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 \begin{matrix}c \\ ◯ \end{matrix} 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

First I describe how to compute spaces of modular forms of weight at least two. I

then I describe 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 enough to compute the odd part of the conjectural order of the Shafarevich-Tate group of any analytic rank 0 optimal quotient of J_{0}(N), with N square-free. I have not determined the factor of 2, 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.

Next, I use these algorithms to investigate the Birch and Swinnerton-Dyer con-

jecture, 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 B. Mazur’s notion of visibility, coupled with explicit computations, to produce lower bounds on the Shafarevich-Tate group. To finish the proof of the conjecture in these cases requires knowing upper bounds on the order of this group; K. Kato has announced such bounds, but they are not yet published.

Professor Hendrik Lenstra Dissertation Committee Chair

To those who inspired me: Kevin Buzzard, Robert Coleman, Hendrik Lenstra, Barry Mazur, Loïc Merel, Ken Ribet, and my family.

Contents

List of Figures vii

List of Tables viii

List of Symbols ix

1 Modular symbols 2 1.1 The definition of modular symbols . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Cuspidal modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Duality between modular symbols and modular forms . . . . . . . . . . . . 3 1.4 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4.1 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.2 The ∗-involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.3 The Atkin-Lehner involutions . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5.1 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Manin symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.1 Conversion between modular and Manin symbols . . . . . . . . . . . 8 1.6.2 Hecke operators on Manin symbols . . . . . . . . . . . . . . . . . . . 10 1.6.3 The cuspidal and boundary spaces in terms of Manin symbols …. 10 1.6.4 Computing the boundary map . . . . . . . . . . . . . . . . . . . . . 11

1.7 The complex torus attached to a modular form . . . . . . . . . . . . . . . . 13 1.7.1 Weight 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Algorithms for modular forms 16 2.1 Computing the space of modular symbols . . . . . . . . . . . . . . . . . . . 16 2.2 Computing the Hecke algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 The dimension of S_{k}(N, ε) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Representing and enumerating Dirichlet characters . . . . . . . . . . . . . . 19 2.5 Decomposing the space of modular symbols . . . . . . . . . . . . . . . . . . 20

2.5.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Efficient computation of Hecke operators on the dual space . . . . . 21 2.5.3 Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5.4 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.5 Sorting and labeling eigenforms . . . . . . . . . . . . . . . . . . . . . 24 2.6 Intersections and congruences . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.7 The rational period mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 The images of cuspidal points . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 The modular degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.10 The modular degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.11 Rational part of the L-function . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.12 Analytic invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.12.1 Extended modular symbols . . . . . . . . . . . . . . . . . . . . . . . 33 2.12.2 Numerically computing period integrals . . . . . . . . . . . . . . . . 33 2.12.3 The W_{N}-trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.12.4 Computing the period map . . . . . . . . . . . . . . . . . . . . . . . 37 2.12.5 Computing special values . . . . . . . . . . . . . . . . . . . . . . . . 38 2.12.6 Computing the real and pure imaginary volume . . . . . . . . . . . . 39 2.12.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Quotients of J_{0}(N) 43

3.1 The torsion subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.1.1 Lower bounds: The cuspidal subgroup of A_{f} . . . . . . . . . . . . . 43 3.1.2 Upper bounds: Counting points mod p . . . . . . . . . . . . . . . . . 44

3.2 The Manin constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.1 The primes that might divide c_{A} . . . . . . . . . . . . . . . . . . . . 44 3.2.2 Numerical evidence for a c_{A} = 1 conjecture . . . . . . . . . . . . . . 46

4 Component groups 47 4.1 Optimal quotients of Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 The special fiber of the Néron model . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Rigid uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.1 Raynaud and van der Put uniformization . . . . . . . . . . . . . . . 49 4.3.2 Some lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4.1 Monodromy description of the component group . . . . . . . . . . . 52 4.5 Optimal quotients of J_{0}(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.5.1 Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.6 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6.1 Table 1: Some large component groups predicted by the Birch and

Swinnerton-Dyer conjecture . . . . . . . . . . . . . . . . . . . . . . . 57 4.6.2 Table 2: Some quotients of J_{0}(N) . . . . . . . . . . . . . . . . . . . 58 4.6.3 Table 3: Some quotients of J_{0}(p) . . . . . . . . . . . . . . . . . . . . 59

5 Visibility of Shafarevich-Tate groups 61 5.1 The Birch and Swinnerton-Dyer conjecture . . . . . . . . . . . . . . . . . . 61 5.2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.3 Prime conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi Bibliography 69

List of Figures

2.1 T-shirt design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

List of Tables

2.1 Volumes of level one cusp forms. . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 CM elliptic curves of weight > 2. . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Volumes of high dimensional abelian varieties. . . . . . . . . . . . . . . . . . 42

5.1 New visible Shafarevich-Tate . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Explanatory factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3 Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4 Component groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.5 Odd square numerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.6 Odd parts of Shafarevich-Tate groups at prime level . . . . . . . . . . . . . 68

List of Symbols

Symbol  Definition  Page
A^{∨}  dual of A  14
B_{k}(N, ε)  boundary modular symbols  3
c_{A}  Manin index of A  44
δ_{A}  modular degree  ??
e  winding element e = {0, ∞}  31
M_{k}(N, ε; K)  modular symbols  3
\overline{M}_{k}(N, ε; K)  extended modular symbols  33
M[I]  ∩_{a∈I} ker(a)  ??
P (X, Y){α, β}  higher weight modular symbol  2
[P (X, Y), (u, v)]  higher weight Manin symbol  8
� (A/K)^{∘}  visible part of �  61
S_{k}(N, ε; K)  cuspidal modular symbols  3
T_{n}  nth Hecke operator  10
V_{k}  homogeneous polys. of degree k in Z[X, Y]  2
W_{n}  nth Atkin-Lehner involution  5
X_{p}(M)  character group of torus of J_{0}(pM) at p  49
α_{t}, β_{t}  degeneracy maps  5
Θ_{f}  rational period mapping  26
σ, τ  σ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ), τ = ( \begin{matrix}0 & −1 \\ 1 & −1 \end{matrix} )  8
Φ_{f}  analytic period mapping  33
Φ_{A,p}  component group of A at p  47
Ω_{A}  real volume  39
⟨ , ⟩  integration pairing  4
∗  conjugation involution  4

Acknowledgements

It is a pleasure to acknowledge my thesis adviser Hendrik Lenstra, who helped me from the beginning. My research was greatly influenced by of Robert Coleman, Barry Mazur, and Ken Ribet, whose advice and direction I greatly appreciate. I thank my collaborators Amod Agashé, Kevin Buzzard, Loïc Merel, David Moulton, and Helena Verrill for keeping me going. Finally, I thank those who answered and asked questions: Matt Baker, Brian Conrad, Janos Ćsirik, Edray Goins, David Jones, Ezra Miller, Bjorn Poonen.

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

Preface

The object of numerical computation is theoretical advance.

–A. O. L. Atkin

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 computational investigations on these objects.

We are fortunate that we now have methods available which 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 [BK75, Cre97a, Mer94]). The Eichler-Selberg trace formula, as extended in [Hij74], can be used to compute characteristic polynomials of Hecke operators and hence gain some information about spaces of modular forms. It is also possible to compute Hecke operators and q-expansions using Brandt matrices and quaternion algebras as in [Koh98, Piz80]; another closely related method relies on the module of enhanced supersingular elliptic curves as exploited by Mestre-Oesterlé [Mes86] and extended by Edixhoven in his Ph.D. thesis. I take the approach of Cremona’s book [Cre97a] and view modular symbols as central.

No single computational method provides complete information about the package

of invariants attached to a modular form. The various methods must be used in concert; for example, the module of supersingular points or quaternion algebras are needed in order to compute component groups, whereas modular symbols are needed to compute the modular degree.

In this thesis, I present methods for explicitly computing with modular forms,

Hecke operators, and the abelian varieties attached to them. In Chapter 5, I give an example of how to coordinate these methods to gain data about real world problems.

