect square at 2. For identification purposes, we remark that the field generated by the Fourier coefficients of f has discriminant 2^{3} · 3^{3} · 2417.

Algorithm 3.37. Let I_{f} be the annihilator of f in the Hecke algebra. The modular kernel ker(θ_{f}) is isomorphic to the cokernel of the natural map S[I_{f}] → Φ_{f}(S) of Diagram 2.1 on page 34. This cokernel can be computed by replacing Φ_{f} by the rational period map Θ_{I_{f}}.

Proof. For concreteness, we give the proof only in the case of weight-two and trivial character. The proof in the general case is similar. Let S = S_{2}(Γ_{0}(N), C) be the complex vector space of weight-two modular forms of level N, and set H = H_{1}(X_{0}(N), Z). The integration pairing S × H → C induces a natural map

Φ_{f} : H → Hom(S[I_{f}], C).

Using the classical Abel-Jacobi theorem, we deduce the following commutative diagram, which has exact columns, but whose rows are not exact.

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ H[I_{f}] \ar[d] \ar[r] & H \ar[d] \ar[r] & Φ_{f}(H) \ar[d] \\ Hom(S, C)[I_{f}] \ar[d] \ar[r] & Hom(S, C) \ar[d] \ar[r] & Hom(S[I_{f}], C) \ar[d] \\ A_{f}^{∨}(C) \ar[d] \ar[r] \ar[rr] & J_{0}(N)(C) \ar[d] \ar[r] & A_{f}(C) \ar[d] \\ 0 & 0 & 0}

By the snake lemma, the kernel of A_{f}^{∨}(C) → A_{f}(C) is isomorphic to the cokernel of the map H[I_{f}] → Φ_{f}(H), which proves the proposition.

Remark 3.38. Suppose E is an optimal quotient of J_{0}(p), with p prime. The surjectivity result in [48] implies that it is possible to efficiently compute the modular degree using only the method of graphs. For more details, see Chapter 4.

3.10 The rational part of L(A_{f}, j)

Let k ≥ 2 be an integer, and let ε : (Z/NZ)^{∗} → C^{∗} be a Dirichlet character such that ε^{2} = 1. This assumption on ε is made only for simplicity; there is no fundamental obstruction to considering arbitrary characters. For the remainder of this section we fix a newform f ∈ S_{k}(N, ε). We will compute certain rational numbers associated to f.

The author was motivated to prove the results of this section after seeing Agashe’s results in the case k = 2 and ε = 1; see [2, Ch. 4].

3.10.1 L-functions

Definition 3.39. The L-series associated to f is the complex-analytic function

L(f, s) := ∑_{n=1}^{∞} a_{n}n^{−s}.

Hecke proved that L(f, s) has an analytic continuation to the whole complex plane. In particular, it makes sense to consider the values L(f, j) where j ∈ {1, 2, …, k − 1} is an integer in the “critical strip.” The general consesus is that these special values have deep arithmetic significance, in the sense that the quotients L(f, j)/ω_{f,j} should be algebraic numbers, where ω_{f,j} is an appropriate period of f, and that these algebraic numbers should encode deep arithmetic properties of the motive attached to f.

For simplicity, especially when doing explicit computations, it is desirable to work exclusively with ratios that are rational numbers instead of algebraic numbers. For this purpose, we consider instead the complex torus A_{f} attached to f, and introduce

L(A_{f}, s) := ∏_{i=1}^{d} L(f_{i}, s),

where f_{1}, …, f_{d} are the distinct Galois-conjugates of f. As we will see, L(A_{f}, j)/Ω_{j} ∈ Q, where Ω_{j} will be defined below.

Though the notation L(A_{f}, s) suggests that there might be a way to attach an Lfunction to a general complex torus, this is definitely not what we have in mind. For our present purposes, the notation L(A_{f}, s) is nothing more than a convenient shorthand for the product of the L-functions attached to the Galois conjugates of f. However, in the case when k = 2 and ε = 1, the L-function L(A_{f}, s) is known to be the canonical L-series associated to the abelian variety A_{f}/Q; see, e.g., the discussion in [20, Sec. 7].

3.10.2 Winding elements

Generalizing Mazur and Merel’s terminology when k = 2, we define winding elements as follows.

Definition 3.40 (Winding element). For 1 ≤ i ≤ k − 1, the ith winding element is

e_{i} := X^{i−1}Y^{k−2−(i−1)}{0, ∞} ∈ M_{k}(N, ε; Z).

For example, when k = 2 there is one winding element e = e_{1} = {0, ∞}. See [44, §2.2] for a topologically motivated discussion of the terminology “winding element.”

3.10. THE RATIONAL PART OF L(A_{F} , J) 53

3.10.3 Real and minus volumes

We briefly review the association of a complex torus to a Galois-conjugacy class of newforms. Consider the space S_{k}(N, ε; Z) of cusp forms in S_{k}(N, ε) whose q-expansion at infinity has Fourier coefficients which lie in Z. Let V be the C-vector space spanned by the Galois conjugates f_{1}, …, f_{d} of f, and choose a Z-basis g_{1}, …, g_{d} for the intersection V ∩S_{k}(N, ε; Z). Then integration via the pairing of Theorem 2.7 against g_{1}, …, g_{d} defines a map S_{k}(N, ε; Z) → C^{d} whose cokernel is A_{f}(C). Viewing A_{f}(C) in this way, the standard measure on C^{d} defines a measure on A_{f}(C).

Because ε^{2} = 1, the complex torus A(C) is equipped with an action of complex conjugation. There are two distinguished additive subgroups of A(C): the subgroup A(R) of elements fixed under complex conjugation, and the subgroup A(C)^{−} of elements sent to their additive inverse by complex conjugation. When j is odd, let Ω_{j} be the measure of the subgroup fixed under conjugation, and when j is even, let Ω_{j} be the measure of the subgroup sent to its inverse under conjugation, times i^{d}, where d is the dimension of A. When j is odd, we call Ω_{j} the real volume; otherwise, we call Ω_{j} the minus volume (see Definition 3.58).

3.10.4 The theorem

We are now prepared to state a theorem that gives a computable expression for the ratio |L(A_{f}, j)/Ω_{j}|. This theorem grew out of joint work with Agashe. It generalizes Cremona’s method for computation L(E, 1)/Ω_{E} when E is an elliptic curve (see [16, §2.8]).

As an immediate corollary of the formula, we see that |L(A_{f}, j)/Ω_{j}| is a rational number. This was already known when f ∈ S_{2}(Γ_{0}(N)) (see [28, §2]). The author remains ignorant as to whether or not the general corollary was known before, or even if the real numbers Ω_{j}, exactly as defined here, had been previously considered. However, rationality of certain related period ratios has been known for some time, due to work of Manin, Shimura, and Hatada. For a clear historical summary of these rationality results see Li’s MathSciNet review of [29]. See also [42, 44].

We take the absolute value of L(A_{f}, j)/Ω_{j} for simplicity only because at present we do not wish to worry about powers of the 4th root of unity i.

Theorem 3.41. Let f ∈ S_{k}(N, ε) be a newform, where k ≥ 2 and ε^{2} = 1, and let j ∈ {1, 2, …, k − 1} be an integer in the critical strip. Let σ = (−1)^{j−1}, and let Θ_{f} be the rational periopd mapping associated to f (see Definition 3.25). Then

|\frac{L(A_{f}, j)}{Ω_{j}} | = [Θ_{f}(S_{k}(N, ε; Z)^{σ}) : Θ_{f}(Te_{j})],

where S_{k}(N, ε; Z)^{σ} denotes the submodule of S_{k}(N, ε; Z) on which the ∗-involution acts as σ, and Ω_{j} is the real or minus volume of A_{f}, as in Section 3.10.3. The right hand expression in the formula is a lattice index, whose definition is given below.

Remark 3.42. In the context of the BSD conjecture, Ω_{A_{f}} = Ω_{1} ·c_{∞}, where c_{∞} is the number of connected components of A_{f}(R).

The theorem involves lattice indexes, which we define as follows.

Definition 3.43. Let V be a finite-dimensional vector space over R. A lattice L ⊂ V is a free abelian group of rank equal to the dimension of V such that RL = V. If L, M ⊂ V are lattices, the lattice index [L : M] ∈ R is the absolute value of the determinant of an automorphism of V taking L isomorphically onto M. For convenience we set [L : M] = 0 for any lattice L and additive abelian group M contained in V and of rank strictly smaller than dim V.

The following fact allows us to compute the lattice the index without using complex numbers.

Lemma 3.44. Suppose τ_{i} : V → W_{i}, i = 1, 2, are surjective linear maps such that ker(τ_{1}) = ker(τ_{2}). Let L and M be lattices in V such that τ_{i}(L) and τ_{i}(M) are both lattices for i = 1, 2. Then

[τ_{1}(L) : τ_{1}(M)] = [τ_{2}(L) : τ_{2}(M)].

Proof. Surjectivity and equality of kernels insures that there is a unique isomorphism ι : W_{1} → W_{2} such that ιτ_{1} = τ_{2}. Let σ be an automorphism of W_{1} such that σ(τ_{1}(L)) = τ_{1}(M). Then

ισι^{−1}(τ_{2}(L)) = ιστ_{1}(L) = ιτ_{1}(M) = τ_{2}(M).

Since conjugation does not change the determinant,

[τ_{2}(L) : τ_{2}(M)] = | det(ισι^{−1})| = | det(σ)| = [τ_{1}(L) : τ_{1}(M)].

Proof of Theorem 3.41. Let Φ = Φ_{f} be the period map M_{k}(N, ε; Z) → C^{d} defined by fixing a basis f_{1}, f_{2}, …, f_{d} of the conjugates of the newform f; thus

Φ(x) = (⟨f_{1}, x⟩, ⟨f_{2}, x⟩, … ⟨f_{d}, x⟩) ∈ C^{d}.

We view C^{d} as an algebra with unit element 1 = (1, …, 1) equipped with an action of the Hecke operators. The operator T_{p} acts as (a_{p}^{(1)}, …, a_{p}^{(d)}), where the components a_{p}^{(j)} are the Galois conjugates of a_{p}. Let Z^{d} ⊂ R^{d} ⊂ C^{d} be the standard submodules.

For brevity, set S = S_{k}(N, ε; Z). Let μ(Φ(S^{σ})) be the measure of a fundamental domain for the lattice Φ(S^{σ}); equivalently, μ(Φ(S^{σ})) is the absolute value of the determinant of a basis for Φ(S^{σ}). Observe that μ(Φ(S^{σ})) = [Z^{d} : Φ(S^{σ})] and |L(A_{f}, j)| = [Z^{d} : Φ(e_{j})Z^{d}].

Let W ⊂ C^{d} be the Z-module spanned by the “columns” of a basis for S_{k}(N, ε; Z)[I_{f}]. More precisely, if g_{1}, …, g_{d} is a basis, then the nth column is the vector (a_{n}(g_{1}), …, a_{n}(g_{d})), where a_{n}(g_{i}) is the coefficient of q^{n} in the q-expansion of g_{i} at infinity. Because Ω_{j} is computed with respect to a basis for S_{k}(N, ε; Z)[I_{f}],

μ(Φ(S^{σ})) = [W : T1] · Ω_{j}.

Observe that S_{k}(N, ε; Z) is saturated, in the sense that there are no nontrivial linear relations between the g_{i} when reduced modulo any prime p. To see this, note that if ∑ a_{i}g_{i} ≡ 0 (mod p), then \frac{1}{p} ∑ a_{i}g_{i} ∈ S_{k}(N, ε; Z) which, if the a_{i} are not all 0, is contrary to our assumption that g_{1}, …, g_{d} are a Z-basis. Because “row rank = column rank”, the same must be true for the “columns” defined in the previous paragraph, so [Z^{d} : W] = 1. It follows that [Z^{d} : T1] = [W : T1].

The following calculation combines together the above observations using properties of the lattice index:

[Φ(S^{σ}) : Φ(Te_{j})] = [Φ(S^{σ}) : Z^{d}] · [Z^{d} : Φ(Te_{j})]
= \frac{1}{[Z^{d} : Φ(S^{σ})]} · [Z^{d} : Φ(Te_{j})]
= \frac{1}{μ(Φ(S^{σ}))} · [Z^{d} : Φ(e_{j})Z^{d}] · [Φ(e_{j})Z^{d} : Φ(Te_{j})]
= \frac{|L(A_{f}, j)|}{μ(Φ(S^{σ}))} · [Φ(e_{j})Z^{d} : Φ(Te_{j})]
= \frac{|L(A_{f}, j)|}{μ(Φ(S^{σ}))} · [Φ(e_{j})Z^{d} : Φ(e_{j})T1]
= \frac{|L(A_{f}, j)|}{|Ω_{j}| · [W : T1]} · [Z^{d} : T1]
= \frac{|L(A_{f}, j)|}{|Ω_{j}|}.

Theorem 3.41 now follows by using Lemma 3.44, to replace Φ by Θ_{f}.

3.10.5 Bounding the denominator of the ratio

In this section we bound the denominators of the ratios appearing in the previous section. We begin with the following lemma, which follows easily from the alternative description of the boundary map given in Proposition 2.25.

Lemma 3.45. For j = 2, …, k − 2 the winding element e_{j} lies in S_{k}(N, ε; Z).

Proof. Recall that e_{j} = P (X, Y){0, ∞} where P (X, Y) = X^{j−1}Y^{k−2−(j−1)}. Since 2 ≤ j ≤ k − 2, it follows that P (1, 0) = P (0, 1) = 0, so Propositison 2.25 implies that e_{j} maps to 0 under the boundary map.

Proposition 3.46. For j = 2, …, k − 2,

\frac{L(A_{f}, j)}{Ω_{j}} ∈ Z.

Proof. This follows from Theorem 3.41 because Θ_{f}(Te_{j}) ⊂ Θ_{f}(S_{k}(N, ε; Z)^{σ}), so the lattice index is an integer.

For the rest of this section, we assume for simplicity that ε = 1.

Lemma 3.47. For j = 1 and j = k − 1, we have for each p ∤ N that

(T_{p} − (1 + p^{k−1}))e_{j} ∈ S_{k}(N, ε; Z).