Chapters 1 and 2 owe a great deal to the publications of Merel and Cremona.

Chapter 1

Modular symbols

Modular symbols are central to most of the algorithms and computations of this

thesis. This chapter closely follows Merel [Mer94] and Cremona [Cre97a]. However, we work with modular forms with character throughout; this makes the theory trickier, but is extremely import in computational applications.

1.1 The definition of modular symbols

Fix a triple (N, k, ε) consisting of a level N, a weight k, and a mod N character

ε : (Z/NZ)^{∗} → C^{∗} such that ε(−1) = (−1)^{k}.

Let M be the torsion-free abelian group generated by symbols {α, β} with α, β ∈

P^{1}(Q) = Q ∪ {∞} subject to the following relations:

{α, β} + {β, γ} + {γ, α} = 0, for all α, β, γ ∈ P^{1}(Q).

Because M is torsion free, {α, α} = 0 and {α, β} = −{β, α}. Let V_{k−2} be the Z-submodule of Z[X, Y] made up of 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 a reduces to \overline{a} ∈ Z/NZ.

Definition 1.1 (Modular symbols). The space M_{k}(N, ε) of modular symbols is the torsionfree quotient of M_{k} ⊗ Z[ε] 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) ⊗ R.

1.2 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 1.2 (Boundary modular symbols). The space of boundary modular symbols is the 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}. The boundary map is

δ : M_{k}(N, ε) → B_{k}(N, ε), δ(P {α, β}) = P {β} − P {α}.

Definition 1.3 (Cuspidal modular symbols). The space S_{k}(N, ε) of cuspidal modular symbols is the kernel of δ:

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

1.3 Duality between modular symbols and modular forms

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

𝔥 := {z ∈ C : im(z) > 0}, and any γ ∈ GL_{2}(Q), define a function f|[γ] on 𝔥 by

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

Definition 1.4 (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 at all cusps in the sense of [DI95, pg. 42].

Definition 1.5 (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 anitholomorphic cusp form z ↦ \overline{f(z)}.

Theorem 1.6 (Duality). 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 [Mer94, Thm. 3].

1.4 Linear operators

1.4.1 Hecke operators

The Hecke operators T_{n}, for n ≥ 1 act on modular symbols and modular forms.

The Hecke algebra T is the subring of the ring of endomorphism of modular symbols or modular forms generated by all T_{n}. For the definition of Hecke operators on modular symbols, see [Mer94, §2]. For example, when n = p is prime,

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.

Proposition 1.7. The pairing of Theorem 1.6 respects the action of the Hecke operators, in the sense that ⟨fT, x⟩ = ⟨f, T x⟩ for all T ∈ T, x ∈ M_{k}(Γ), and f ∈ S_{k}(Γ) ⊕ \overline{S}_{k}(Γ).

1.4.2 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}.

We are extremely fortunate to have the following proposition, because modular symbols correspond to both holomorphic and non-holomorphic cusp forms.

Proposition 1.8. The ∗-involution is well defined.

Proof. M_{k}(N, ε) is the 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 ∗ is well defined, it suffices to check that ∗(γx − ε(γ)x) is of the form γ′y − ε(γ′)y. 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, it’s character is \overline{ε} instead of ε. The pairing of Theorem 1.6 is the restriction of a pairing on the full spaces without character, and we have the following proposition.

Proposition 1.9. We have

⟨f^{∗}, x^{∗}⟩ = \overline{⟨f, x⟩}.

Definition 1.10 (Plus one quotient). The plus one quotient M_{k}(N, ε)_{+} is the torsionfree quotient of M_{k}(N, ε) by the relations x^{∗} − x = 0 for all x ∈ M_{k}(N, ε). The minus one quotient is defined similarly.

WARNING 1.11. The choice of ∗ above agrees with [Cre97a, §2.1.3], but not with [Mer94, §1.6].

1.4.3 The Atkin-Lehner involutions

In this section we assume that k is even and ε^{2} = 1. To each divisor d of N such

that (d, N/d) = 1 there is an Atkin-Lehner involution W_{d} on 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; let g = ( \begin{matrix}dx & y \\ Nz & dw \end{matrix} ) and define W_{d}(x) := g(x)/d^{\frac{k−2}{2}} . For example, when d = N we have g = ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ). There is also an Atkin-Lehner involution, also denoted W_{d}, that acts on modular forms by W_{d}(f) = f|[W_{d}]_{k}. These involutions are compatible with the integration pairing: ⟨W_{d}(f), x⟩ = ⟨f, W_{d}(x)⟩.

1.5 Degeneracy maps

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

different levels. These are useful in investigating level lowering and raising questions, and in computing kernels of certain natural maps between Jacobians of modular curves.

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: There is a mistake in [Mer94, §2.6]: The quotient “Γ_{1}(N)\Γ_{1}(M)T ”

should be replaced by “Γ_{1}(N)\T Γ_{1}(M)”.

Proposition 1.12. 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 · [Γ_{0}(M) : Γ_{0}(N)].

Proof. To show that α_{t} is well defined, we must show that for x ∈ M_{k}(N, ε) and γ =

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N), that 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 show 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 ∑_{T γ_{i}} ε′(γ_{i})^{−1}γ_{i}x

= t ∑_{T γ_{i}} x

= t · [Γ_{0}(N) : Γ_{0}(M)] · x.

Definition 1.13 (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.

Remark 1.14. The map β_{t}∘α_{t} can not be multiplication by a scalar since M_{k}(M, ε′) usually has smaller dimension than M_{k}(N, ε).

Remark 1.15. The proposition implies that β_{t} is injective. For each prime p there is also a map β_{t,p} : M_{k}(M, ε′; F_{p}[ε]) → M_{k}(N, ε; F_{p}[ε]). When p does not divide t·[Γ_{0}(M) : Γ_{0}(N)], the proposition shows that β_{t,p} is injective. However, β_{t} ⊗ F_{p} need not be injective for all p. For example, suppose M = 14, N = 28, and ε = 1. Then there are basis with respect to which the matrix of β_{1} is

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

The row vector (0, 0, 0, 1, 1) is in the kernel of the mod 2 reduction of this matrix.

Algorithm 1.16. Let M be a positive divisor of N and t a divisor of N/M. This algorithm computes a set D_{t} of representatives for the orbit space Γ_{0}(M)\T Γ_{0}(N). 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), i.e., ab′ ≡ ba′ (mod 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.

1.5.1 Compatibility

Suppose that the characteristic of the base field is zero. 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}.

1.6 Manin symbols

From the definition given in Section 1.1, it is not even clear that M_{k}(N, ε) is of

finite rank. The Manin symbols provide a finite presentation of M_{k}(N, ε) that is much more useful from a computational point of view.

Definition 1.17 (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 torsion-free quotient of M by

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

Theorem 1.18. 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 some matrix such that (u, v) ≡ (c, d) (mod N).

Proof. In [Mer94, §1.2, §1.7] it is proved that φ ⊗ C is an isomorphism, so φ is injective and well defined. The discussion in Section 1.6.1 below shows that every element in M_{k}(N, ε) is a Z-linear combination of elements in the image, so φ is surjective as well.

1.6.1 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; in practice it is more efficient to compute this involution using modular symbols. It is thus essential to convert between these two representations. The conversion from Manin to modular symbols is straightforward, and follows immediately from Theorem 1.18. The conversion back is accomplished using the method used to prove Theorem 1.18; 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 [Shi94, pg. 20]; in practice it 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}.

1.6.2 Hecke operators on Manin symbols

Thoerem 2 of [Mer94] 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 of Cremona.

As in [Cre97a, §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 1.19. 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 [Cre97a, §2.4]. The general case follows from [Mer94, Theorem 2] applied to the set S of [Mer94, §3] by observing that when p is an odd prime S′_{p} is empty.

1.6.3 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 [Mer94, §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 1.20. 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 1.21. 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} )].

1.6.4 Computing the boundary map

In this section we describe how to efficiently compute the map M_{k}(N, ε) →

B_{k}(N, ε) given in the previous section. Henceforth, we will denote this map by δ. Our algorithm generalizes [Cre97a, §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 present the algorithm first, and then prove the results used by the algorithm in testing equivalence.