Proof. This is a standard calculation; see, e.g., [16, §2.8] for the case when k = 2.

Proposition 3.48. Let j ∈ {1, …, k − 1}, and let n be the order of the image in A_{f}(C) of the modular symbol e_{j}, so n = 1 if j ∉ {1, k − 1}. Then

\frac{L(A_{f}, j)}{Ω_{j}} ∈ \frac{1}{n}Z.

Proof. Let x denote the image of e_{j} ∈ A_{f}(C), and set I = Ann(x) ⊂ T. Though we write A_{f}(C) here and below, we will always work within the subgroup of A_{f}(C) generated by the image of M_{k}(N, ε; Z) under the period map.

First we check that the Hecke operators all act as scalars on x. Since f is a newform, the Hecke operators T_{p}, for p | N, act as 0 or ±p^{k/2−1} on f, and hence in the same way on A_{f}(C) (see, e.g., the end of section 6 of [21]). If p ∤ N, Lemma 3.47 shows that T_{p}(x) = (1+p^{k−1})x.

Let C = Zx denote the cyclic subgroup of A_{f}(C) generated by x, so n is the order of C. Since the Hecke operators act as scalars on C, we are pleased to find that there is an injection T/I ↪ C which sends T_{p} to T_{p}(x).

Setting S = S_{k}(N, ε; Z) and applying Theorem 3.41 we find that

±\frac{L(A_{f}, j)}{Ω_{j}} = [Θ_{f}(S^{+}) : Θ_{f}(Te)]
= [Θ_{f}(S^{+}) : Θ_{f}(Ie)] · [Θ_{f}(Ie) : Θ_{f}(Te)]
= [Θ_{f}(S^{+}) : IΘ_{f}(e)] · [IΘ_{f}(e) : TΘ_{f}(e)]
= \frac{[Θ_{f}(S^{+}) : IΘ_{f}(e)]}{[TΘ_{f}(e) : IΘ_{f}(e)]}.

To conclude that

\frac{[Θ_{f}(S^{+}) : IΘ_{f}(e)]}{[TΘ_{f}(e) : IΘ_{f}(e)]} ∈ \frac{1}{n}Z

we make two observations. By the construction of A_{f}(C), the ideal I consists of those elements of T that send Θ_{f}(e) into Θ_{f}(S^{+}), so [Θ_{f}(S^{+}) : IΘ_{f}(e)] ∈ Z. Second, there is a surjective map

T/I → \frac{TΘ_{f}(e)}{IΘ_{f}(e)}

sending t to tΘ_{f}(e), so [TΘ_{f}(e) : IΘ_{f}(e)] divides n = #C = #(T/I).

Remark 3.49 (Historical notes). In the special case when k = 2, the modular symbol e_{1} corresponds to (0)−(∞) ∈ J_{0}(N). In this situation, Manin proves at the bottom of page 28 of [38] that (0)−(∞) ∈ J_{0}(N)(Q), and asserts in the footnote to [38, Cor. 3.6] that (0)−(∞) has finite order. Based on observations such as a special case of the above proposition, he declares: “These explicit formulas have the structure predicted by the Birch-Swinnerton-Dyer conjectures.”

The main result of this section was inspired by a weaker result of Agashe, which can be found in Chapter 4 of [2]. Agashe considers only the case k = 2 and replaces n by the order of the subgroup of J_{0}(N)(\overline{Q}) generated by all cusps.

3.11 The Manin constant

In this section k = 2 and ε = 1; we sometimes omit k and ε from the notation. The assumption that k = 2 will be essential, because we do not know how to define a Manin constant in other weights, let alone bound it.

Consider the optimal quotient A of J_{0}(N) corresponding to a newform f on Γ_{0}(N) of weight 2. Let I_{A} be the kernel of the natural map from the Hecke algebra to End(A). The Manin constant c_{A} of A is the lattice index

c_{A} := [S_{2}(Γ_{0}(N); Z)[I_{A}] : H^{0}(A, Ω_{A/Z})]

taken inside of S_{2}(Γ_{0}(N); Q). Though, a priori, c_{A} is a rational number, the work of [31] implies that c_{A} ∈ Z (see, e.g., [3]).

Generalizing a theorem of Mazur, we prove that c_{A} is a unit in Z[\frac{1}{2m}], where m is the largest square dividing N. Essentially no new ideas beyond what Mazur used are involved. We then conjecture that c_{A} = 1, and give supporting numerical evidence.

For related results involving modular “building blocks” for J_{1}(N), we refer the reader to [26, §4].

3.11.1 The primes that might divide c_{A}

In the special case dim A = 1, the Manin constant is the classical Manin constant of A, and in [41] Mazur proved that c_{A} is a unit in Z[\frac{1}{2m}]. We generalize his proof to obtain the analogous result in dimension greater than 1.

Theorem 3.50. Let A be the new optimal quotient of J_{0}(N) corresponding to a newform f. Then the Manin constant c_{A} is a unit in Z[\frac{1}{2m}], where m is the largest square dividing N.

Proof. The reader is strongly recommended to keep the proof of Proposition 3.1 in [41] at hand while reading the following argument.

Let π denote the map J_{0}(N) → A; let A denote the Néron model of A over R := Z[\frac{1}{2m}], and J the Néron model of J_{0}(N) over R. Let X be the minimal proper regular model for X_{0}(N) over R. As in Mazur’s proof in [41], consider the diagram

H^{0}(A, Ω_{A}) ⟶^{π^{∗}} H^{0}(J , Ω_{J}) ≅ H^{0}(X , Ω_{X}^{reg}) ⟶^{q-exp} R[[q]].  (3.2)

(Note that “Ω_{X}^{reg}” is not defined to be the usual sheaf of differentials; see, e.g., the discussion in [40, pg. 67].) The map π^{∗} must be an inclusion, by [41, Cor. 1.1]. To show that the Manin constant is a unit in R, it suffices to check that the image of H^{0}(A, Ω_{A}) in R[[q]] is saturated, in the sense that the cokernel is torsion free; indeed, the image of S_{2}(Γ_{0}(N); R)[I] is saturated and S_{2}(Γ_{0}(N); R)[I] ⊗ Q = q-exp(π^{∗}(H^{0}(A, Ω_{A}))) ⊗ Q.

For the image of H^{0}(A, Ω_{A}) in R[[q]] to be saturated means that the quotient D is torsion free. Let ℓ be a prime not dividing 2m; tensoring

0 → H^{0}(A, Ω_{A}) ⟶^{q-exp} R[[q]] → D → 0

with F_{ℓ} we obtain

0 → D[ℓ] → H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] → D ⊗ F_{ℓ} → 0.

Here we have used that Tor^{1}(D, F_{ℓ}) is the ℓ-torsion in D, and that Tor^{1}(−, F_{ℓ}) vanishes on the torsion-free group R[[q]]. (Alternatively, we could have used the snake lemma.) To show that D[ℓ] = 0, it suffices to prove that the map Ψ : H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] is injective.

Since ℓ ≠ 2 and A is an optimal quotient, [41, Cor 1.1] gives an exact sequence

0 → H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) → H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) → 0

where B is the Néron model of ker(J → A). In particular, H^{0}(B/Z_{ℓ}, Ω_{B/Z_{ℓ}}) is torsion free, so

H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ} → H^{0}(J /Z_{ℓ}, Ω_{J /Z_{ℓ}}) ⊗ F_{ℓ} ≅ H^{0}(X /Z_{ℓ}, Ω_{X /Z_{ℓ}}^{reg}) ⊗ F_{ℓ}
≅ H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}}^{reg})

is injective. (The last isomorphism is by [40, Prop. 3.3, pg. 68].) We also remark that

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} ≅ H^{0}(A/Z_{ℓ}, Ω_{A/Z_{ℓ}}) ⊗ F_{ℓ},

because Z_{ℓ} is torsion free, hence flat over R. Thus the map

H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}}^{reg})

is injective.

If ℓ ∤ N, then injectivity of Ψ now follows from the q-expansion principle, which asserts that the q-expansion map H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}}^{reg}) → F_{ℓ}[[q]] is injective.

Suppose ℓ does divide N, and let ω ∈ ker(Ψ). Since ℓ | N and ℓ ∤ 2m, we have that ℓ || N; thus X /F_{ℓ} breaks up into a union of two irreducible components, and the q-expansion principle implies only that ω vanishes on the irreducible component containing the cusp ∞. However, since A is new and corresponds to a single eigenform, ω is an eigenvector for the involution W_{N} (since f and all of its conjugates are). Since W_{N} permutes the two components, ω must be 0 on all X /F_{ℓ}. Therefore ω = 0, and hence Ψ is injective.

3.11.2 Numerical evidence for the c_{A} = 1 conjecture

In the paper [24], the authors show that c_{A} = 1 for 28 two-dimensional optimal quotients of J_{0}(N) (see Section 3.12.8). The non-square-free levels treated are:

N = 3^{2} · 7, 3^{2} · 13, 5^{3}, 3^{3} · 5, 3 · 7^{2}, 5^{2} · 7, 2^{2} · 47, 3^{3} · 7.

In every case, c_{A} = 1.

Conjecture 3.51 (Agashe). Let A be an optimal quotient of J_{0}(N), and let c_{A} be the corresponding Manin constant. Then c_{A} = 1.

3.12 Analytic invariants

Fix a newform

f = ∑_{n≥1} a_{n}q^{n} ∈ S_{k}(N, ε),

and assume that ε^{2} = 1.

Remark 3.52. Our assumption that ε^{2} = 1 does not imply that f has totally real Fourier coefficients. There is an eigenform in S_{2}(24, ε) whose Fourier coefficients are not totally real, where ε is one of the characters of conductor 8.

Let K_{f} = Q(… a_{n} …) and let f_{1}, …, f_{d} be the Galois conjugates of f, where d = [K_{f} : Q]. As in Section 2.7, we consider the complex torus A_{f} attached to f. In this section we describe how to compute the torus A_{f} and the special values at the critical integers 1, 2, …, k − 1 of the L function L(A_{f}, s) associated to A_{f}. (See 3.39 for the definition of L(A_{f}, s).)

Let

f = ∑_{n≥1} a_{n}q^{n} ∈ M_{k}(N, ε)

be a modular form (we do not assume that f is an eigenform). We recall the integration pairing of Theorem 2.7:

⟨ , ⟩ : M_{k}(N, ε) × M_{k}(N, ε) ⟶ C
⟨f, P {α, β}⟩ = 2πi ∫_{α}^{β} f(z)P (z, 1)dz.

Let I_{f} ⊂ T be the kernel of the map T → K_{f} sending T_{n} to a_{n}. The integration pairing gives rise to the period mapping

Φ_{f} : M_{k}(N, ε) → Hom_{C}(S_{k}(N, ε)[I_{f}], C),

and A_{f} = Hom_{C}(S_{k}(N, ε)[I_{f}], C)/Φ_{f}(S_{k}(N, ε)) is the cokernel.

3.12.1 Extended modular symbols

For the purposes of computing periods, it is advantageous to extend the notion of modular symbols to allows symbols of the form P {z, w} where z and w are now arbitrary elements of 𝔥^{∗} = 𝔥 ∪ P^{1}(Q). The free abelian group \overline{M}_{k} of extended modular symbols is spanned by such symbols, and is of uncountable rank over Z. However, it is still equipped with an action of Γ_{0}(N) and we can form the largest torsion-free quotient \overline{M}_{k}(N, ε) of \overline{M}_{k} by the relations γx = ε(γ)x for γ ∈ Γ_{0}(N).

The integration pairing extends to \overline{M}_{k}(N, ε). There is a natural embedding ι : M_{k}(N, ε) ↪ \overline{M}_{k}(N, ε) which respects the pairing in the sense that ⟨f, ι(x)⟩ = ⟨f, x⟩. In many cases it is advantageous to replace x ∈ M_{k}(N, ε) first by ι(x), and then by an equivalent sum ∑ y_{i} of symbols y_{i} ∈ \overline{M}_{k}(N, ε). The period ⟨f, x⟩ is then replaced by the equivalent sum of periods ∑⟨f, y_{i}⟩. The latter is frequently much easier to approximate numerically.

3.12.2 Numerically computing period integrals

Consider a point α in the upper half plane and any one of the (extended) modular symbols X^{m}Y^{k−2−m}{α, ∞}. Given a cusp form g = ∑_{n≥1} b_{n}q^{n} ∈ S_{k}(N, ε) and an integer m ∈ {0, 1, …, k − 2}, we find that

⟨g, X^{m}Y^{k−2−m}{α, ∞}⟩ = 2πi ∫_{α}^{i∞} g(z)z^{m}dz = 2πi ∑_{n=1}^{∞} b_{n} ∫_{α}^{i∞} e^{2πinz}z^{m}dz.  (3.3)

The reversal of summation and integration is justified because the imaginary part of α is positive so that the sum converges absolutely. This is made explicit in the following lemma, which can be proved using repeated integration by parts.

Lemma 3.53.

∫_{α}^{i∞} e^{2πinz}z^{m}dz = e^{2πinα} ∑_{s=0}^{m} (\frac{(−1)^{s}α^{m−s}}{(2πin)^{s+1}} ∏_{j=(m+1)−s}^{m} j) .  (3.4)

The following proposition is the higher weight analogue of [16, Prop. 2.1.1(5)].

Proposition 3.54. For any γ ∈ Γ_{0}(N), P ∈ V_{k−2} and α ∈ 𝔥^{∗} the following holds:

P {∞, γ(∞)} = P {α, γ(α)} + (P − ε(γ)γ^{−1}P){∞, α}  (3.5)
= ε(γ)(γ^{−1}P){α, ∞} − P {γ(α), ∞}.  (3.6)

Proof. By definition, if x ∈ M_{k}(N, ε) is a modular symbol and γ ∈ Γ_{0}(N) then γx = ε(γ)x; in particular, ε(γ)γ^{−1}x = x, so

P {∞, γ(∞)} = P {∞, α} + P {α, γ(α)} + P {γ(α), γ(∞)}
= P {∞, α} + P {α, γ(α)} + ε(γ)γ^{−1}(P {γ(α), γ(∞)})
= P {∞, α} + P {α, γ(α)} + ε(γ)(γ^{−1}P){α, ∞}
= P {α, γ(α)} + P {∞, α} − ε(γ)(γ^{−1}P){∞, α}
= P {α, γ(α)} + (P − ε(γ)γ^{−1}P){∞, α}.