Algorithm 1.22. 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 1.23 and Algorithm 1.24, 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 1.26, 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 [Cre97a], the relations between cusps involve only the

trivial character, so they do not force any cusp classes to vanish. He gives the following two criteria for equivalence.

Proposition 1.23 (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 [Cre97a], and the second is Lemma 3.2 of [Cre92].

Algorithm 1.24. 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 1.23 in [Cre97a].

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 Gauss elimination. This is far too inefficient to be useful in practice because the number of Γ_{1}(N) cusp classes is huge. Instead, we give a quick test to determine whether or not a cusp vanishes modulo the ε-relations.

Lemma 1.25. 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 [Shi94, 1.38] the map SL_{2}(Z) → SL_{2}(Z/NZ) is surjective. By the Euclidean algorithm, there exists 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 1.26. Let N be a positive integer and ε a Dirichlet character modulo 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 1.25 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 1.23

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 1.23 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.

The possible α in Proposition 1.26 can be enumerated 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 or minus 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 1.22 can be modified to deal with this situation as follows. Given a cusp x = ( \begin{matrix}u \\ v \end{matrix}), proceed as in Algorithm 1.22 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 1.22 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}.

1.7 The complex torus attached to a modular form

Fix integers N ≥ 1, k ≥ 2, and let ε be a mod N Dirichilet character. For the rest

of this section assume that ε^{2} = 1. We construct a Hecke and complex conjugation invariant lattice in S, hence a complex torus J_{k}(N, ε) equipped with an action of Hecke operators and an R-structure. The reader may wish to compare our construction to the one given by Shimura in [Shi59], where in addition he observes that the Petterson pairing gives his torus the structure of an abelian variety. When k = 2, the torus comes from an abelian variety 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 [BK90]) on motifs attached to modular forms.

Let S = S_{k}(N, ε) (resp., S = S_{k}(N, ε)) be the associated space of cuspidal modular

symbols (resp., forms). The Hecke algebra T acts in a way compatible with the integration pairing ⟨ , ⟩ : S × S → C, which we view as a T-module homomorphism Φ : S → S^{∗} = Hom(S, C) called the period mapping. Because ε^{2} = 1, the ∗-involution preserves S.

Proposition 1.27. The period mapping Φ is injective and the image of Φ is a lattice in S^{∗}.

Proof. By Theorem 1.6, 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 defined \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 1.9 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} ≅ Hom_{R}(S_{R}, R) ⊕ Hom_{R}(S_{R}, iR) ≅ Hom_{C}(S, C).

Finally, observing that S is by definition torsion free completes the proof.

Thus we have a torus J_{k}(N, ε) that fits into an exact sequence

0 ⟶ S ⟶^{Φ} Hom(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(S_{f}, C) obtaind by composing Φ with restriction to S_{f}. Set

A_{f} := Hom(S_{f}, C)/Φ_{f}(S).

Associate an abelian subvariety of J to f as follows. Let I_{f} = Ann_{T}(f) be the

annihilator of f in the Hecke algebra, and set

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

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

The following diagram summarizes the tori just defined; its columns are exact but

its rows are not.

\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(S, C)[I_{f}] \ar[r] \ar[d] & Hom(S, C) \ar[r] \ar[d] & Hom(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}  (1.1)

1.7.1 Weight 2

When k = 2 and ε = 1 the above is just Shimura’s classical association of an

abelian variety to a modular form; see [Shi94, Thm. 7.14] and [Shi73]. In this case A_{f} and A_{f}^{∨} are abelian varieties that are defined over Q and are dual to each other. Furthermore A_{f} is an optimal quotient of J, in the sense that the kernel of J → A_{f} is connected.

Chapter 2

Algorithms for modular forms

2.1 Computing the space of modular symbols

Definition 2.1. 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 2.2. Given a positive integer N, a Dirichlet character ε, and an integer k ≥ 2 this algorithm computes M_{k}(N, ε; K). It uses the Manin-symbols description of M_{k}(N, ε; K) given in Theorem 1.18. We compute the quotient presentation of Theorem 1.18 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 algebra over Z/NZ.

2. Let σ act on Manin symbols as in Section 1.6; 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 1.6; 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 Gauss elimination, we compute V_{3}/W_{3}; we then have 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 1.18 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 Gauss 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 2.3. In implementing the above algorithm, the reader should take 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 2.4. To compute the +1 quotient M_{k}(N, ε; K)^{+}, it is necessary to modify Step 2 of Algorithm 2.2 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 −1 quotient we include the sums x + xI. Note, as in the remarks in the proof of Algorithm 2.2, we can not add in the I relations in Step 1 because I and σ do not commute.

Algorithm 2.5. 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).

1. Using Algorithm 2.2 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) which equals L.

3. To compute the submodule S_{k}(N, ε) of L, we use the algorithm of Section 1.6.4 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 2.3, 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 2.5. Remark 2.6 ([To be removed.). ] I have not thought through how to perform Steps 2 or 3 in case O ≠ Z. In particular, I do not know how to compute the O-module kernel of δ, except using a silly algorithm that reduces everything to working over Z.

2.2 Computing the Hecke algebra

In this chapter we give a formula for the dimension of S_{k}(Γ_{1}(N), ε) and an upper

bound on the number of Hecke operators needed to generate the Hecke algebra as a Zmodule. The dimension formula follows by specializing a general formula of Hijikata to the case of the identity operator, and the bound on Hecke operators is really just a useful restatement of a theorem in ??.

Let Γ be a subgroup of SL_{2}(Z) that contains Γ_{1}(N) for some N. Let M_{k}(Γ) be

the space of weight-k modular forms for Γ, and let T ⊂ End(M_{k}(Γ)) be the corresponding Hecke algebra. This section contains a bound r such that the Hecke operators T_{n}, with n ≤ r, generate T as a Z-module. This result was suggested to the author by Ribet and Agashé.

For any subring R of C, denote by M_{k}(Γ; R) the space of modular forms for Γ

with Fourier coefficients in R. The following proposition is well-known.

Proposition 2.7. For any ring R, there is a perfect pairing

T_{R} ⊗_{R} M_{k}(N; R) → R, (T, f) ↦ a_{1}(T f),

where T_{R} = T ⊗_{Z} R. We have (T_{n}, f) = a_{n}(f) where T_{n} is the nth Hecke operator.

Let μ denote the index of Γ in SL_{2}(Z).

Theorem 2.8 (Sturm). Let λ be a prime ideal in the ring O of integers in some number field. If f ∈ M_{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 [Stu87].

Denote by ⌈x⌉ the smallest integer ≥ x.

Proposition 2.9. If f ∈ M_{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 M_{k}(Γ) ↪ C[[q]] → C[[q]]/(q^{r+1}) is injective. Because C is a flat Z-module and M_{k}(Γ; Z) ⊗ C = M_{k}(Γ), it suffices to show that the map F : M_{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 2.8 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.

2.3. THE DIMENSION OF S_{K}(N, ε) 19

Theorem 2.10. 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 2.7, and suppose that f ∈ M_{k}(N, F_{p}) pairs to 0 with each of T_{1}, …, T_{r}. Then by Proposition 2.7, a_{m}(f) = a_{1}(T_{m}f) = 0 in F_{p} for each m, 1 ≤ m ≤ r. Theorem 2.8 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}} M_{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.

2.3 The dimension of S_{k}(N, ε)

Explicit formulas for dim S_{k}(Γ), with Γ a congruence subgroup, were given by

Shimura in [Shi94, Thms. 2.23–2.25]. I don’t know if these methods generalize to give a formula for dim S_{k}(N, ε). However, an extremely tedious specialization of Hijikata’s trace formula (see [Hij74]) to the case n = 1 yields an explicit formula for tr(T_{1}) = dim S_{k}(N, ε); source code is available from the author.

2.4 Representing and enumerating Dirichlet characters

A Dirichlet character is a homomorphism ε : (Z/NZ)^{∗} → C^{∗}. We have the

following lemma, whose proof is well-known.