The second equality in the statement of the proposition now follows easily.

In the classical case of weight two and trivial character, the error term (P −ε(γ)γ^{−1}P){∞, α} vanishes. In general this term does not vanish, instead perturbing the analogues of the formulas found in [16, 2.10].

Algorithm 3.55. Given a triple γ ∈ Γ_{0}(N), P ∈ V_{k−2} and g ∈ S_{k}(N, ε) (as a q-expansion to some precision) this algorithm computes the period integral ⟨g, P {∞, γ(∞)}⟩. Express γ as ( \begin{matrix}a & b \\ cN & d \end{matrix} ) ∈ Γ_{0}(N) and take α = \frac{−d+i}{cN} in Proposition 3.54. Replacing γ by −γ if necessary, we find that the imaginary parts of α and γ(α) = \frac{a+i}{cN} are both equal to 1/(cN) which is positive. Equation 3.3 and Lemma 3.53 can now be used to compute the period integrals of Proposition 3.54.

With the goal of computing period lattices in mind, it is reassuring to know that every element of S_{k}(N, ε) can be written as a linear combination of symbols of the form P {∞, γ(∞)}. The author asked Helena Verrill if this is the case and she was eventually able to prove that it is; the proof is given below. In the special case of weight two and trivial character, this is the assertion, which was proved by Manin [38], that the group homomorphism Γ_{0}(N) → H_{1}(X_{0}(N), Z) sending γ to {0, γ(0)} is surjective. When the weight is greater than two, we have not found any similar group-theoretic statement.

Proposition 3.56. Any element of S_{k}(N, ε) can be written in the form

∑_{i=1}^{n} P_{i}{∞, γ_{i}(∞)}

with P_{i} ∈ V_{k−2} and γ_{i} ∈ Γ_{0}(N). Moreover, P_{i} and γ_{i} can be chosen so that ∑ ε(γ_{i})P_{i} = ∑ γ_{i}^{−1}P_{i}.

Proof.^{1} First recall the definition of the spaces M, M_{k} = V_{k−2} ⊗ M and M_{k}(N, ε) = M_{k}/I (see Section 2.1). Let I = I_{N,ε} be the ideal in the group ring of Γ_{0}(N) generated by all elements of the form ε(γ) − γ for γ ∈ Γ_{0}(N).

Suppose v ∈ S_{k}(N, ε). Use the relation {α, β} = {∞, β} − {∞, α} ∈ M to see that any v is the image of an element ṽ ∈ M_{k} of the form

ṽ = ∑_{β∈Q} P_{β} ⊗ {∞, β} ∈ M_{k}

with only finitely many P_{β} nonzero. The boundary map δ lifts in a natural way to V_{k−2}⊗M, as illustrated.

\xymatrix{ & I(V_{k−2} ⊗ M) \ar[r] \ar[d] & I(V_{k−2} ⊗ B) \ar[d] \\  & V_{k−2} ⊗ M \ar[r]^{δ̃} \ar[d] & V_{k−2} ⊗ B \ar[d] \\ S_{k}(N, ε) \ar@{^{(}->}[r] & M_{k}(N, ε) \ar[r]^{δ} & B_{k}(N, ε)}

Our assumption that δ(v) = 0 implies that δ̃(ṽ) ∈ I(V_{k−2} ⊗ B). So there are Q_{γ,β} ∈ V_{k−2}, for γ ∈ Γ_{0}(N) and β ∈ P^{1}(Q), only finitely many nonzero, such that

δ̃(ṽ) = ∑_{γ,β}(ε(γ) − γ)(Q_{γ,β} ⊗ {β}).

1The author thanks Helena Verrill for permission to reproduce her proof here.

We now use a summation trick.

∑_{β∈Q} δ̃(ṽ) = P_{β} ⊗ {β} − P_{β} ⊗ {∞} = ∑_{γ,β} ε(γ)Q_{γ,β} ⊗ {β} − (γQ_{γ,β}) ⊗ {γβ}
= ∑_{γ,β} ε(γ)Q_{γ,β} ⊗ {β} − (γQ_{γ,γ^{−1}β}) ⊗ {β}
= ∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}) ⊗ {β}.

Equating terms we deduce that for β ≠ ∞,

P_{β} = ∑_{γ} ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}.

Using this expression for P_{β} and that ε(γ)γ^{−1} acts trivially on M_{k}(N, ε) we find that

v = ∑_{β} P_{β}{∞, β} = ∑_{γ,β}(ε(γ)Q_{γ,β} − γQ_{γ,γ^{−1}β}){∞, β}
= ∑_{γ,β} ε(γ)Q_{γ,β} − ε(γ)γ^{−1}(γQ_{γ,γ^{−1}β}){∞, β}
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, β} − ε(γ)Q_{γ,γ^{−1}β}{γ^{−1}∞, γ^{−1}β}
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, β} − ε(γ)Q_{γ,β}{γ^{−1}∞, β}
= ∑_{γ,β} ε(γ)Q_{γ,β}{∞, γ^{−1}∞}.

This is of the desired form.

Unlike the case of weight two and trivial character, Proposition 3.56 does not give generators for S_{k}(N, ε). This is because not every element of the form P {∞, γ(∞)} must lie in S_{k}(N, ε). However, if γP = P then P {∞, γ(∞)} does lie in S_{k}(N, ε). It would be interesting to know under what circumstances S_{k}(N, ε) is generated by symbols of the form P {∞, γ(∞)} with γP = P. This sometimes fails for k odd; for example, when k = 3 the condition γP = P implies that γ ∈ Γ_{0}(N) has an eigenvector with eigenvalue 1, hence is of finite order. When k is even the author can see no obstruction to generating S_{k}(N, ε) using such symbols.

3.12.3 The W_{N}-trick

In this section we assume that k is even. Consider the involution W_{N} defined in Section 2.4.3. This is an involution that acts on both modular symbols and modular forms. The follow proposition shows how to compute ⟨g, P {∞, γ(∞)}⟩ under certain restrictive assumptions. It generalizes the main result of [17] to higher weight. (Compare also [25].)

Proposition 3.57. Let g ∈ S_{k}(N, ε) be a cusp form which is an eigenform for the Atkin-Lehner involution W having eigenvalue w ∈ {±1}. Then for any γ ∈ Γ_{0}(N) and any P ∈ V_{k−2}, with the property that γP = ε(γ)P, we have for any α ∈ 𝔥 the following formula:

⟨g, P {∞, γ(∞)}⟩ =

⟨g, w \frac{P (Y, −NX)}{N^{k/2−1}}\{W (α), ∞\} + (P − w \frac{P (Y, −NX)}{N^{k/2−1}})\{i/\sqrt{N}, ∞\} − P \{γ(α), ∞\}⟩.

Here W (α) = −1/(Nα).

Proof. By Proposition 3.54 our condition on P implies that P {∞, γ(∞)} = P {α, γ(α)}. The steps of the following computation are described below.

⟨g, P {α, γ(α)}⟩

= ⟨g, P \{α, i/\sqrt{N}\} + P \{i/\sqrt{N}, W (α)\} + P \{W (α), γ(α)\}⟩
= ⟨g, w \frac{W (P)}{N^{k/2−1}} \{W (α), i/\sqrt{N}\} + P \{i/\sqrt{N}, W (α)\} + P \{W (α), γ(α)\}⟩
= ⟨g, (w \frac{W (P)}{N^{k/2−1}} − P)\{W (α), i/\sqrt{N}\} + P \{W (α), ∞\} − P \{γ(α), ∞\}⟩
= ⟨g, w \frac{W (P)}{N^{k/2−1}} \{W (α), ∞\} + (P − w \frac{W (P)}{N^{k/2−1}} )\{i/\sqrt{N}, ∞\} − P \{γ(α), ∞\}⟩.

For the first step, we break the path into three paths. In the second step, we apply the W -involution to the first term, and use that the action of W is compatible with the pairing ⟨ , ⟩. The third step involves combining the first two terms and breaking up the third. In the final step, we replace \{W (α), i/\sqrt{N}\} by \{W (α), ∞\} + \{∞, i/\sqrt{N}\} and regroup.

A good choice for α is α = γ^{−1} (\frac{b}{d} + \frac{i}{d\sqrt{N}} ), so that W (α) = \frac{c}{d} + \frac{i}{d\sqrt{N}}. This maximizes the minimum of the imaginary parts of α and W (α).

Let γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N). A polynomial P for which γ(P) = P is given by

P (X, Y) = (cX^{2} + (d − a)XY − bY^{2})^{\frac{k−2}{2}} .

This formula was obtained by viewing V_{k−2} as the (k − 2)th symmetric product of the two-dimensional space on which Γ_{0}(N) acts naturally. For example, observe that since det(γ) = 1 the symmetric product of two eigenvectors for γ is an eigenvector in V_{2} having eigenvalue 1. For the same reason, if ε(γ) ≠ 1, there is often no polynomial P (X, Y) such that γ(P) = ε(γ)P. When this is the case, first choose γ so that ε(γ) = 1.

Since the imaginary parts of the terms i/\sqrt{N}, α and W (α) in the proposition are all relatively large, the sums appearing in Equation 3.3 converge quickly if d is small. Let us emphasize, that it is extremely important to choose γ in Proposition 3.57 with d small, otherwise the series will converge very slowly.

3.12.4 Computing the period mapping

Let I ⊂ T be the kernel of the map T → K_{f} sending T_{n} to a_{n}. As in Section 3.7, let Θ_{f} = Θ_{I} be the rational period mapping associated to f. We have a commutative diagram

\xymatrix{M_{k}(N, ε) \ar[dr]_{Θ_{f}} \ar[rr]^{Φ_{f}} &  & Hom_{C}(S_{k}(N, ε)[I], C) \\  & \frac{M_{k}(N, ε)}{ker(Φ_{f})} \ar@{-->}[ur]^{i_{f}}}

Using Algorithm 3.26, we can compute Θ_{f} so to compute Φ_{f} we need to compute i_{f}. Let g_{1}, …, g_{d} be a basis for the Q-vector space S_{k}(N, ε; Q)[I]. We will compute the period mapping with respect to the basis of Hom_{Q}(S_{k}(N, ε; Q)[I], C) dual to this basis. Choose elements x_{1}, …, x_{d} ∈ M_{k}(N, ε) with the following properties:

1. Using Proposition 3.54 or Proposition 3.57 it is possible to compute the period integrals ⟨g_{i}, x_{j}⟩, i, j ∈ {1, … d} efficiently.

2. The 2d elements v + ∗v and v − ∗v for v = Θ_{f}(x_{1}), …, Θ_{f}(x_{d}) span a space of dimension 2d.

Given this data, we can compute

i_{f}(v + ∗v) = 2Re(⟨g_{1}, x_{i}⟩, …, ⟨g_{d}, x_{i}⟩)

and

i_{f}(v − ∗v) = 2iIm(⟨g_{1}, x_{i}⟩, …, ⟨g_{d}, x_{i}⟩).

We break the integrals into real and imaginary parts because this increases the precision of our answers. Since the vectors v_{n} + ∗v_{n} and v_{n} − ∗v_{n}, n = 1, …, d span \frac{M_{k}(N,ε)}{ker(Φ_{f})} we have computed i_{f}.

It is advantageous when possible to find symbols x_{i} satisfying the conditions of Proposition 3.57. This is usually possible when d is very small, but in practice we have had problems doing this when d is large, for example with 131k2B, in which case the dimension is 10.

3.12.5 Computing special values

For s = 1, …, k − 1 we have
L(f, s) = \frac{−2π^{s−1}i^{s−1}}{(s − 1)!} · ⟨f, X^{s−1}Y^{k−1−s}\{0, ∞\}⟩,  (3.7)

L(A_{I}, s) = ∏_{i=1}^{d} L(f_{i}, s).  (3.8)

Let

e_{i} := X^{i−1}{0, ∞}

denote the ith winding element. In section 3.12.4 we computed the period map Φ_{f} with respect to a basis g_{1}, …, g_{d} for S_{k}(N, ε; Q)[I]. Upon writing f as a K_{f}-linear combination α_{1}g_{1} + ⋯ + α_{d}g_{d} we find that

⟨f, e_{i}⟩ = ⟨α_{1}g_{1} + ⋯ + α_{d}g_{d}, e_{i}⟩
= α_{1}⟨g_{1}, e_{i}⟩ + ⋯ + α_{d}⟨g_{d}, e_{i}⟩
= α_{1}Φ_{f}(e_{i})_{1} + ⋯ + α_{d}Φ_{f}(e_{i})_{d}

Here Φ_{f}(e_{i})_{j} denotes the jth coordinate of Φ_{f}(e_{i}). Finally using Equation 3.7 we compute the special value.

3.12.6 The real and minus volume associated to A_{f}

Fix a choice of basis g_{1}, …, g_{d} for the free Z-module S_{k}(N, ε; Z)[I], where I is the annihilator in the Hecke algebra of our fixed newform f.

For any x ∈ S_{k}(N, ε) we have, by Proposition 2.11,

\overline{Φ_{f}(x)} = (\overline{⟨g_{1}, x⟩}, …, \overline{⟨g_{d}, x⟩})
= (⟨g_{1}^{∗}, x^{∗}⟩, …, ⟨g_{d}^{∗}, x^{∗}⟩)
= (⟨g_{1}, x^{∗}⟩, …, ⟨g_{d}, x^{∗}⟩) ∈ Φ_{f}(S_{k}(N, ε)),

so complex conjugation leaves invariant the period lattice

Λ_{f} = Φ_{f}(S_{k}(N, ε)) ⊂ Hom_{C}(S_{k}(N, ε)[I], C).

Fix a Z-basis for S_{k}(N, ε; Z)[I], thus making an identification Hom_{C}(S_{k}(N, ε)[I], C) ≅ C^{d}. The above observation implies that A_{f}(C) ≅ C^{d}/Λ_{f} is equipped with an action of complex conjugation. Our choice of basis defines a real-valued measure μ on A_{f}(C), coming from the standard measure on C^{d}. The measure does not depend on the choice of Z-basis.

Definition 3.58 (Real and minus volume). The real measure Ω_{f}^{+} is the measure μ(A_{f}(R)). The minus measure Ω_{f}^{−} is the measure μ(A_{f}(C)^{−}) times i^{d}, where A_{f}(C)^{−} is the set of points in A_{f}(C) on which complex conjugation acts as −1.

Thus, in connection with Section 3.10.3, |Ω_{f}^{+}| = |Ω_{1}| and |Ω_{f}^{−}| = |Ω_{2}|.

Algorithm 3.59. To compute Ω_{f}^{+} and Ω_{f}^{−}, proceed as follows. Using Algorithm 3.26, compute S_{k}(N, ε)/ Ker(Φ_{f}). Next, compute a basis for the kernel (S_{k}(N, ε)/ Ker(Φ_{f}))^{+} of the map induced by the ∗-involution. Using Section 3.12.4 compute the image of this basis under i_{f}; this is a basis for Λ_{f}^{+}. The determinant of this latter basis then gives the measure (Ω_{f}^{+})^{0} of the identity component A_{f}(R)^{0} of A_{f}(R). Finally Ω_{f}^{+} = c_{∞}^{+} · (Ω_{f}^{+})^{0}, where the number c_{∞}^{+} of real components can be computed using the algorithm in Section 3.12.7

Remark 3.60 (Alternative method). Suppose s is an integer in the set {1, …, k−1}, and let σ = + or σ = −, depending on whether s is odd or even, respectively. Section 3.10 contains a formula for the ratio L(A_{f}, s)/Ω_{f}^{σ}. When this ratio is nonzero, Ω_{f}^{σ} can be determined by computing L(A_{f}, s)/Ω_{f}^{σ} and L(A_{f}, s), using Section 3.12.5.

Remark 3.61. When k = 2 and ε is trivial, A_{f} has the structure of abelian variety over Q. The quantity Ω_{f}^{+} above is related to the quantity Ω_{A} appearing in the Birch and Swinnerton-Dyer conjecture [67] for A_{f}. The latter quantity is the measure of A_{f}(R) with respect to a basis of integral differentials on the Néron model of A_{f} over Spec(Z). The two quantities are related by the Manin constant, which the author conjectures is always 1 (see Section 3.11).

3.12.7 The component groups c_{∞}^{+} and c_{∞}^{−}

Assume in this section that f has totally real Fourier coefficients and continue to assume that ε^{2} = 1.

Definition 3.62. Let c_{∞}^{+} be the number of components of the topological space A_{f}(R). Let c_{∞}^{−} be the number of components of A_{f}(C)^{−}, where A_{f}(C)^{−} is the set of points z ∈ A_{f}(C) such that \overline{z} = −z.

Proposition 3.63. Let \overline{C} be the map induced by complex conjugation on Λ_{f}/2Λ_{f} = Λ_{f} ⊗ F_{2}. Then

c_{∞}^{+} = c_{∞}^{−} = 2^{dim(ker(\overline{C}−1))−d},

where d is the dimension of A_{f}.

Proof. We must compute the order of the component group

Ψ = \frac{A_{f}(R)}{A_{f}(R)^{0}} = \frac{(C^{d}/Λ_{f})^{+}}{R^{d}/Λ_{f}^{+}} ,

where R^{d}/Λ_{f}^{+} is the identity component because it is the continuous image of the connected set R^{d}. For v ∈ C^{d} denote by \overline{v} its complex conjugate and by [v] its image in C^{d}/Λ_{f}. Suppose [v] ∈ (C^{d}/Λ_{f})^{+}; this means that [v] = [\overline{v}], so since v + \overline{v} ∈ R^{d} we have

2[v] = [v] + [\overline{v}] ∈ R^{d}/Λ_{f}^{+},

so Ψ is annihilated by 2. Thus there is λ ∈ Λ_{f} so that 2v + λ ∈ R^{d}, and so v + \frac{1}{2}λ ∈ R^{d}, i.e., v can be written as an element of \frac{1}{2}Λ_{f} plus an element of R^{d}. This means that Ψ is generated by the image of (\frac{1}{2}Λ_{f}/Λ_{f})^{+}. Thus

Ψ ≅ \frac{(\frac{1}{2}Λ_{f}/Λ_{f})^{+}}{(\frac{1}{2}Λ_{f} ∩ R^{d})/Λ_{f}^{+}} ≅ \frac{(Λ_{f}/2Λ_{f})^{+}}{Λ_{f}^{+}/2Λ_{f}^{+}}

Consequently

dim_{F_{2}} Ψ = dim(Λ_{f}/2Λ_{f})^{+} − dim Λ_{f}^{+}/2Λ_{f}^{+} = dim(ker(\overline{C} − 1)) − d.

Here Λ_{f}^{+}/2Λ_{f}^{+} has dimension d because Λ_{f}^{+} is a lattice in R^{d}, hence a free Z-module of rank d.

The argument for c_{∞}^{−} proceeds in the same way, and results in the same answer because

dim(ker(\overline{C} − 1)) = dim(ker(\overline{C} + 1)).

To compute C on Λ_{f}, use Algorithm 3.26 to compute the action of ∗ on

S_{k}(N, ε)/ Ker(Φ_{f}) ≅ Λ_{f}.

3.12.8 Examples

Jacobians of genus-two curves

The author is among the the six authors of [24], who gather empirical evidence for the BSD conjecture for Jacobian of genus two curves. Of the 32 Jacobians considered, all but four are optimal quotients of J_{0}(N) for some N. The methods of this section can be used to compute Ω_{f}^{+} for the Jacobians of these 28 curves. Using explicit models for the genus two curves, the authors of [24] computed the measure of A with respect to a basis for the Néron differentials of A. In all 28 cases our answers agreed to the precision computed. Thus in these cases we have numerically verified that the Manin constant equals 1.

The first example considered in [24] is the Jacobian A = J_{0}(23) of the modular curve X_{0}(23). This curve has as a model

y^{2} + (x^{3} + x + 1)y = −2x^{5} − 3x^{2} + 2x − 2

from which one can compute the BSD Ω_{A} = 2.7328.... The following is an integral basis of cusp forms for S_{2}(23).

g_{1} = q − q^{3} − q^{4} − 2q^{6} + 2q^{7} + ⋯
g_{2} = q^{2} − 2q^{3} − q^{4} + 2q^{5} + q^{6} + 2q^{7} + ⋯

The space M_{2}(23; Q) of modular symbols has dimension five and is spanned by {−1/19, 0}, {−1/17, 0}, {−1/15, 0}, {−1/11, 0} and {∞, 0}. The submodule S_{2}(23; Z) has rank four and has as basis the first four of the above five symbols. Choose γ_{1} = ( \begin{matrix}8 & 1 \\ 23 & 3 \end{matrix} ) and γ_{2} = ( \begin{matrix}6 & 1 \\ 23 & 4 \end{matrix} ) and let x_{i} = {∞, γ_{i}(∞)}. Using the W_{N}-trick (see Section 3.12.3) we compute the period integrals ⟨g_{i}, x_{j}⟩ using 97 terms of the q-expansions of g_{1} and g_{2}, and obtain

\begin{matrix}⟨g_{1}, x_{1}⟩ ∼ −1.3543 + 1.0838i, & ⟨g_{1}, x_{2}⟩ ∼ −0.5915 + 1.6875i \\ ⟨g_{2}, x_{1}⟩ ∼ −0.5915 − 0.4801i, & ⟨g_{2}, x_{2}⟩ ∼ −0.7628 + 0.6037i \end{matrix}

Using 97 terms we already obtain about 14 decimal digits of accuracy, but we do not reproduce them all here. We next find that

⟨g_{1}, x_{1} + x_{1}^{∗}⟩ ∼ 2Re(−1.3543 + 1.0838i) = 2.7086,

and so on. Upon writing each generator of S_{2}(23) in terms of x_{1} + x_{1}^{∗}, x_{1} − x_{1}^{∗}, x_{2} + x_{2}^{∗} and x_{2} − x_{2}^{∗} we discover that the period mapping with respect to the basis dual to g_{1} and g_{2} is (approximately)

\begin{matrix}\{−1/19, 0\} & ↦ & ( 0.5915 − 1.6875i, & 0.7628 − 0.6037i) \\ \{−1/17, 0\} & ↦ & (−0.5915 − 1.6875i, & −0.7628 − 0.6037i) \\ \{−1/15, 0\} & ↦ & (−1.3543 − 1.0838i, & −0.5915 + 0.4801i) \\ \{−1/11, 0\} & ↦ & (−1.5256, & 0.3425) \end{matrix}

Working in S_{2}(23) we find S_{2}(23)^{+} is spanned by {−1/19, 0} − {−1/17, 0} and {−1/11, 0}. Using the algorithm of Section 3.12.6, we find that there is only one real component so

Ω_{I}^{+} ∼ | \begin{matrix}1.1831 & 1.5256 \\ −1.5256 & 0.3425 \end{matrix} | = 2.7327...

To greater precision we find that Ω_{f}^{+} ∼ 2.7327505324965. This agrees with the value in [24]; since the Manin constant is an integer, it must equal 1.

Table 3.1: Volumes associated to level one cusp forms.

\begin{matrix}k & Ω^{+} & Ω^{−} \\ 12 & 0.002281474899 & 0.000971088287i \\ 16 & 0.003927981492 & 0.000566379403i \\ 18 & 0.000286607497 & 0.023020042428i \\ 20 & 0.008297636952 & 0.0005609325015i \\ 22 & 0.002589288079 & 0.0020245743816i \\ 24 & 0.000000002968 & 0.0000000054322i \\ 26 & 0.003377464512 & 0.3910726132671i \\ 28 & 0.000000015627 & 0.0000000029272i \end{matrix}

Level one cusp forms

In the following two sections we consider several specific examples of tori attached to modular forms of weight greater than two.

Let k ≥ 12 be an even integer. Associated to each Galois conjugacy class of normalized eigenforms f, there is a torus A_{f} over R. The real and minus volume of the first few of these tori are displayed in Table 3.1. For weights 24 and 28 we give Ω^{−}/i so that the columns will line up nicely. In each case, 97 terms of the q-expansion were used.

The volumes appear to be much smaller than the volumes of weight two abelian varieties. The dimension of each A_{f} is 1, except for weights 24 and 28 when the dimension is 2.

CM elliptic curves of weight greater than two

Let f be a rational newform with “complex multiplication”, in the sense that “half” of the Fourier coefficients of f are zero. For our purposes, it is not necessary to define complex multiplication any more precisely. Experimentally, it appears that the associated elliptic A_{f} has rational j-invariant. As evidence for this we present Table 3.2, which includes the analytic data about every rational CM form of weight four and level ≤ 197. The computations of Table 3.2 were done using at least 97 terms of the q-expansion of f. The rationality of j could probably be proved by observing that the CM forces A_{f} to have extra automorphisms.

In these examples, the invariants c_{4} and c_{6} are unrecognizable to the author; in contrast, in weight 2 these invariants are (expected to be) integers (see [16, 2.14]).

Some abelian varieties of large dimension

In Table 3.3, we give the volumes of five abelian varieties of dimension greater than 1. In each case, at least 200 terms of the q-expansions were used.

3.12. ANALYTIC INVARIANTS 69

Table 3.2: CM elliptic curves of weight > 2.

E  j  Ω^{+}  Ω^{−}  c_{4}  c_{6}
9k4A  0 0.2095 0.1210i  0.0000  −56626421686.2951
32k4A 1728 0.2283 0.2283i  −3339814.8874  0.0000
64k4D 1728 0.1614 0.1614i  53437038.1988  0.0000
108k4A 0 0.0440 0.0762i  −14699.2655  24463608892439.7456
108k4C 0 0.0554 0.0960i  1608.7743  6115643810955.1724
121k4A −2^{15} 0.0116 0.0385i 85659519816.8841 25723073306989527.1216
144k4E 0 0.0454 0.0262i  81.1130 −549788016394046.1396
27k6A 0 0.0110 0.0191i 0.0000 97856189971744203.7795
32k6A 1728 0.0199 0.0199i −58095643136.7658  8.0094

Table 3.3: Volumes of higher dimensional abelian varieties.

A dim  Ω^{+}  Ω^{−}
79k2B 5  10  209i
83k2B 6  22  41
131k2B 10  51  615
11k4A 2  0.0815 0.0212
17k4B 3  0.0047 0.0007i

Chapter 4

Component groups of optimal quotients

Let A be an abelian variety over the rational numbers Q. The Birch and Swinnerton-Dyer conjecture supplies a formula for the order of the Shafarevich-Tate group of A. A key step in computing this order is to find each of the Tamagawa numbers c_{p} of A. The Tamagawa numbers are defined as follows, where the definition of Néron model and component group is given below.

Definition 4.1 (Tamagawa number). Let p be a prime, let A be a Néron model of A over the p-adic integers Z_{p}, and let Φ_{A,p} be the component group of A at p. Then the Tamagawa number c_{p} of A is the order of the group Φ_{A,p}(F_{p}) of F_{p}-rational points of Φ_{A,p}(\overline{F}_{p}).

Remark 4.2. We warn the reader that the Tamagawa number is defined in a different way in some other papers. The definitions are equivalent.

In this chapter we present a method for computing the Tamagawa numbers c_{p}, up to a power of 2, under the hypothesis that A has purely toric reduction at p. Such A are plentiful among the modular abelian varieties; for example, if A is a new optimal quotient of J_{0}(N) and p exactly divides N, then A is purely toric at p.

In Sections 4.1–4.5 we state and prove an explicit formula involving component groups of fairly general abelian varieties. Then in Section 4.6 we turn to quotients of modular Jacobians J_{0}(N). We give several tables and issue a conjecture and a question.

The results of this chapter were inspired by a letter that Ribet wrote to Mestre, in which he treats the case when A is an elliptic curve.

4.1 Main results

4.1.1 Néron models and component groups

Let A be an abelian variety over a finite extension K of the p-adic numbers Q_{p}. Let O be the ring of integers of K, let 𝔪 be its maximal ideal, and let k = O/𝔪 be the residue class field.

Definition 4.3 (Néron model). A Néron model of A is a smooth commutative group scheme A over O such that A is its generic fiber and A satisfies the Néron mapping property: the restriction map

Hom_{O}(S, A) ⟶ Hom_{K}(S_{K}, A)

is bijective for all smooth schemes S over O.