Lemma 2.11. 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.

It is necessary to agree upon a 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} 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 matrix with respect to a canonically chosen, but unnatural, basis.

The matrix representing a character ε can be viewed as a vector with r entries in

Z/nZ, where m 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 runs over (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 n′th root of unity, where n′ 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 2.12. Suppose p is an odd prime. The group of mod p Dirichlet characters 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 classes of mod p Dirichlet characters are in bijection with the divisors d of p − 1 = #(Z/pZ)^{∗}.

The quadratic mod p character is denoted [(p − 1)/2]. We denote the quadratic

mod 2p character by [0, 0, (p − 1)/2]; the quadratic mod 4p character is denoted [(p − 1)/2, 0, (p − 1)/2]. If n ≥ 2, then the nontrivial mod 2^{n} character that factors through (Z/4Z)^{∗} is denoted [(2^{n−2} − 1)/2, 0].

2.5 Decomposing the space of modular symbols

Let M_{k}(N, ε) be the space of modular symbols of level N and character ε over

K = Q(ε). In this section we describe how to decompose the new part of M_{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 [AL70], it is then possible to construct a basis for S.

The algorithm is, for the most part, a straightforward generalization of the method

used by Cremona [Cre97a] to enumerate the Q-rational weight two newforms corresponding to modular elliptic curves. Nevertheless, we present several nonobvious tricks which we learned in the course of doing computation and which greatly speed up the algorithm. One key idea is to work in the space dual to modular symbols as described in the next section.

2.5.1 Duality

Let M_{k}(N, ε)^{⊥} denote Hom(M_{k}(N, ε), K) equipped with its natural right T-

action:

(φT)(x) = φ(T x).

The natural pairing

⟨ , ⟩ : M_{k}(N, ε)^{⊥} × M_{k}(N, ε) → K  (2.1)

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

Viewing the elements T ∈ T as sitting inside M_{k}(N, ε), the transpose map T ↦ T^{t}

allows us to view M_{k}(N, ε)^{⊥} as a left T-module.

Proposition 2.13. Let V ⊂ M_{k}(N, ε)^{new} be an irreducible new T-submodule and set I = Ann_{T} V. Then M_{k}(N, ε)^{⊥}[I] is isomorphic to V as a T ⊗ Q-module.

The degeneracy maps α_{t} and β_{t} of Section 1.5 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 M_{k}(N, ε)^{⊥} are defined as in Definition 1.13.

Algorithm 2.14. This algorithm computes a decomposition of M_{k}(N, ε)^{⊥ new}. Using Algorithm ?? compute M_{k}(N, ε). Then compute the maps β_{t} using Algorithm 1.16 and intersect the transposes of their kernels in order to obtain M_{k}(N, ε)^{⊥ new}. Compute the boundary map δ : M_{k}(N, ε) → B_{k}(N, ε) using Algorithm 1.22. Using the Hecke operators, Algorithm 2.16, and Proposition 2.13, cut out the cuspidal submodule S_{k}(N, ε)^{⊥ new}. Set p = 2 and perform the following steps.

1. Using Algorithm 2.16, compute a matrix A representing the Hecke operator T_{p} on M_{k}(N, ε)^{⊥ 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)). Use the following criteria to determine if V_{f} is irreducible:

(a) If p is greater than the Sturm bound (Section ??) then V_{f} must be irreducible.

(b) If some element T ∈ T has characteristic polynomial on V_{f} the square of an irreducible polynomial, then V_{f} is irreducible. Determine if V_{f} is cuspidal by checking if V_{f} is contained in S_{k}(N, ε)^{⊥ new} computed above. If V_{f} is cuspidal and some element T ∈ T has characteristic polynomial on V_{f} the square of an irreducible polynomial, then V_{f} is irreducible.

4. If V_{f} is irreducible, record V_{f} in a list and continue with the next factor of the characteristic polynomial. If we can not show that V_{f} is irreducible, repeat step 1 with M_{k}(N, ε)^{⊥ new} replaced by V_{f}.

2.5.2 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} on an invariant subspace V ⊂ M_{k}(N, ε)^{⊥}. A naive way to compute the right action of T_{p} on V is to compute a matrix representing T_{p} on M_{k}(N, ε), transpose to obtain T_{p} on M_{k}(N, ε)^{⊥}, and then restrict to V using Gaussian elimination. To compute T_{p} on M_{k}(N, ε), observe that M_{k}(N, ε) 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 1.6.2. In practice, d = dim V will often be much less than n; it is then possible 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 ↪ M_{k}(N, ε)^{⊥} we obtain a surjection M_{k}(N, ε) → 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}}. We describe in detail below a definite way to carry out this computation using matrices.

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 2.15. 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}. This results in a tremendous savings; usually dim V is small and d is large.

Algorithm 2.16. Let T be any linear transformation [[of the ambient space]] 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 Gausselimination 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.

2.5.3 Eigenvectors

Once a T-simple subspace of M^{∗} has been identified, the following algorithm can

be used to write down an eigenvector defined over an extension of the base field.

Algorithm 2.17. Let A be an n × n matrix over a 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 randomly chosen v ∈ K^{n} the vector g(A)v is an eigenvector of A with eigenvalue α. Compute the vector g(A)v by computing Av, A(Av), A(A(Av)) and then summing 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.

Proof. By the Cayley-Hamilton theorem [Lan93, 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)).

2.5.4 Eigenvalues

In this section we give an algorithm for computing the q-expansion of one of

the newforms corresponding to a factor of M_{k}(N, ε)^{new}. This is a generalization of the algorithm described in [Cre97a, §2.9].

Algorithm 2.18. Given a factor V ⊂ S_{k}(N, ε)^{⊥ new} as computed by Algorithm 2.14 this algorithm computes the q-expansion of one of the corresponding Galois conjugate newforms.

1. Using Algorithm 2.16 compute the action of the ∗-involution (Section 1.4) on V. Then compute the +1 eigenspace V^{+} ⊂ V.

2. Find a Hecke operator T ∈ T such that the characteristic polynomial of the matrix A of T acting on V^{+} is irreducible. Such a Hecke operator T must exist by the primitive element theorem [Lan93, V.4].

3. Using Algorithm 2.5.3 compute an eigenvector e for A over an extension of K.

4. Because e is an eigenvector and the pairing given in Equation 2.1 respects the Hecke action, we have that for any Hecke operator T_{n} and element w ∈ M_{k}(N, ε), that

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

Choose w so that 1) w is a freely generating Manin symbols (see Algorithm ??) and 2) ⟨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.

The beauty of this algorithm is that when w is a freely generating Manin sym-

bol the computation of T_{p}w = ∑_{x∈R_{p}} wx is very quick, requiring only summing 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 recurences 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}}.

2.5.5 Sorting and labeling eigenforms

In Section 2.5.4 we saw how to associate to each new factor a sequence a_{n} of Hecke

eigenvalues. These can be used to sort and hence label the factors. This is essential so that we can refer to the factors in our explicit investigations.

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_{5}), …

where the trace is from K[f] down to Q. Sort the eigenforms by ordering the sequences in dictionary order with minus coming before plus. Note that we included tr(a_{1}) so that this ordering would gather together factors of the same dimension.

When k = 2 and the character is trivial we use a different and somewhat compli-

cated ordering because it extends the notation for elliptic curves that was introduced in the second edition of [Cre97a] 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, thus 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.

2.6 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 2.19. 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)}

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 nontorsion in Λ/(Λ_{A} + Λ_{B}) could map to 0, because if it did then A ∩ B wouldn’t be finite. The lemma follows.

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

Proposition 2.20. 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 2.21. Using this proposition the author constructed the t-shirt design in Figure 2.1.

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 congruences between infinite Fourier expansions; the key advantage of the algorithm is that it only involves finite exact computations and avoids any need to compute q-expansions.

Proposition 2.22. 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 ℘).

                    shirt.eps

Figure 2.1: T-shirt design

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}) is 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.

2.7 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 computing the image of cuspidal points on J(N, k, ε).

Definition 2.23. Let D := Hom(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\}}.

Algorithm 2.24. This algorithm computes Φ_{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(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 acts 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 Φ_{I} are the same.

Example 2.25. 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 Φ_{I}((a, b, c, d, e)^{t}) = (a − c, b − c).

Lemma 2.26.

dim M_{k}(N, ε; K)[I] = dim Hom(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 square 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 2.27. Suppose M_{k}(N, ε; K)[I] ⊂ S_{k}(N, ε; K), and let P : M_{k}(N, ε; K) → Hom(S_{k}(N, ε; C)[I], C) be the classical period map induced by the integration pairing. Then ker(P) = ker(Φ_{I}).

Proof. Since P (M_{k}(N, ε; O) is known to be an O-lattice in the complex vector space Hom(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 2.26 the images im(P) and im(Φ_{I}) have the same dimension, hence ker(P) and ker(Φ_{I}) also have the same dimension. It thus suffices to prove the inclusion ker(Φ_{I}) ⊂ ker(P). Suppose Φ_{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.

2.8 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(S_{k}(N, ε; C), C).

A cuspidal point of

J = J(N, k, ε) := \frac{Hom(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 Manin proved (see [Man72]) that the cuspidal point {0, ∞} is a torsion point in J(Q), so its order gives a lower bound on J(Q)_{tor}.

Algorithm 2.28 (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 2.5 compute M_{k}(N, ε; O) and S_{k}(N, ε; O). Using Algorithm 2.24, compute the rational period mapping Φ_{I}. Then the cuspidal subgroup is the subgroup of Φ_{I}(S_{k}(N, ε; O)) generated by the elements Φ_{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 Φ_{I}(X^{i}Y^{k−2−i}{α, β}) in the quotient of Φ_{I}(M_{k}(N, ε; O) by Φ_{I}(S_{k}(N, ε; O)).

Example 2.29. This example continues Example 2.25. 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 Φ_{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

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

Thus Φ_{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.

2.9 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 compue the modular degree of the torus A_{f} attached to f.

Definition 2.30. The modular map is the map θ_{f} : A_{f}^{∨} → A_{f} of Diagram 1.1. The modular degree m_{f} of f (or of A_{f}) is the degree of this map.

When k = 2, θ_{f} is a polarization so ([Mil86, Thm. 13.3]) the degree of θ is a square.

Proposition 2.31. Let E/Q be a modular elliptic curve of conductor N which is an optimal quotient of J_{0}(N). Then δ_{f} is the square of the usual modular degree, which is the least degree of a map X_{0}(N) → E.

When k ≠ 2, the degree of θ need not be a perfect square. For example, their is a

one-dimensional quotient A_{f} associated to form f ∈ S_{4}(10) such that m_{f} = 2 · 5.

Algorithm 2.32. The modular kernel is the cokernel of the natural map S[I_{f}] → Φ_{f}(S) of Diagram 1.1.

Proof. Use the snake lemma.

2.10 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 compue the modular degree of the torus A_{f} attached to f.

Definition 2.33. The modular map is the map θ_{f} : A_{f}^{∨} → A_{f} of Diagram 1.1. The modular degree m_{f} of f (or of A_{f}) is the degree of this map.

When k = 2, θ_{f} is a polarization so ([Mil86, Thm. 13.3]) the degree of θ is a square.

Proposition 2.34. Let E/Q be a modular elliptic curve of conductor N which is an optimal quotient of J_{0}(N). Then δ_{f} is the square of the usual modular degree, which is the least degree of a map X_{0}(N) → E.

When k ≠ 2, the degree of θ need not be a perfect square. For example, their is a

one-dimensional quotient A_{f} associated to form f ∈ S_{4}(10) such that m_{f} = 2 · 5.

Algorithm 2.35. The modular kernel is the cokernel of the natural map S[I_{f}] → Φ_{f}(S) of Diagram 1.1.

Proof. Use the snake lemma.

2.11 Rational part of the L-function

Let M(Q) = M(N, Q) and extend Φ_{f} to a map M(Q) → C. Then Φ_{f} has a

rational structure in the following sense.

Lemma 2.36. Let φ_{1}, …, φ_{n} be a Q-basis for Hom(M(Q), Q)[I_{f}] and set

Ψ = φ_{1} × ⋯ × φ_{n} : M(Q) → Q^{n}.

Then n = 2d and ker(Ψ) = ker(Φ_{f}).

Proof. This result is due to Shimura ([Shi77]), but we sketch a proof. To compute the dimension of Hom(M(Q), Q)[I_{f}] we may first tensor with C. Let \overline{S}_{2} denote the weight 2 anti-holomorphic cusp forms and E_{2} the weight 2 Eisenstein series for Γ_{0}(N). Then M(C) is isomorphic as a T-module to S_{2} ⊕ \overline{S}_{2} ⊕ E_{2} (prop. 9 of [Mer94] and the Eichler-Shimura embedding). Because of the Peterson inner product, the dual Hom(M(C), C) is also isomorphic as a T-module to S_{2} ⊕\overline{S}_{2} ⊕E_{2}. Since f is new, by the Atkin-Lehner theory,

(S_{2} ⊕ \overline{S}_{2} ⊕ E_{2})[I_{f}] = S_{2}[I_{f}] ⊕ \overline{S}_{2}[I_{f}]

has complex dimension 2d, which gives the first assertion.

Next note that ker(Φ_{f}) ⊗ C ⊂ ker(Ψ) ⊗ C because each map x ↦ ⟨f_{i}, x⟩ lies

in Hom(M(Q), C)[I_{f}] and ker(Ψ) ⊗ C is the intersection of the kernels of all maps in Hom(M(Q), C)[I_{f}]. By Theorem 4.14 the image of Φ_{f} is a lattice, so dim_{Q} ker(Φ_{f}) = dim_{Q} M(Q) − 2d. Since Ψ is the intersection of the kernels of n = 2d independent linear functionals φ_{1}, …, φ_{n}, ker(Ψ) also has dimension dim M(Q) − 2d. Since the dimensions are the same and there is an inclusion, we have an equality ker(Φ_{f})⊗C = ker(Ψ)⊗C which forces ker(Φ_{f}) = ker(Ψ).

Let V be a finite dimensional vector space over R. A lattice L ⊂ V is a free

abelian group of rank = dim V such that RL = V. If L, M ⊂ V are lattices, the lattice index [L : M] is the absolute value of the determinant of an automorphism of V taking L isomorphically onto M. Extend the definition to the case when M has rank strictly smaller than dim V by defining [L : M] = 0.

Lemma 2.37. Suppose τ_{i} : V → W_{i}, i = 1, 2 are surjective linear maps such that ker(τ_{1}) = ker(τ_{2}). Then

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

Proof. Surjectivety 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 doesn’t change the determinant,

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

2.11. RATIONAL PART OF THE L-FUNCTION 31

Let S_{2}(N, Z) be the space of cusp forms whose q-expansion at infinity hass integer

coefficients. Let Ω_{f}^{0} be the measure of the identity component of A_{f}(R) with respect to an integral basis for S_{f}(Z) = S_{2}(N, Z)[I_{f}]. Let e = {0, i∞} ∈ M(N, Z) denote the winding element.

Theorem 2.38. Let Ψ be as in Lemma 2.36. Then

±\frac{L(A_{f}, 1)}{Ω_{f}^{0}} = [Ψ(S(N, Z)^{+}) : Ψ(Te)]

Proof. Let Φ = Φ_{f} be the period map defined by a basis f_{1}, …, f_{d} of conjugate newforms. The codomain of Φ, which we identify with C^{d}, is an algebra with unit element 1 = (1, …, 1) equipped with an action of the Hecke operators: T_{p} acts as (a_{p}^{(1)}, …, a_{p}^{(d)}) where the components are the Galois conjugates of a_{p}. Let Z^{d} ⊂ R^{d} ⊂ C^{d} be the usual submodules. Let Vol(S^{+}) be the volume of a fundamental domain for the real lattice Φ(S^{+}) = Φ(S(N, Z)^{+}). Observe that Vol(S^{+}) = [Z^{d} : Φ(S^{+})] and |L(A_{f}, 1)| = [Z^{d} : Φ(e)Z^{d}]. Let W ⊂ C^{d} be the Z-module spanned by the columns of a basis for S_{f}(Z). Because Ω_{f}^{0} is computed with respect to a basis for S_{f}(Z),

Vol(S^{+}) = [W : T1] · Ω_{f}^{0}.