The Néron mapping property implies that A is unique up to a unique isomorphism, so we will refer without hesitation to “the” Néron model of A.

The closed fiber A_{k} of A is a group scheme over k, which need not be connected; denote by A_{k}^{0} the connected component containing the identity. There is an exact sequence

0 ⟶ A_{k}^{0} ⟶ A_{k} ⟶ Φ_{A} ⟶ 0,

where Φ_{A} a finite étale group scheme over k. Equivalently, Φ_{A} may be viewed as a finite abelian group equipped with an action of Gal(\overline{k}/k).

Definition 4.4 (Component group). The component group of an abelian variety A over a local field K is the group scheme Φ_{A} = A_{k}/A_{k}^{0} defined above.

4.1.2 Motivating problem

This chapter is motivated by the problem of computing the groups Φ_{A,p} attached to quotients A of Jacobians of modular curves X_{0}(N). When A has semistable reduction, Grothendieck and Mumford described the component group in terms of a monodromy pairing on certain free abelian groups. When A = J = J_{0}(N) is the Jacobian of X_{0}(N), this pairing can be explicitly computed, hence the component group Φ_{J} can also be computed; this has been done in many cases in [40] and [23].

Suppose now that A = A_{f} is an optimal quotient of J_{0}(N) that is attached to a newform f, so that the kernel of the map π : J → A is connected. There is a natural map π_{∗} : Φ_{J} → Φ_{A}. We wish to compute the image and the order of the cokernel of π_{∗}.

4.1.3 The main result

We now state our main result more precisely, necessarily supressing some of the definitions of the terms used until later. Suppose π : J → A is an optimal quotient, with J a Jacobian with semistable reduction and A having purely toric reduction. We express the component group of A in terms of the monodromy pairing associated to J.

Let m_{A} = \sqrt{deg(θ_{A})}, where θ_{A} : A^{∨} → A is induced by the canonical principal polarization of J arising from the θ-divisor. Let X_{J} be the character group of the toric part of the closed fiber of the Néron model of J. Let L be the saturation of the image of X_{A} in X_{J}. The monodromy pairing induces a map α : X_{J} → Hom(L, Z). Let Φ_{X} be the cokernel of α and m_{X} = [α(X_{J}) : α(L)] be the order of the finite group α(X_{J})/α(L). We obtain the equality

\frac{#Φ_{A}}{m_{A}} = \frac{#Φ_{X}}{m_{X}} .

Using the snake lemma, one see that Φ_{X} is isomorphic to the image of the natural map Φ_{J} → Φ_{A}, and the above formula implies that the cokernel of the map Φ_{J} → Φ_{A} has order m_{A}/m_{X}.

If the optimal quotient J → A arises from a modular form on Γ_{0}(N), then the quantities m_{A}, m_{X} and Φ_{X} can be explicitly computed, hence we can compute #Φ_{A}.

4.2 Optimal quotients of Jacobians

Let J be a Jacobian, and let θ_{J} be the canonical principal polarization arising from the θ-divisor. Recall that an optimal quotient of J is an abelian variety A and a surjective map π : J → A whose kernel is an abelian subvariety B of J. Denote by J^{∨} and A^{∨} the abelian varieties dual to J and A, respectively. Upon composing the dual of π with θ_{J}^{∨} = θ_{J}, we obtain a map

A^{∨} ⟶^{π^{∨}} J^{∨} ⟶^{θ_{J}} J.

Proposition 4.5. The map A^{∨} → J is injective.

Proof. Since θ_{J} is an isomorphism it suffices to prove that π^{∨} is injective. Since the dual of π^{∨} is (π^{∨})^{∨} = π and π is surjective, the map π^{∨} must have finite kernel. Thus A^{∨} → C = im(π^{∨}) is an isogeny. Let G denote the kernel of this isogeny, and dualize. By [50, §11] we have the following two commutative diagrams:

\xymatrix{G \ar[r] & A^{∨} \ar@{->>}[r] \ar[dr]_{π^{∨}} & C \ar[d] & \underrightarrow{dualize} & A & C^{∨} \ar[l] & G^{∨} \ar[l] \\  &  & J^{∨} &  &  & J, \ar[u]_{φ} \ar[ul]^{π}}

where G^{∨} is the Cartier dual of G. Since G^{∨} is finite, ker(φ) is of finite index in ker(π). Since ker(π) is an abelian variety, as a group it is divisible. But a divisible group has no nontrivial finite-index subgroups (divisibility is a property inherited by quotients, and nonzero finite groups are not divisible). Thus ker(φ) = ker(π), so G^{∨} = 0. It follows that G = 0.

Henceforth we will abuse notation and denote the injection A^{∨} → J by π^{∨}. The kernel of θ_{A} equals the intersection of A^{∨} and B = ker(π), as depicted in the following diagram:

\xymatrix{A^{∨} ∩ B \ar[r] \ar[d] & B \ar[d] \\ A^{∨} \ar@{^{(}->}[r]^{π^{∨}} \ar[dr]_{θ_{A}} & J \ar[d]^{π} \\  & A.}

Since θ_{A} is a polarization, the degree # ker(θ_{A}) of θ_{A} is a perfect square (see [50, Thm. 13.3]). Recall that the modular degree is the integer

m_{A} = \sqrt{# ker(θ_{A})}.

For an algorithm to compute m_{A}, see Section 3.9 and Corollary 4.23.

4.3 The closed fiber of the Néron model

Let K be a finite extension of Q_{p} with ring of integers O and residue class field k. Let A be an abelian variety over K and denote its Néron model by A. Let Φ_{A} be the group of connected components of the closed fiber A_{k}. This group is a finite étale group scheme over k; equivalently, it is a finite abelian group equipped with an action of Gal(\overline{k}/k). There is an exact sequence of group schemes

0 → A_{k}^{0} → A_{k} → Φ_{A} → 0.

The group scheme A_{k}^{0} is an extension of an abelian variety B of some dimension a by a group scheme C; we have a diagram

\xymatrix{ & 0 \ar[d] \\  & T \ar[d] \\ 0 \ar[r] & C \ar[r] \ar[d] & A_{k}^{0} \ar[r] & B \ar[r] & 0 \\  & U \ar[d] \\  & 0}

with T a torus of dimension t and U a unipotent group of dimension u. The abelian variety A is said to have purely toric reduction if t = dim A, and have semistable reduction if u = 0.

Definition 4.6 (Character group of torus). The character group

X_{A} = Hom\overline{k}(T_{/\overline{k}}, G_{m}_{/\overline{k}})

is a free abelian group of rank t contravariantly associated to A.

As discussed in, e.g., [53], if A is semistable there is a monodromy pairing X_{A}×X_{A^{∨}} → Z and an exact sequence

0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A} → 0.

4.4 Rigid uniformization

In this section we review the rigid analytic uniformization of a semistable abelian variety over a finite extension K of the maximal unramified extension Q_{p}^{ur} of Q_{p}. We use this uniformization to prove that if A has purely toric reduction, and ϕ : A^{∨} → A is a symmetric isogeny (as defined below), then

deg(ϕ) = (# coker(X_{A} → X_{A^{∨}}))^{2}.

We also prove some lemmas about character groups.

It is possible to prove the assertions we will need without recourse to rigid uniformization, as Ahmed Abbes has pointed out to the author.

4.4.1 Raynaud’s uniformization

Theorem 4.7 (Raynaud). If A is a semistable abelian variety, its universal covering (as defined in [14]) is isomorphic to an extension G of an abelian variety B with good reduction by a torus T. The covering map from G to A is a homomorphism, and its kernel is a twisted free abelian group Γ of finite rank.

This may be summarized by the diagram

\xymatrix{ & Γ \ar[d] \\ T \ar[r] & G \ar[r] \ar[d] & B \\  & A,}

which we call the uniformization cross of A.

Remark 4.8. The group Γ can be identified with the character group X_{A^{∨}} of the previous and latter sections.

The uniformization cross of the dual abelian variety A^{∨} is

\xymatrix{ & Γ^{∨} \ar[d] \\ T^{∨} \ar[r] & G^{∨} \ar[r] \ar[d] & B^{∨} \\  & A^{∨},}

where Γ^{∨} = Hom(T, G_{m}), where T^{∨} = Hom(Γ, G_{m}), and the morphisms Γ^{∨} → G^{∨} and T^{∨} → G^{∨} are the one-motif duals of the morphisms T → G and Γ → G, respectively. For more details see, e.g., [14].

To avoid confusion when considering the uniformization of more than one abelian variety, we will often denote the objects T, G, Γ, and B connected with A by T_{A}, G_{A}, Γ_{A}, and B_{A}, respectively.

Example 4.9 (Tate curve). If E/Q_{p} is an elliptic curve with split multiplicative reduction, then the uniformization is E = G_{m}/q^{Z} where q = q(j) is obtained by inverting the expression for j as a function of q(z) = e^{2πiz}.

4.4.2 Some lemmas

Let π : J → A be an optimal quotient, assume that J has semistable reduction, and that A has purely toric reduction.

Lemma 4.10. The map Γ_{J} → Γ_{A} induced by π is surjective.

Proof. Since G_{J} is simply connected, π induces a map G_{J} → T_{A} and a map Γ_{J} → Γ_{A}. Because π is surjective and T_{A} is a torus, the map G_{J} → T_{A} is surjective. Upon applying the snake lemma to the following diagram, we obtain a surjective map from B = ker(π) to M = coker(Γ_{J} → Γ_{A}):

\xymatrix{ & Γ_{J} \ar[r] \ar[d] & Γ_{A} \ar[r] \ar[d] & M \ar[r] \ar[d] & 0 \\  & G_{J} \ar[r] \ar[d] & T_{A} \ar[d] \ar[r] & 0 \\ B \ar[r] & J \ar[r]^{π} & A.}

Since π : J → A is an optimal quotient, the kernel B is connected. Thus M must also be connected. Since M is discrete it follows that M = 0.

Abelian varieties with purely toric reduction

Assume that A has purely toric reduction. Then B = 0, and the uniformization cross is simply

Γ

T

A.

Definition 4.11 (Symmetric isogeny). A symmetric isogeny φ : A^{∨} → A is an isogeny such that the map φ^{∨} : A^{∨} → (A^{∨})^{∨} = A is equal to φ.

Let φ : A^{∨} → A be a symmetric isogeny. Denote by φ_{t} : T^{∨} → T and φ_{a} : Γ^{∨} → Γ the maps induced by φ.

Proposition 4.12. There is an exact sequence

0 → ker(φ_{t}) → ker(φ) → coker(φ_{a}) → 0,

and ker(φ_{t}) is the Cartier dual of coker(φ_{a}).

Proof. Since φ is an isogeny we obtain the following diagram:

\xymatrix{0 \ar[r] \ar[d] & Γ^{∨} \ar[r]^{φ_{a}} \ar[d] & Γ \ar[r] \ar[d] & coker(φ_{a}) \ar[d] \\ ker(φ_{t}) \ar[r] \ar[d] & T^{∨} \ar[d] \ar[r]^{φ_{t}} & T \ar[r] \ar[d] & 0 \\ ker(φ) \ar[r] & A^{∨} \ar[r]^{φ} & A.}

The snake lemma then gives the claimed exact sequence.

For the second assertion, observe that if we take one-motif duals of every object in the diagram

\xymatrix{ & Γ^{∨} \ar[r]^{φ_{a}} \ar[d] & Γ \ar[d] \ar[r] & coker(φ_{a}) \\ ker(φ_{t}) \ar[r] & T^{∨} \ar[r]^{φ_{t}} & T}

we obtain the following diagram:

\xymatrix{ & T & T^{∨} \ar[l]_{φ_{a}^{∨}} & coker(φ_{a})^{∨} \ar[l] \\ ker(φ_{t})^{∨} & Γ \ar[l] \ar[u] & Γ^{∨}. \ar[l]^{φ_{t}^{∨}} \ar[u]}

Since φ is symmetric, φ_{a}^{∨} = φ_{t}, so

ker(φ_{t}) = coker(φ_{a})^{∨}.

Lemma 4.13. # ker(φ) = # coker(φ_{a})^{2}

Proof. Use the exact sequence of Proposition 4.12 together with the observation that the order of a finite group scheme equals the order of its Cartier dual.

4.5 The main theorem

Let π : J → A be an optimal quotient, with J a Jacobian having semistable reduction and A an abelian variety having purely toric reduction. Let X_{A}, X_{A^{∨}}, and X_{J} denote the character groups of the toric parts of the closed fibers of the abelian varieties A, A^{∨}, and J, respectively.

4.5.1 Description of the component group in terms of the monodromy pairing

Recall that there is a pairing X_{A} × X_{A^{∨}} → Z called the monodromy pairing. We have an exact sequence

0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A} → 0.

If J is a Jacobian then J is canonically self-dual via the θ-polarization, so the monodromy pairing on J can be viewed as a pairing X_{J} × X_{J} → Z, and there is an exact sequence

0 → X_{J} → Hom(X_{J}, Z) → Φ_{J} → 0.

Example 4.14 (Tate curve). Suppose E = G_{m}/q^{Z} is a Tate curve over Q_{p}^{ur}. The monodromy pairing on X_{E} = q^{Z} is

⟨q, q⟩ = ord_{p}(q) = − ord_{p}(j).

Thus Φ_{E} is cyclic of order − ord_{p}(j).

Proof of the main theorem

We now prove the main theorem. Let π : J → A be an optimal quotient, and let θ : A^{∨} → A denote the induced polarization. Let π_{∗}, π^{∗}, θ_{∗}, and θ^{∗} be the maps induced on character groups by the various functorialities, as indicated in the following two key diagrams:

\xymatrix{A^{∨} \ar@{^{(}->}[r]^{π^{∨}} \ar[dr]_{θ} & J \ar@{->>}[d]^{π} & X_{A} \ar@{^{(}->}[r]^{π^{∗}} \ar[dr]^{θ^{∗}} & X_{J} \ar@{->>}[d]^{π_{∗}} \\  & A &  & X_{A^{∨}}. \ar[ul]^{θ_{∗}}}

The surjectivity of π_{∗} was proved in Lemma 4.10. The injectivity of π^{∗} follows because

θ_{∗}π_{∗}π^{∗} = θ_{∗}θ^{∗} = deg(θ) ≠ 0,

and multiplication by a nonzero integer on a free abelian group is injective.