Because S_{2}(N, Z) is saturated, [Z^{d} : W] = 1 so [Z^{d} : T1] = [W : T1]. The following calculation involves lattices in R^{d}:

[Φ(S^{+}) : Φ(Te)] = [Φ(S^{+}) : Z^{d}] · [Z^{d} : Φ(Te)]
= \frac{1}{[Z^{d} : Φ(S^{+})]} · [Z^{d} : Φ(Te)]
= \frac{1}{Vol(S^{+})} · [Z^{d} : Φ(e)Z^{d}] · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(S^{+})} · [Φ(e)Z^{d} : Φ(Te)]
= \frac{|L(A_{f}, 1)|}{Vol(S^{+})} · [Φ(e)Z^{d} : Φ(e)T1]
= \frac{|L(A_{f}, 1)|}{Ω_{f}^{0} · [W : T1]} · [Z^{d} : T1]
= \frac{|L(A_{f}, 1)|}{Ω_{f}^{0}}.

The theorem now follows from lemmas 2.36, 2.37, and the fact that f has real Fourier coefficients so L(A_{f}, 1) ∈ R hence |L(A_{f}, 1)| = ±L(A_{f}, 1).

Corollary 2.39. Let n_{f} be the order of the image in A_{f}(Q) of the point (0) − (∞) ∈ J_{0}(N)(Q). Then

\frac{L(A_{f}, 1)}{Ω_{f}^{0}} ∈ \frac{1}{n_{f}} Z.

Proof. Let x denote the image of (0) − (∞) ∈ A_{f}(Q) and set I = Ann(x) ⊂ T. Since f is a newform the Hecke operators T_{p} for p|N act as 0 or ±1 on A_{f}(Q) (end of section 6 of [DI95]). If p ∤ N a standard calculation (section 2.8 of [Cre97a]) combined with the Abel-Jacobi theorem shows that T_{p}(x) = (p + 1)x. Let C = Zx denote the (finite, by Manin-Drinfeld) cyclic subgroup of A_{f}(Q) generated by x, so n_{f} is the order of C. There is an injection T/I ↪ C sending T_{p} to T_{p}(x). By the theorem, we have

±L(A_{f}, 1)/Ω_{f}^{0} = [Ψ(S^{+}) : Ψ(Te)]
= [Ψ(S^{+}) : Ψ(Ie)] · [Ψ(Ie) : Ψ(Te)]
= [Ψ(S^{+}) : IΨ(e)] · [IΨ(e) : TΨ(e)]
= \frac{[Ψ(S^{+}) : IΨ(e)]}{[TΨ(e) : IΨ(e)]} ∈ \frac{1}{n_{f}} Z.

The final inclusion follows from two observations. By Abel-Jacobi, I is exactly those elements of T which send Ψ(e) into Ψ(S^{+}), so [Ψ(S^{+}) : IΨ(e)] ∈ Z. Second, there is a surjective map

T/I → \frac{TΨ(e)}{IΨ(e)}

sending t to tΨ(e), so [TΨ(e) : IΨ(e)] divides n_{f} = |C| = |T/I|.

2.12 Analytic invariants

Let

f = ∑_{n≥1} a_{n}q^{n} ∈ S_{k}(N, ε)

be a newform, and assume that ε^{2} = 1. 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 1.7, 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 associated to f.

Recall that the L-series associated to f is

L(f, s) := ∑_{n=1}^{∞} a_{n}n^{−s},

and that Hecke proved that L(f, s) has an analytic continuation to the whole complex plane. Set

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

When k = 2 and ε = 1, this is the canonical L-series associated to the abelian variety A_{f}/Q.

Let

g = ∑_{n≥1} a_{n}q^{n} ∈ M_{k}(N, ε)

be a modular form (we do not assume that g is an eigenform). Recall (??) the integration pairing:

⟨ , ⟩ : 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(S_{k}(N, ε)[I_{f}], C),

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

2.12.1 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 respect 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.

2.12.2 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 cuspform 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.  (2.2)

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 2.40.

∫_{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) .  (2.3)

The following proposition is the higher weight analogue of [Cre97a, Prop. 2.1.1(5)].

Proposition 2.41. 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}  (2.4)
= ε(γ)(γ^{−1}P){α, ∞} − P {γ(α), ∞}.  (2.5)

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 (or rendering apparently unrepairable) the analogues of the formulas found in [Cre97a, 2.10].

Algorithm 2.42. 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 2.41. Replacing γ by −γ if necessary, we find that the imaginary parts of α and γ(α) = \frac{a+i}{cN} are both 1/cN > 0. Equation 2.2 and Lemma 2.40 can now be used to compute the period integrals of Proposition 2.41.

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(∞)}. In the special case of weight two and trivial character, this is the assertion (proved by Manin [Man72]) 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 2.43. 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 ??). 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).

1Helena Verrill found this proof!

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}).

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 2.43 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 whether S_{k}(N, ε) is generated by symbols of the form P {∞, γ(∞)} with γP = P. When k is odd this is clearly not the case: 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.

2.12.3 The W_{N}-trick

In this section we assume that k is even. Consider the involution W_{N} defined

in Section ??. 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 [Cre97b] to higher weight.

Proposition 2.44. Let g ∈ S_{k}(N, ε) be a cuspform 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 2.41 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 − 2th symmetric product of the twodimensional 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 2.2 converge quickly if d is small. Let us emphasize, that it is extremely important to choose γ in Proposition 2.44 with d small, otherwise the series will converge very slowly.

2.12.4 Computing the period map

Let I ⊂ T be the kernel of the map T → K_{f} sending T_{n} to a_{n}. As in Section 2.7,

let Θ_{f} be the rational period mapping associated to f. We have a commutative diagram

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

Using Algorithm 2.24, we can compute Θ_{f} so to compute Φ_{f} we need to compute ι. 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(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 2.41 or Proposition 2.44 it is possible to efficiently compute the period integrals ⟨g_{i}, x_{j}⟩, i, j ∈ {1, … d}.

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

Given this data, we can compute

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

and

ι(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 ι.

It is advantageous when possible to find symbols x_{i} satisfying the conditions of

Proposition 2.44. 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.

2.12.5 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, ∞\}⟩,  (2.6)

L(A_{I}, s) = ∏_{i=1}^{d} L(f_{i}, s).  (2.7)

For ease of notation let

e_{i} := X^{s−1}{0, ∞}

denote the ith winding element. In section 2.12.4 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 2.6 we compute the special value.

2.12.6 Computing the real and pure imaginary volume

By Proposition 1.9, for any x ∈ S_{k}(N, ε) we have that

\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)).

Therefore complex conjugation leaves the period lattice Λ_{f} = Φ_{f}(S_{k}(N, ε)) invariant, which means that A_{f} = C^{d}/Λ_{f} is equipped with an action of complex conjugation. We can thus define the real and imaginary volumes of A_{f}. Let g_{1}, …, g_{d} be a Z-basis for S_{k}(N, ε; Z)[I_{f}]. This choice equips A_{f} = C^{d}/Λ_{f} with a real-valued measure μ. We define the real volume Ω_{f}^{+} = μ(A_{f}(R)) to be the measure of the points of A_{f} invariants under complex conjugation. The imaginary volume Ω_{f}^{−} = μ(A_{f}(C)^{−})i^{d} is the measure of those points anti-invariant under complex conjugation, multiplied by the power d = dim A_{f} of i.

Suppose s is an integer in the set {1, …, k − 1}. If s is odd then the ratio

L(A_{f}, s)/Ω_{f}^{+} is a rational number. When s is even the ratio L(A_{f}, s)/Ω_{f}^{−} is an integer times a power of 2. This is proved in Section 2.11.