Let

α : X_{J} → Hom(π^{∗}X_{A}, Z)

be the map defined by the monodromy pairing restricted to X_{J} × π^{∗}X_{A}.

Lemma 4.15. ker(π_{∗}) = ker(α)

Proof. Suppose x ∈ ker(π_{∗}), and let y = π^{∗}z with z ∈ X_{A}. Then

⟨x, y⟩ = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩ = 0,

so x ∈ ker(α). Next let x ∈ ker(α). Then for all z ∈ X_{A},

0 = ⟨x, π^{∗}z⟩ = ⟨π_{∗}x, z⟩,

so π_{∗}x is in the kernel of the monodromy map

X_{A^{∨}} → Hom(X_{A}, Z).

Since X_{A^{∨}} and Hom(X_{A}, Z) are free of the same finite rank and the cokernel is torsion, the monodromy map is injective. Thus π_{∗}x = 0 and x ∈ ker(π_{∗}).

Lemma 4.16. There is an exact sequence

X_{J} → Hom(π^{∗}X_{A}, Z) → Φ_{A} → 0.

Proof. Lemma 4.15 gives the following commutative diagram with exact rows

\xymatrix{0 \ar[r] & X_{J}/ ker(α) \ar[d]^{≅} \ar[r] & Hom(π^{∗}X_{A}, Z) \ar[r] \ar[d]^{≅} & coker(α) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & X_{A^{∨}} \ar[r] & Hom(X_{A}, Z) \ar[r] & Φ_{A} \ar[r] & 0.}

By Lemma 4.15, the first vertical map is an isomorphism. The second is an isomorphism because it is induced by the isomorphism π^{∗} : X_{A} → π^{∗}X_{A}. It follows that coker(α) ≅ Φ_{A}, as claimed.

Let L be the saturation of π^{∗}X_{A} in X_{J}; thus π^{∗}X_{A} is a finite-index subgroup of L and the quotient X_{J}/L is torsion free. For L of finite index in L, define the modular degree of L to be

m_{L} = [α(X_{J}) : α(L)],

and the component group of L to be

Φ_{L} = coker(X_{J} → Hom(L, Z)).

When L = L and A is fixed, we often slightly abuse notation and write m_{X} = m_{L} and Φ_{X} = Φ_{L}. We think of m_{X} and Φ_{X} as the character group “modular degree and component group” of A.

Lemma 4.17. Choose a subgroup L of finite index in L. The rational number \frac{#Φ_{L}}{m_{L}} is

independent of the choice of L.

Proof. Suppose L′ is another finite index subgroup of L, and let n = [L : L′]. Here n is a rational number, the lattice index of L′ in L. Since α is injective when restricted to L, it follows that

m_{L′} = [α(X_{J}) : α(L′)] = [α(X_{J}) : α(L)] · [α(L) : α(L′)] = m_{L} · n.

Similarly, #Φ_{L′} = #Φ_{L} · n.

Recall that m_{A} = \sqrt{deg(θ)} and
Φ_{A} ≅ coker(X_{A^{∨}} → Hom(X_{A}, Z)),

where m_{A} is the modular degree of A and Φ_{A} is the component group of A.

Theorem 4.18. For any subgroup L of finite index in L, the following relation holds:

\frac{#Φ_{A}}{m_{A}} = \frac{#Φ_{L}}{m_{L}} .

Proof. By Lemma 4.17 we may assume that L = π^{∗}X_{A}. With this choice of L, Lemma 4.16 asserts that Φ_{L} ≅ Φ_{A}. By Lemma 4.15, properties of the index, and Lemma 4.13 we have

m_{L} = [α(X_{J}) : α(L)]
= [π_{∗}(X_{J}) : π_{∗}(L)]
= [X_{A^{∨}} : π_{∗}(π^{∗}X_{A})]
= [X_{A^{∨}} : θ^{∗}X_{A}]
= # coker(θ^{∗})
= \sqrt{deg(θ)} = m_{A}.

Proposition 4.19.

image(Φ_{J} → Φ_{A}) ≅ Φ_{L}.

Proof. Since π^{∗}X_{A} ⊂ L ⊂ X_{J}, an application of Lemma 4.16 gives the following commutative diagram with exact rows:

[figure: X_{J}; Hom(X_{J}, Z); Φ_{J}; 0; X_{J}; Hom(L, Z); Φ_{L}; 0; X_{J}; Hom(π^{∗}X_{A}, Z); Φ_{A}; 0.]

The map Hom(L, Z) → Hom(π^{∗}X_{A}, Z) is an isomorphism, so the map Φ_{L} → Φ_{A} is injective. Thus

image(Φ_{J} → Φ_{A}) ≅ image(Φ_{J} → Φ_{L}).

The cokernel of Hom(X_{J}, Z) → Hom(L, Z) surjects onto the cokernel of Φ_{J} → Φ_{L}. Using the exact sequence

0 → L → X_{J} → X_{J}/L → 0,

we find that

coker(Hom(X_{J}, Z) → Hom(L, Z)) ⊂ Ext^{1}(X_{J}/L, Z).

Because L is saturated, the quotient X_{J}/L is torsion free, so the indicated Ext^{1} group vanishes. Thus the map Φ_{J} → Φ_{L} is surjective, from which the proposition follows.

The following corollary follows from Theorem 4.18 and Proposition 4.19.

Corollary 4.20.

# coker(X_{J} → X_{A}) = \frac{m_{A}}{m_{L}} .

Remark 4.21. A non-obvious consequence of this corollary is that

m_{L} | m_{A}.

4.6 Optimal quotients of J_{0}(N)

We now summarize some facts about J_{0}(N) that will be used in our numerical computations. Some of these facts were discussed in greater generality in the previous chapters of this thesis.

4.6.1 Modular curves and semistability

Let X_{0}(N) be the modular curve associated to the subgroup Γ_{0}(N) of SL_{2}(Z) that consists of those matrices which are upper triangular modulo N. Initially, X_{0}(N) is constructed as a Riemann surface as the quotient

Γ_{0}(N)\({z : z ∈ C, Im(z) > 0} ∪ P^{1}(Q)).

With some work, we find that X_{0}(N) has a canonical structure of algebraic curve over Q.

Suppose that p is a prime divisor of N such that N/p is coprime to p. We write p || N. In this situation, it is well-known that the Jacobian J_{0}(N) of X_{0}(N) has semistable reduction at p.

4.6.2 Newforms and optimal quotients

The Hecke algebra

T = Z[… T_{n} …] ⊂ End(J_{0}(N))

is a commutative ring of endomorphisms of J_{0}(N) of Z-rank equal to the dimension J_{0}(N). The character group X_{J_{0}(N)} of J_{0}(N) at p is equipped with a functorial action of T. The Hecke algebra T also acts on the complex vector space S = S_{2}(Γ_{0}(N), C) of cusp forms.

A newform f is an eigenform normalized so that the coefficient of q in the Fourier expansion of f at the cusp ∞ is 1, and such that f is not a modular form of any level N′ | N, with N′ a proper divisor of N.

Let f be a newform, and associate to f the ideal I_{f} of the Hecke algebra T of elements which annihilate f. Then O_{f} = T/I_{f} is an order in the ring of integers of the totally real number field K_{f} obtained by adjoining the Fourier coefficients of f to Q. The quotient

A_{f} = J_{0}(N)/I_{f}J_{0}(N)

is an optimal quotient of J_{0}(N) of dimension equal to [K_{f} : Q]. It is purely toric at p, since p || N.

4.6.3 Homology and the modular degree

Let H = H_{1}(X_{0}(N), Z) be the integral homology of the complex algebraic curve X_{0}(N). Integration defines a T-equivariant nondegenerate pairing S ×H → C. This pairing induces a map α : H → Hom_{C}(S, C).

4.6. OPTIMAL QUOTIENTS OF J_{0}(N) 81

Theorem 4.22. We have the following commutative diagram of T-modules:

\xymatrix{H[I_{f}] \ar@{^{(}->}[r] \ar@{^{(}->}[d] & H \ar@{->>}[r] \ar@{^{(}->}[d] & α(H) \ar[d] \ar@{^{(}->}[d] \\ Hom_{C}(S, C)[I_{f}] \ar@{^{(}->}[r] \ar@{->>}[d] & Hom_{C}(S, C) \ar@{->>}[r] \ar@{->>}[d] & Hom_{C}(S[I_{f}], C) \ar@{->>}[d] \\ A_{f}^{∨}(C) \ar@{^{(}->}[r] \ar[rr]_{θ_{A}} & J(C) \ar@{->>}[r] & A_{f}(C)}

Proof. This can be deduced from [61]. See also Section 2.7.

Corollary 4.23. m_{A}^{2} = [α(H) : α(H[I_{f}])].

Proof. Recall that m_{A} is by definition equal to \sqrt{deg(θ_{A})}. The kernel of an isogeny between complex tori is isomorphic to the cokernel of the induced map on lattices. The corollary now follows from the diagram of Theorem 4.22, which indicates that the index [α(H) : α(H[I_{f}])] is the cokernel of the map H[I_{f}] → α(H).

For more details, see Section 3.9.

4.6.4 Rational points of the component group (Tamagawa numbers)

Let Frob_{p} : X_{J} → X_{J} denote the map induced by the Frobenius automorphism. We have Frob_{p} = −W_{p}, where W_{p} is the map induced by the Atkin-Lehner involution on J_{0}(p). Let f be a newform, A = A_{f} the corresponding optimal quotient, and w_{p} the sign of the eigenvalue of W_{p} on f.

Proposition 4.24.

Φ_{A}(F_{p}) = \begin{cases}Φ_{A}(\overline{F}_{p}) & if w_{p} = −1, \\ Φ_{A}(\overline{F}_{p})[2] & if w_{p} = 1. \end{cases}

Proof. If w_{p} = −1, then Frob_{p} = 1 and the Gal(\overline{F}_{p}/F_{p})-action of Φ_{A}(\overline{F}_{p}) is trivial. In this case Φ(F_{p}) = Φ(\overline{F}_{p}). Next suppose w_{p} = 1. Recall that we have an exact sequence

0 → X_{A^{∨}} → Hom(X_{A}, Z) → Φ_{A} → 0.

Since W_{p} acts as +1 on f, it also acts as +1 on each of the four modules A, X_{A}, Hom(X_{A}, Z), and Φ_{A}. Thus Frob_{p} = −W_{p} acts as −1 on Φ_{A}. Since the subgroup of 2-torsion elements of a finite abelian group equals the subgroup of elements fixed under −1, it follows that Φ_{A}(F_{p}) = Φ_{A}(\overline{F}_{p})[2].

WARNING: When we extend this result to the whole of J_{0}(N), it is necessary to be exceedingly careful! The action of Frob_{p} = T_{p} need not be by ±1, even though it must be by an involution of order 2. For example, the component group of J_{0}(65) at 5 is cyclic of order 42. The action of Frob_{5} is by multiplication by −13. Note that (−13)^{2} = 1 (mod 42). The fixed points of multiplication by −13 is the order 14 subgroup of Z/42Z.

4.7 Computations

Using the algorithms of Chapter 3, we can enumerate the optimal quotients A_{f} of J_{0}(N) and compute the modular degree m_{A}. The method of graphs (see [47]) and quaternion algebras (see [32]) can be used to compute X = X_{J_{0}(N)} with its T-action and the monodromy pairing. We can then compute the following three modules: the saturated submodule L = ⋂_{t∈I_{f}} ker(t) of X, the character group modular degree m_{X} = m_{L}, and Φ_{X} = Φ_{L}. By Theorem 4.18 we obtain

#Φ_{A} = #Φ_{X} · \frac{m_{A}}{m_{X}} .

Using this method, we have computed #Φ_{A} in a number of cases. We give tables that report on some of these computations in Secton 4.7.2. In the next section we discuss a conjecture and a question, which were both suggested by our numerical computations.

4.7.1 Conjectures and questions

Suppose that N = pM with (p, M) = 1. Let

H_{new} = ker (H_{1}(X_{0}(N), Z) ⟶ H_{1}(X_{0}(M), Z) ⊕ H_{1}(X_{0}(M), Z)),

where the map is induced by the two natural degeneracy maps X_{0}(N) → X_{0}(M).

The Hecke algebra T acts on H_{new}, and also on the submodule H_{new}[I_{f}] of those elements that are annihilated by I_{f}. Integration defines a map α : H_{new} → Hom(S[I_{f}], C). Define the p-new homology modular degree m_{H} by

m_{H}^{2} = [α(H_{new}) : α(H_{new}[I_{f}])].

We expect that there is a very close relationship between m_{X} and m_{H}.

Question 4.25. Is m_{X} equal to m_{H}?

The following conjecture offers a refinement of some of the results of [40].

Conjecture 4.26 (Refined Eisenstein conjecture). Let p be a prime and let f_{1}, …, f_{n} be a set of representatives for the Galois-conjugacy classes of newforms in S_{2}(Γ_{0}(p)). Let A_{1}, …, A_{n} be the optimal quotients associated to f_{1}, …, f_{n}, respectively. Then for each i, i = 1, …, n, we have

#A_{i}(Q)_{tor} = #Φ_{A_{i}}(\overline{F}_{p}) = #Φ_{A_{i}}(F_{p}).

Furthermore,

#Φ_{J_{0}(p)}(\overline{F}_{p}) = ∏_{i=1}^{d} #Φ_{A_{i}}(\overline{F}_{p}).

We have verified Conjecture 4.26 for all p ≤ 757, and, up to a power of 2, for all p < 2000.

Remark 4.27. It is tempting to guess that, e.g., the natural map

Φ_{J_{0}(113)}(\overline{F}_{p}) → ∏_{i=1}^{4} Φ_{A_{i}}(\overline{F}_{p})

is an isomorphism. Two of the Φ_{A_{i}}(\overline{F}_{p}) have order 2, so the product ∏ Φ_{A_{i}}(\overline{F}_{p}) can not be a cyclic group. However, the groups Φ_{J_{0}(p)}(\overline{F}_{p}) are known to be cyclic for all primes p.

4.7.2 Tables

We have computed component groups of many optimal quotients A_{f} of J_{0}(N). In this section we provide tables, which hint at the data we have gathered. Our notation for optimal quotients is described in Section 1.3.1. See also Table 1.6.

Table 4.1: Component groups at low level

Table 4.1 gives the component groups of the quotients A_{f} of J_{0}(N) for N ≤ 106. The column labeled d contains the dimensions of the A_{f}, and the column labeled #Φ_{A,p} contains a list of the orders of the component groups of A_{f}, one for each divisor p of N, ordered by increasing p. An entry of “?” indicates that p^{2} | N, so our algorithm does not apply. A component group order is starred if the Gal(\overline{F}_{p}/F_{p})-action is nontrivial.

Table 4.2–4.3: Big component groups

Using the algorithms described in Section 3.10, we computed the rational numbers L(A, 1)/Ω_{A} for every optimal quotient A that is attached to a newform of level ≤ 1500. There are exactly 5 optimal quotients A such that the numerator of L(A, 1)/Ω_{A} is nonzero and divisible by a prime > 10^{9}. The Birch and Swinnerton-Dyer conjecture predicts that these large prime divisors must divide either #Φ_{A} or the Shafarevich-Tate group of A. This is the case, as Table 4.3 shows.

Table 4.4: Quotients of J_{0}(N)

Table 4.4 contains all of the invariants involved in the computation of component groups for each of the newform optimal quotients of levels 65, 66, 68, and 69.

Table 4.5: Quotients of J_{0}(p)^{−}

We computed the quantities m_{A}, m_{X} and Φ_{X} for each abelian variety A = A_{f} associated to a newform of prime level p with p ≤ 757. The results are as follows:

1. In all cases m_{A} = m_{X}, so the map Φ_{J} → Φ_{A} is surjective.

2. Φ_{A} = 1 whenever the sign of the Atkin-Lehner involution w_{p} on A is 1.

3. ∏ #Φ_{A}(\overline{F}_{p}) = #Φ_{J}(\overline{F}_{p})

Table 4.5 lists those A of level ≤ 631 for which w_{p} = −1, along with the order of the corresponding component group.

84 CHAPTER 4. COMPONENT GROUPS OF OPTIMAL QUOTIENTS

Table 4.1: Component groups at low level

A  d #Φ_{A,p}
11A 1 5
14A 1 6^{∗}, 3
15A 1 4^{∗}, 4
17A 1 4
19A 1 3
20A 1 ?, 2^{∗}
21A 1 4, 2^{∗}
23A 2 11
24A 1 ?, 2^{∗}
26A 1 3^{∗}, 3
26B 1 7, 1^{∗}
27A 1 ?
29A 2 7
30A 1 4^{∗}, 3, 1^{∗}
31A 2 5
32A 1 ?
33A 1 6^{∗}, 2
34A 1 6, 1^{∗}
35A 1 3^{∗}, 3
35B 2 8, 4^{∗}
36A 1 ?,?
37A 1 1^{∗}
37B 1 3
38A 1 9^{∗}, 3
38B 1 5, 1^{∗}
39A 1 2^{∗}, 2
39B 2 14, 2^{∗}
40A 1 ?, 2
41A 3 10
42A 1 8, 2^{∗}, 1^{∗}
43A 1 1^{∗}
43B 2 7
44A 1 ?, 1^{∗}
45A 1 ?, 1^{∗}
46A 1 10^{∗}, 1
47A 4 23
48A 1 ?, 2
49A 1 ?
50A 1 1^{∗},?
50B 1 5,?

A  d #Φ_{A,p}
51A 1 3, 1^{∗}
51B 2 16^{∗}, 4
52A 1 ?, 2^{∗}
53A 1 1^{∗}
53A 1 1^{∗}
53B 3 13
54A 1 3^{∗},?
54B 1 3,?
55A 1 2, 2^{∗}
55B 2 14^{∗}, 2
56A 1 ?, 1
56B 1 ?, 1^{∗}
57A 1 2^{∗}, 1^{∗}
57B 1 2, 2^{∗}
57C 1 10, 1^{∗}
58A 1 2^{∗}, 1^{∗}
58B 1 10, 1^{∗}
59A 5 29
61A 1 1^{∗}
61B 3 5
62A 1 4, 1^{∗}
62B 2 66^{∗}, 3
63A 1 ?, 1^{∗}
63B 2 ?, 3
64A 1 ?
65A 1 1^{∗}, 1^{∗}
65B 2 3^{∗}, 3
65C 2 7, 1^{∗}
66A 1 2^{∗}, 3, 1^{∗}
66B 1 4, 1^{∗}, 1^{∗}
66C 1 10, 5, 1
67A 1 1
67B 2 1^{∗}
67C 2 11
68A 2 ?, 2^{∗}
69A 1 2, 1^{∗}
69B 2 22^{∗}, 2
70A 1 4, 2^{∗}, 1^{∗}
71A 3 5
71B 3 7

A  d #Φ_{A,p}
72A 1 ?,?
73A 1 2
73B 2 1^{∗}
73C 2 3
74A 2 9^{∗}, 3
74B 2 95, 1^{∗}
75A 1 1^{∗},?
75B 1 1,?
75C 1 5,?
76A 1 ?, 1^{∗}
77A 1 2^{∗}, 1^{∗}
77B 1 3^{∗}, 2
77C 1 6, 3^{∗}
77D 2 2, 2^{∗}
78A 1 16^{∗}, 5^{∗}, 1
79A 1 1^{∗}
79B 5 13
80A 1 ?, 2
80B 1 ?, 2^{∗}
81A 2 ?
82A 1 2^{∗}, 1^{∗}
82B 2 28, 1^{∗}
83A 1 1^{∗}
83B 6 41
84A 1 ?, 1^{∗}, 2^{∗}
84B 1 ?, 3, 2
85A 1 2^{∗}, 1
85B 2 2^{∗}, 1^{∗}
85C 2 6, 1^{∗}
86A 2 21^{∗}, 3
86B 2 55, 1^{∗}
87A 2 5, 1^{∗}
87B 3 92^{∗}, 4
88A 1 ?, 1^{∗}
88B 2 ?, 2^{∗}
89A 1 1^{∗}
89B 1 2
89C 5 11
90A 1 2^{∗},?, 3
90B 1 6,?, 1^{∗}

A  d #Φ_{A,p}
90C 1 4,?, 1
91A 1 1^{∗}, 1^{∗}
91B 1 1, 1
91C 2 7, 1^{∗}
91D 3 4^{∗}, 8
92A 1 ?, 1^{∗}
92B 1 ?, 1
93A 2 4^{∗}, 1^{∗}
93B 3 64, 2^{∗}
94A 1 2, 1^{∗}
94B 2 94^{∗}, 1
95A 3 10, 2^{∗}
95B 4 54^{∗}, 6
96A 1 ?, 2
96B 1 ?, 2^{∗}
97A 3 1^{∗}
97B 4 8
98A 1 2^{∗},?
98B 2 14,?
99A 1 ?, 1^{∗}
99B 1 ?, 1
99C 1 ?, 1^{∗}
99D 1 ?, 1^{∗}
100A 1 ?,?
101A 1 1^{∗}
101B 7 25
102A 1 2^{∗}, 2^{∗}, 1^{∗}
102B 1 6^{∗}, 6, 1^{∗}
102C 1 8, 4, 1
103A 2 1^{∗}
103B 6 17
104A 1 ?, 1^{∗}
104B 2 ?, 2
105A 1 1, 1, 1
105B 2 10^{∗}, 2^{∗}, 2
106A 1 4^{∗}, 1^{∗}
106B 1 5^{∗}, 1
106C 1 24, 1^{∗}
106D 1 3, 1^{∗}

4.7. COMPUTATIONS 85

Table 4.2: Big L(A, 1)/Ω_{A}

A dim N L(A, 1)/Ω_{A} · Manin constant
1154E 20 2 · 577  2^{?} · 85495047371/17^{2}
1238G 19 2 · 619  2^{?} · 7553329019/5 · 31
1322E 21 2 · 661  2^{?} · 57851840099/331
1382D 20 2 · 691  2^{?} · 37 · 1864449649/173
1478J 20 2 · 739  2^{?} · 7 · 29 · 1183045463/5 · 37

Table 4.3: Big component groups

A  p w #Φ_{X}  m_{X}  #Φ_{A}(\overline{F}_{p})
1154E 2 − 17^{2}  2^{24}  2^{?} · 17^{2} · 85495047371
577 + 1  2^{26} · 85495047371  2^{?}
1238G 2 − 5 · 31  2^{26}  2^{?} · 5 · 31 · 7553329019
619 + 1  2^{28} · 7553329019  2^{?}
1322E 2 − 331  2^{28}  2^{?} · 331 · 57851840099
661 + 1  2^{32} · 57851840099  2^{?}
1382D 2 − 173  2^{29}  2^{?} · 37 · 173 · 1864449649
691 + 1  2^{31} · 37 · 1864449649  2^{?}
1478J 2 − 5 · 37  2^{31}  2^{?} · 5 · 7 · 29 · 37 · 1183045463
739 + 1 2^{33} · 7 · 29 · 1183045463  2^{?}

86 CHAPTER 4. COMPONENT GROUPS OF OPTIMAL QUOTIENTS Table 4.4: Component groups of quotients of J_{0}(N)

A  dim p w_{p} #Φ_{X} m_{X} m_{A} #Φ_{A}
65A 1 5 + 1  2  2  1
13 + 1  2  1
65B 2 5 + 3  2^{2}  2^{2}  3
13 − 3  2^{2}  3
65C 2 5 − 7  2^{2}  2^{2}  7
13 + 1  2^{2}  1
66A 1 2 + 1  2  2^{2}  2
3 − 3  2^{2}  3
11 + 1  2^{2}  1
66B 1 2 − 2  2  2^{2}  2^{2}
3 + 1  2^{2}  1
11 + 1  2^{2}  1
66C 1 2 − 1  2 2^{2} · 5 2 · 5
3 − 1 2^{2} 5
11 − 1 2^{2} · 5  1
68A 2 17 + 2  2 · 3 2 · 3  2
69A 1 3 − 2  2  2  2
23 + 1  2  1
69B 2 3 + 2  2 2 · 11 2 · 11
23 − 2 2 · 11  2

4.7. COMPUTATIONS 87

Table 4.5: Component groups of quotients of J_{0}(p)^{−}

A  d #Φ_{A}
11A 1  5
17A 1 2^{2}
19A 1  3
23A 2 11
29A 2  7
31A 2  5
37B 1  3
41A 3 2 · 5
43B 2  7
47A 4 23
53B 3 13
59A 5 29
61B 3  5
67A 1  1
67C 2 11
71A 3  5
71B 3  7
73A 1  2
73C 2  3
79B 5 13
83B 6 41
89B 1  2
89C 5 11
97B 4 2^{3}
101B 7 5^{2}
103B 6 17
107B 7 53
109A 1  1
109C 4 3^{2}
113A 1  2
113B 2  2
113D 3  7
127B 7 3 · 7
131B 10 5 · 13
137B 7 2 · 17
139A 1  1
139C 7 23
149B 9 37
151B 3  1
151C 6 5^{2}

A  d #Φ_{A}
157B 7  13
163C 7  3^{3}
167B 12 83
173B 10 43
179A 1  1
179C 11 89
181B 9 3 · 5
191B 14 5 · 19
193C 8  2^{4}
197C 10 7^{2}
199A 2  1
199C 10 3 · 11
211A 2  5
211D 9  7
223C 12 37
227B 2  1
227C 2  1
227E 10 113
229C 11 19
233A 1  2
233C 11 29
239B 17 7 · 17
241B 12 2^{2} · 5
251B 17 5^{3}
257B 14 2^{6}
263B 17 131
269C 16 67
271B 16 3^{2} · 5
277B 3  1
277D 9  23
281B 16 2 · 5 · 7
283B 14 47
293B 16 73
307A 1  1
307B 1  1
307C 1  1
307D 1  1
307E 2  3
307F 9  17
311B 22 5 · 31

A  d #Φ_{A}
313A 2  1
313C 12 2 · 13
317B 15 79
331D 16 5 · 11
337B 15 2^{2} · 7
347D 19 173
349B 17 29
353A 1  2
353B 3  2
353D 14 2 · 11
359D 24 179
367B 19 61
373C 17 31
379B 18 3^{2} · 7
383C 24 191
389A 1  1
389E 20 97
397B 2  1
397C 5  11
397D 10  3
401B 21 2^{2} · 5^{2}
409B 20 2 · 17
419B 26 11 · 19
421B 19 5 · 7
431B 1  1
431D 3  1
431F 24 5 · 43
433A 1  1
433B 3  1
433D 16 2^{2} · 3^{2}
439C 25 73
443C 1  1
443E 22 13 · 17
449B 23 2^{4} · 7
457C 20 2 · 19
461D 26 5 · 23
463B 22 7 · 11
467C 26 233
479B 32 239
487A 2  1

A  d #Φ_{A}
487B 2  3
487C 3  1
487D 16  3^{3}
491C 29 5 · 7^{2}
499C 23  83
503B 1  1
503C 1  1
503D 3  1
503F 26 251
509B 28 127
521B 29 2 · 5 · 13
523C 26 3 · 29
541B 24 3^{2} · 5
547C 25 7 · 13
557B 1  1
557D 26 139
563A 1  1
563E 31 281
569B 31 2 · 71
571A 1  1
571B 1  1
571C 2  1
571D 2  1
571F 4  1
571I 18 5 · 19
577A 2  3
577B 2  1
577C 3  1
577D 18  2^{4}
587C 31 293
593B 1  2
593C 2  1
593E 27 2 · 37
599C 37 13 · 23
601B 29 2 · 5^{2}
607D 31 101
613C 27 3 · 17
617B 28 2 · 7 · 11
619B 30 103
631B 32 3 · 5 · 7

Bibliography

[1] A. Agashé, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369–374.

[2] A. Agashe, The Birch and Swinnerton-Dyer formula for modular abelian varieties of analytic rank 0, U. C. Berkeley Ph.D. thesis (2000).

[3] A. Agashe and W. A. Stein, On the generalized manin constant for quotients of J_{0}(N), in preparation.