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 [Tat95] for A_{f}. The latter quantity is the volume 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 so-called Manin constant, which is conjecturally 1 (see Section 3.2).

2.12.7 Examples

Jacobians of genus two curves

The authors of [FpS^{+}99] 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 [FpS^{+}99] computed the volume 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 one.

The first example considered in [FpS^{+}99] is the Jacobian A = J_{0}(23) of the mod-

ular 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 integral basis of cusp forms for S_{2}(23) can be found using the method described in Section ??:

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 2.12.3) 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 don’t 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 Section ?? 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 [FpS^{+}99]; since the Manin constant is an integer, it must equal 1.

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 pure imaginary volumes of the first few of these tori are displayed in Table 2.1^{2}. (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.

2It didn’t take more than three minutes to compute any number in this table

Table 2.1: Volumes of 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}

Table 2.2: 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

CM elliptic curves of weight greater than two

Let f be a rational newform with complex multiplication. Experimentally, it

appears that the associated elliptic A_{f} has rational j-invariant. As evidence for this we present Table 2.2, which includes the analytic data about every rational CM form of weight four and level ≤ 197. The computations of Table 2.2 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 unreckognizable to the author; in

contrast, in weight 2 these invariants are (expected to be) integers (see [Cre97a, 2.14]).

Some abelian varieties of large dimension

In Table 2.3, 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.

42 CHAPTER 2. ALGORITHMS FOR MODULAR FORMS

Table 2.3: Volumes of high 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 3

Quotients of J_{0}(N)

In this chapter k = 2 and ε = 1; we sometimes omit k and ε from the notation.

3.1 The torsion subgroup

Modular symbols can be used to obtain both upper and lower bounds on the

rational torsion subgroup of A_{f}.

3.1.1 Lower bounds: The cuspidal subgroup of A_{f}

Let s be the projection onto S_{2}(N; Q) in the T-module decomposition M_{2}(N; Q) =

S_{2}(N; Q) ⊕ E_{2}(N; Q).

Theorem 3.1 (Abel-Jacobi). Let [α], [β] be cusps of X_{0}(N) with α, β ∈ P^{1}(Q). Then the point [α] − [β] ∈ J_{0}(N) is 0 if and only if s({α, β}) ∈ S_{2}(N; Z).

Definition 3.2. The cuspidal subgroup of J_{0}(N) is the group generated by the differences of all cusps. The cuspdial subgroup of A_{f} is the image of the cuspidal subgroup of J_{0}(N).

Proposition 3.3. The cuspidal subgroup of J_{0}(N) is a finite group isomorphic to the subgroup C of S_{2}(N; Q)/S_{2}(N; Z) generated by the images of modular symbols {α, β} with α, β ∈ P^{1}(Q).

Corollary 3.4. The cuspidal subgroup of A_{f} is isomorphic to the image of C in

Φ_{f}(S_{2}(N; Q))/Φ_{f}(S_{2}(N; Z)).

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}.

For the case k > 2, see Section 2.8.

44 CHAPTER 3. QUOTIENTS OF J_{0}(N)

3.1.2 Upper bounds: Counting points mod p

Let f be a newform. The Hecke algebra T acts through a quotient \overline{T} on the

subspace of S_{2}(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 A_{f}.

Proposition 3.5.

N_{p} = χ_{p}(p + 1)

Proof. ^{1} We have

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

If the characteristic polynomial of Frob on an ℓ-adic Tate module is F (t), and the characteristic polynomial of T_{p} on differentials is f(t), then we have f(t) = x^{−g}F (x), where t = x + (p/x). In other words, Eichler-Shimura gives an easy conversion between f and F. Since it’s a general fact that #J(F_{p}) = F (1), we have

#J(F_{p}) = f(p + 1) = constant term of f(x + p + 1).

Theorem 3.6. Suppose p ∤ N. Then the kernel of the reduction map A_{f}(Q)_{tor} → A_{f}(F_{p}) is killed by p. If p > 2 then the kernel is trivial.

Thus we obtain an upper bound on #A_{f}(Q)_{tor}. This upper bound is not in general

sharp, in fact it is unchanged if A_{f} is replaced by any abelian variety isogeneous to A_{f}.

3.2 The Manin constant

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). 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. We then conjecture that c_{A} = 1, and give supporting numerical evidence.

3.2.1 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 [Maz78] 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 3.7. 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.

1Matt Baker told the author this proof.

Proof. Let π denote the map J_{0}(N) → A; let A denote the Neron model of A over R := Z[\frac{1}{2m}], and J the Neron model of J_{0}(N) over R. Let X be the smooth locus a minimal proper regular model for X_{0}(N) over R. Consider the diagram

H^{0}(A, Ω_{A}) ⟶^{π^{∗}} H^{0}(J , Ω_{J}) ≅ H^{0}(X , Ω_{X}) ⟶^{q-exp} R[[q]]  (3.1)

The map π^{∗} must be an inclusion because H^{0}(A, Ω_{A}) is torsion-free and π^{∗} is an

inclusion after tensoring with C. 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 = 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]]. To show that D[ℓ] = 0, it suffices to prove that the map Θ : H^{0}(A, Ω_{A}) ⊗ F_{ℓ} → F_{ℓ}[[q]] is injective.

Since ℓ ≠ 2 and A is optimal, [Maz78, 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 = 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 /F_{ℓ}, Ω_{X /F_{ℓ}})

is injective. 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}})

is injective.

If ℓ ∤ N, then injectivity of Θ now follows from the q-expansion principle, which

asserts that the q-expansion map H^{0}(X /F_{ℓ}, Ω_{X /F_{ℓ}}) → F_{ℓ}[[q]] is injective.

Suppose ℓ does divide N, and let ω ∈ ker(Θ). 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 singe 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 Θ is injective.

46 CHAPTER 3. QUOTIENTS OF J_{0}(N)

3.2.2 Numerical evidence for a c_{A} = 1 conjecture

In the paper [FpS^{+}99], the authors show that c_{A} = 1 for 28 two-dimensional

optimal quotients of J_{0}(N) (see Section 2.12.7). 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 3.8 (Agashé, Stein). Let A be an optimal quotient of J_{0}(N), and let c_{A} be the corresponding Manin constant. Then c_{A} = 1.

Chapter 4

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, 𝔪 its maximal ideal and k = O/𝔪 the residue class field. The Néron model of A is a smooth commutative group scheme A over O such that A is its generic fiber and satisfying the property: the restriction map

Hom_{O}(S, A) ⟶ Hom_{K}(S_{K}, A)

is bijective for all smooth schemes S over O. The closed fiber A_{k} 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

with Φ_{A} a finite étale group scheme over k, i.e., a finite abelian group equipped with an action of Gal(\overline{k}/k).

In this chapter we study the groups Φ_{A} attached to quotients A of Jacobians of

modular curves X_{0}(N). When A has semistable reduction, Grothendieck described the component group in terms of a monodromy pairing on certain free abelian groups. When A = J is the Jacobian of X_{0}(N), this pairing can be explicitly computed, hence so can Φ_{J}; this has been done in many cases in [Maz77] and [Edi91]. Suppose now that A = A_{f} is a quotient of J corresponding to a newform f, so the kernel of the map π : J → A is connected. There is a natural map π_{∗} : Φ_{J} → Φ_{A}. We describe how to compute the image and the order of the cokernel of π_{∗}.

We now state our main result more precisely. Suppose π : J → A is an optimal

quotient, with J a semistable Jacobian and A purely toric. 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. Let X_{J} be the character group of the toric part of the special fiber of J. Let L be the saturation of the image of X_{A} in X_{J}. The monodromy pairing defines 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 prove that

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

More precisely, Φ_{X} is the image of Φ_{J} → Φ_{A} and the cokernel of 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 so can |Φ_{A}|. Having done this, we present some tables and discuss the conjectures that they suggest.