[4] A. O. L. Atkin and J. Lehner, Hecke operators on Γ_{0}(m), Math. Ann. 185 (1970), 134–160.

[5] B. Birch, Atkin and the Atlas Lab, Proceedings of the conference in honor of A. O. L. Atkin held at the University of Illinois, Chicago, IL, September 1995, Amer. Math. Soc., Providence, RI, 1998, pp. 13–20.

[6] B. J. Birch and W. Kuyk (eds.), Modular functions of one variable. IV, Springer-Verlag, Berlin, 1975, Lecture Notes in Mathematics, Vol. 476.

[7] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I, Birkhäuser Boston, Boston, MA, 1990, pp. 333–400.

[8] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990.

[9] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp. 24 (1997), no. 3-4, 235–265, http://www.maths.usyd.edu.au:8000/u/magma/.

[10] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q, or Wild 3-adic exercises, (2000), http://www.math.harvard.edu/HTML/Individuals/Richard Taylor.html.

[11] A. Brumer, The rank of J_{0}(N), Astérisque (1995), no. 228, 3, 41–68, Columbia University Number Theory Seminar (New York, 1992).

[12] K. Buzzard and W. A. Stein, Modularity of some icosahedral Galois representations, in preparation.

[13] H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, (1977), 69–78. Lecture Notes in Math., Vol. 627.

[14] R. Coleman, The monodromy pairing, Asian Math. Journal (1999).

[15] J. E. Cremona, Modular symbols for Γ_{1}(N) and elliptic curves with everywhere good reduction, Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 199–218.

[16] , Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[17] , Computing periods of cusp forms and modular elliptic curves, Experiment. Math. 6 (1997), no. 2, 97–107.

[18] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, to appear in Experiment. Math.

[19] H. Darmon, Wiles’ theorem and the arithmetic of elliptic curves, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 549–569.

[20] H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math. 490 (1997), 81–100.

[21] F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem, Providence, RI, 1995, pp. 39–133.

[22] N. Dummigan, Period ratios of modular forms, to appear in Math. Annalen.

[23] B. Edixhoven, L’action de l’algèbre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est “Eisenstein”, Astérisque (1991), no. 196–197, 7–8, 159–170 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).

[24] E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. of Comp. (2000).

[25] D. Goldfeld, On the computational complexity of modular symbols, Math. Comp. 58 (1992), no. 198, 807–814.

[26] J. González and J-C. Lario, Q-curves and their Manin ideals, preprint (2000).

[27] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320.

[28] B. H. Gross, L-functions at the central critical point, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 527–535.

[29] K. Hatada, Multiplicity one theorem and modular symbols, J. Math. Soc. Japan 33 (1981), no. 3, 445–470.

[30] H. Hijikata, Explicit formula of the traces of Hecke operators for Γ_{0}(N), J. Math. Soc. Japan 26 (1974), no. 1, 56–82.

[31] N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Princeton University Press, Princeton, N.J., 1985.

[32] D. R. Kohel, Hecke module structure of quaternions, preprint (1998).

[33] V. A. Kolyvagin, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp. 429–436.

[34] , On the structure of Shafarevich-Tate groups, Algebraic geometry (Chicago, IL, 1989), Springer, Berlin, 1991, pp. 94–121.

[35] V. A. Kolyvagin and D. Y. Logachev, Finiteness of � over totally real fields, Math. USSR Izvestiya 39 (1992), no. 1, 829–853.

[36] S. Lang, Algebra, third ed., Addison-Wesley Publishing Co., Reading, Mass., 1993.

[37] W-C. Li, Newforms and functional equations, Math. Ann. 212 (1975), 285–315.

[38] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66.

[39] B. Mazur, Courbes elliptiques et symboles modulaires, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 414, Springer, Berlin, 1973, pp. 277–294. Lecture Notes in Math., Vol. 317.

[40] , Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978).

[41] , Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162.

[42] , On the arithmetic of special values of L functions, Invent. Math. 55 (1979), no. 3, 207–240.

[43] , Visualizing elements of order three in the Shafarevich-Tate group, preprint (1999).

[44] B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25 (1974), 1–61.

[45] L. Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations (Berlin), Springer, 1994, pp. 59–94.

[46] , L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de J_{0}(p), J. Reine Angew. Math. 477 (1996), 71–115.

[47] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata) (1986), 217–242.

[48] J.-F. Mestre and J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance m-ième, J. Reine Angew. Math. 400 (1989), 173–184.

[49] J. S. Milne, Étale cohomology, Princeton University Press, Princeton, N.J., 1980.

[50] , Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150.

[51] , Arithmetic duality theorems, Academic Press Inc., Boston, Mass., 1986.

[52] A. Pizer, An algorithm for computing modular forms on Γ_{0}(N), J. Algebra 64 (1980), no. 2, 340–390.

[53] K. A. Ribet, On modular representations of Gal(\overline{Q}/Q) arising from modular forms, Invent. Math. 100 (1990), no. 2, 431–476.

[54] , Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987–88, Birkhäuser Boston, Boston, MA, 1990, pp. 259–271.

[55] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68.

[56] , Euler Systems, Princeton University Press, Spring 2000, Annals of Mathematics Studies 147, http://math.Stanford.EDU/˜ rubin/weyl.html.

[57] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), no. 2, 419–430.

[58] , An introduction to Kato’s Euler systems, Galois Representations in Arithmetic Algebraic Geometry, Cambridge University Press, 1998, pp. 379–460.

[59] I. R. Shafarevich, Exponents of elliptic curves, Dokl. Akad. Nauk SSSR (N.S.) 114 (1957), 714–716.

[60] G. Shimura, Sur les intégrales attachées aux formes automorphes, J. Math. Soc. Japan 11 (1959), 291–311.

[61] , On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), no. 3, 523–544.

[62] , Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1.

[63] V. V. Šokurov, Modular symbols of arbitrary weight, Funkcional. Anal. i Priložen. 10 (1976), no. 1, 95–96.

[64] W. A. Stein, HECKE: The modular symbols calculator, Software (available online) (1999).

[65] G. Stevens, Arithmetic on modular curves, Birkhäuser Boston Inc., Boston, Mass., 1982.

[66] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280.

[67] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 306, 415– 440.

[68] C. Viola, Arithmetic theory of elliptic curves. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME)., Springer-Verlag, Berlin, 1997 (English).

[69] D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), no. 3, 372–384.

Index

L-series, 52, 52 q-expansion principle, 58 Étale group scheme, 73

Agashe, 2, 3, 37, 52, 53, 57, 58

conjecture of, 58 Algorithm for computing

an eigenvector, 43 congruences, 47 coset representatives, 26 cuspidal subgroup, 49 cusps, 30 decomposition of space of modular sym-

bols, 41 eigenvalues, 44 equivalent cusps, 31 Hecke operators on the dual, 42, 43 integral modular symbols, 37 modular kernel, 51 period integrals, 60 rational part of L(A, j), 53 rational period mapping, 48 real and minus volumes, 65 space of modular symbols, 36

Analytic invariants, 59 Antiholomorphic cusp forms, 21 Atkin-Lehner involution, 23, 28, 63, 81

and integration pairing, 24 and ordering eigenforms, 45

Birch and Swinnerton-Dyer conjecture, see

BSD conjecture Bloch and Kato conjecture, 4, 33 Bound of

Kato, 5 Kolyvagin, 5 Sturm, 38, 42 Boundary map, 30 Boundary modular symbols, 21

and Manin symbols, 30 BSD conjecture, 2, 2, 12, 35

and Ω_{A}, 53, 66 and � , 5 and L(A, j)/Ω_{j}, 35 and component groups, 70 generalization of, 4 in higher dimensions, 2 is still unknown, 2 Manin’s remarks on, 56 predicted order of � , 10 predicts invisible elements, 6 predicts large component groups, 83 statement of, 3 verification of, 4

Canonical polarization, 71 Cartier dual, 72 Character group of torus, 71, 73, 73, 76 Closed fiber of Néron model, 73 Complex torus, 32, 59

dual of, 33 in weight two, 34

Component group, 70, 71, 71 and character group, 78 archimedean, 66 geometric, 7 rational points of, 70 table of, 84–87

Conductor of Dirichlet character, 39 Congruences

and BSD conjecture, 2 and lower bounds on � , 6 between q-expansions, 47 between elliptic curves, 6 computed using homology, 47 computing, 45

Conjecture about modular degree, 82 Agashe and Stein, 58 Birch and Swinnerton-Dyer, see BSD

conjecture Bloch and Kato, 4, 33 Kani, 6 refined Eisenstein, 82 Shimura and Taniyama, 2 that Manin constant equals 1, 58

Continued fractions, 28 Coset representatives, 26 Cremona, 19, 29, 30, 40, 53 Cusp forms, 21

antiholomorphic, 21 Cuspidal modular symbols, 21

and Manin symbols, 30 Cuspidal points, 49 Cusps

and boundary map, 30 criterion for vanishing, 31

Degeneracy maps, 24, 41

compatibility, 27 Dimension of S_{k}(N, ε), 40 Dirichlet character, 20, 24, 32, 36–38

and cusps, 31 conductor of, 39

Eichler-Shimura relation, 50 Eigenforms

computing, 44 sorting and labeling, 44

Euler system, 4, 5 Explanatory factor, 10 Extended modular symbols, 59, 60

Fourier coefficients, 23, 47

Genus-two curves, 67 Grothendieck, 71

Hecke, 52 Hecke algebra, 3, 22, 33, 51, 57, 65, 80, 82 and congruences, 47 and cuspidal subgroup, 49 and integration pairing, 33 and rational period mapping, 47 bounds torsion, 50 computation of, 37 generators as module, 37 generators as ring, 37

Hecke operators, 1, 22, 32, 37, 41, 44, 48,

54, 56 and degeneracy maps, 41 computation on subspace of dual, 42 on Manin symbols, 29 respect pairing, 22

Heegner points, 5 Heilbronn matrices, 44

Index of lattices, 54 Integration pairing, 21, 59, 80

and complex torus, 33 and extended modular symbols, 59

Invisible elements of � , 6, 10

Jacobian, 2, 6, 72 is principally polarized, 76 of X_{0}(N), 70, 71, 80 of genus-two curve, 67 semistable, 71, 76 visibility in, 6

Kani, 6

Lattice, 54 Lattice index, 3, 54 Level of modular symbols, 20 Local-to-global principle, 3 Logan, 13

Manin, 35, 36, 49, 53, 61 comment on BSD, 56 trick of, 28

Manin constant, 3, 57, 58, 67

conjecture about, 58 Manin symbols, 27, 27, 28, 30, 36, 42, 44

and boundary space, 30 and cuspidal subspace, 30 and Hecke operators, 29 and Manin’s trick, 27 conversion to modular symbols, 28

Manin’s trick, 27, 28 Mazur, 2–5, 13, 52, 57 Merel, 4, 19, 22, 29, 30, 52 Mestre, 14, 70 Method of graphs, 82 Minus volume, 53, 65, 68 Minus-one quotient, 23, 32 Modular curve, 80 Modular degree, 50, 72

and character group, 78 Modular forms, 23, 27, 33, 34, 62

and Atkin-Lehner involution, 24 and BSD, 2 associated complex torus, 32 associated subtorus, 33 congruences between, 6 duality with modular symbols, 21

Modular map, 50 Modular symbols, 3, 19, 20, 28, 40, 59, 62

computing, 35 conversion to Manin symbols, 28 duality with modular forms, 21 finite presentation of, 27 minus-one quotient of, 23 new and old subspace of, 25 plus-one quotient of, 23 relations satisfied by, 20

Monodromy pairing, 71, 76, 77, 82 Motifs, 4, 33 Mumford, 71

Néron model, 57, 70, 71, 73

closed fiber of, 73 New modular symbols, 25 New subspace, 41

Old modular symbols, 25 Old subspace, 41 One-motif dual, 74 Operators

∗-involution, 23 Atkin-Lehner, 23

Hecke, 22 Optimal quotient, 2, 34, 51, 58, 72, 76, 80

and semistable reduction, 74 component groups of, 70 dual map is injective, 72 of J_{0}(N), 80 Tamagawa numbers of, 70

Ordering of eigenforms, 44

Period integrals

algorithm for computing, 60 Period lattice, 61 Period mapping, 33, 54, 65

computation of, 64 is injective, 33

Petersson pairing, 32 Plus-one quotient, 23, 32 Polarization, 72 Projective line modulo N, 26 Purely toric reduction, 8, 73–75, 80

Quaternion algebras, 82

Rational part of L(A, j), 52 Rational period mapping, 47, 64 Raynaud, 74 Real volume, 53, 65, 68 Ribet, 4, 6, 13, 37, 70 Rigid uniformization, 73

Saturated, 57 Semistable reduction, 73, 74

and uniformization, 73 Shafarevich-Tate group, 3, 6, 83

first invisible example, 13 invisible elements of, 6 visibility at higher level, 6 visible part of, 5

Shimura, 32, 34, 53 Shimura-Taniyama conjecture, 2 Snake lemma, 46, 51, 58, 72, 75, 76 Special value L(A, j)

computing, 64 rational part of, 52

Star involution, 23, 33 and integration pairing, 23

is well defined, 23 Sturm bound, 38, 42 Symmetric isogeny, 73, 75

T-shirt design, 46 Table of

� at prime level, 15 big L(A, 1)/Ω_{A}, 85 big component groups, 85 CM elliptic curves of weight > 2, 69 component groups at low level, 84 component groups at prime level, 87 component groups of explanatory fac-

tors, 18 component groups of quotients, 86 explanatory factors, 17 factorizations, 17 new visible � , 16 odd invisible � _{E}, 12 volumes of higher dimensional abelian

varieties, 69 volumes of level one cusp forms, 68

Tamagawa numbers, 3, 4, 70 Tate curve

and monodromy pairing, 77 component group of, 77 uniformization of, 74

Toric part, 71 Torsion subgroup, 3

lower bounds on, 50 upper bounds on, 50

Uniformization cross, 74, 75

Verrill, 61 Visibility

at higher level, 6 existence theorem, 7 in some Jacobian, 6 of � , 5

Weight of modular symbols, 20 Winding element, 52, 65