4.1 Optimal quotients of Jacobians

Let J be a Jacobian equipped with its canonical principal polarization θ_{J}. 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. Dualizing π and composing with θ′_{J} = θ_{J} we obtain a map A′ ⟶^{π′} J′ ⟶^{θ_{J}} J.

Proposition 4.1. 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, π′ must have finite kernel. Thus A′ → C = im(π′) is an isogeny. Let G denote the kernel, and dualize. By [Mil86, §11] we have have

\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}}

with G′ the Cartier dual of G. Since G′ is finite, ker(φ) ⊂ ker(π) is of finite index. Since ker(π) is an abelian variety it is divisible. Thus ker(φ) = ker(π) so G′ = 0 and hence G = 0.

We denote the map A′ → J by π′. The kernel of θ_{A} measures the intersection of A′ and B = ker(θ_{A}) inside of J as shown 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}

Because θ_{A} is a polarization, | ker(θ_{A})| is a square [Mil86, Theorem 13.3]. The congruence modulus is the integer

m_{A} = \sqrt{| ker(θ_{A})|}.

4.2 The special 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/K be an abelian variety and denote its Néron model by A. Let Φ_{A} be the group of connected components of the special fiber A_{k}. This group is a finite étale group scheme over k, i.e., 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 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 of a U a unipotent group of dimension u. The abelian variety A is said to have purely toric reduction if t = dim A, and is semistable if u = 0. The character group X_{A} = Hom(T , G_{m}) is a free abelian group of rank t contravariantly associated to A. 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.

4.3 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 to prove that if A is purely toric and ϕ : A′ → A is a symmetric isogeny, then

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

We also prove a some lemmas about character groups.

4.3.1 Raynaud and van der Put uniformization

Theorem 4.2 (Raynaud). If A is a semistable Abelian variety, its universal covering 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 following diagram,

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

which we call the uniformization cross of A. The group Γ can be identified with the character group X_{A} of the previous section. 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}) and T′ = Hom(Γ, G_{m}) and the morphisms Γ′ → G′ and T′ → G′ are the one-motive duals of the morphisms T → G and Γ → G, respectively. For more details see, e.g., [Col99]. Example 4.3 (Tate curve). If E/Q_{p} is an elliptic curve with 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}.

4.3.2 Some lemmas

Let π : J → A be an optimal quotient, with J semistable and A purely toric.

Lemma 4.4. The map Γ_{J} → Γ_{A} induced by π is surjective.

Proof. Because 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. The snake lemma applied to the following diagram gives 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}

Because π is optimal, B is connected so M must also be connected. Since M is discrete it follows that M = 0.

Purely toric abelian varieties

Assume that A is purely toric. Then B = 0, and the uniformization cross becomes

Γ

T

A Let φ : A′ → A be a symmetric isogeny, i.e., φ′ : A′ → (A′)′ = A is equal to φ. Denote by φ_{t} : T′ → T and φ_{a} : Γ′ → Γ the induced maps.

Proposition 4.5. There is an exact sequence

0 → ker(φ_{t}) → ker(φ) → coker(φ_{a}) → 0,

and ker(φ_{t}) is dual to coker(φ_{a}).

Proof. Since φ is an isogeny we have the following diagram:

\xymatrix{0 \ar[r] \ar[d] & Γ′ \ar[r] \ar[d] & Γ \ar[r] \ar[d] & coker(φ_{a}) \ar[d] \\ ker(φ_{t}) \ar[r] \ar[d] & T′ \ar[d] \ar[r] & 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 the one-motive dual of the diagram

\xymatrix{ & Γ′ \ar[r] \ar[d] & Γ \ar[d] \ar[r] & coker(φ_{a}) \\ ker(φ_{t}) \ar[r] & T′ \ar[r]^{φ} & T}

is the diagram

\xymatrix{ & T & T′ \ar[l]_{φ′} & coker(φ_{a})′ \ar[l] \\ ker(φ_{t})′ & Γ \ar[l] \ar[u] & Γ′ \ar[l] \ar[u]}

Since φ is symmetric, φ′ = φ and so

ker(φ_{t}) = coker(φ_{a})′.

Lemma 4.6. | ker(φ)| = | coker(φ_{a})|^{2}

Proof. The order of a finite group scheme equals the order of its dual.

4.4 The main theorem

Let π : J → A be an optimal quotient, with J a semistable Jacobian and A purely

toric. Let X_{A}, X_{A′}, and X_{J} denote the character groups of the toric parts of the special fibers.

4.4.1 Monodromy description of the component group

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 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 4.7 (Tate curve). Suppose E = G_{m}/q^{Z} is a Tate curve. 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. The key diagrams are

\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 4.4. The injectivity of π^{∗} follows because

θ_{∗}π_{∗}π^{∗} = θ_{∗}θ^{∗} = deg(θ) ≠ 0,

and multiplication by deg(θ) 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 4.8. 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 rank and the cokernel is torsion, the monodromy map is injective. Thus π_{∗}x = 0 and x ∈ ker(π_{∗}).

Lemma 4.9. There is an exact sequence

X_{J} → Hom(π^{∗}X_{A}, Z) → Φ_{A} → 0.

Proof. Lemma 4.8 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 4.8, 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}, i.e., [L : π^{∗}X_{A}] is finite and X_{J}/L is

torsion free. Suppose L is of finite index in L. Define the congruence modulus of L

m_{L} = [α(X_{J}) : α(L)]

and the component group by

Φ_{L} = coker(X_{J} → Hom(L, Z)).

When L = L we often set m_{X} = m_{L} and Φ_{X} = Φ_{L} and think of m_{X} and Φ_{X} as the character group “congruence modulus and component group.”

Lemma 4.10. The rational number \frac{|Φ_{L}|}{m_{L}} does not depend on the choice of L.

Proof. If L′ is another choice let n = [L : L′] ∈ Q. Then since α is injective when restricted to L,

m_{L′} = [α(X_{J}) : α(L′)] = [α(X_{J}) : α(L)] · [α(L) : α(L′)] = m_{L} · n

and similarly |Φ_{L′}| = |Φ_{L}| · n.

Recall that we defined

m_{A} = \sqrt{deg(θ)}
Φ_{A} = coker(X_{A′} → Hom(X_{A}, Z))

Theorem 4.11. For any L of finite index in L the following relation holds:

\frac{|Φ_{A}|}{m_{A}} = \frac{|Φ_{L}|}{m_{L}}.

Proof. By Lemma 4.10 we may assume that L = π^{∗}X_{A}. With this choice of L, Lemma 4.9 says that Φ_{L} ≅ Φ_{A}. By Lemma 4.8, properties of the index, and Lemma 4.6 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 4.12.

image(Φ_{J} → Φ_{A}) ≅ Φ_{L}.

Proof. Because π^{∗}X_{A} ⊂ L ⊂ X_{J}, by Lemma 4.9 we obtain a 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 Φ_{L} → Φ_{A} is injective, hence

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) = 0,

where Ext^{1} vanishes because L is saturated so that X_{J}/L is torsion free. Thus the cokernel of Φ_{J} → Φ_{L} is 0, from which the proposition follows.

The following corollary follows from Theorem 4.11 and Proposition 4.12.

Corollary 4.13.

| coker(X_{J} → X_{A})| = \frac{m_{A}}{m_{L}} .

As a consequence, m_{L}|m_{A}.

4.5. OPTIMAL QUOTIENTS OF J_{0}(N) 55

4.5 Optimal quotients of J_{0}(N)

Let X_{0}(N)/Q be the modular curve associated to the congruence subgroup Γ_{0}(N) ⊂

SL_{2}(Z) of matrices which are upper triangular modulo N. Let p be a prime divisor of N which is coprime to M = N/p. The Jacobian J = J_{0}(N) of X_{0}(N) has semistable reduction at p. The Hecke algebra

T = Z[… T_{n} …] ⊂ End(J)

is a commutative ring of endomorphisms of J of Z-rank = dim J. The character group X_{J} is equipped with a functorial action of T. The Hecke algebra T also act on the cusp forms

S = S_{2}(Γ_{0}(N), C).

A newform f is an eigenform normalized so that the coefficient of q in the expansion of f at the cusp ∞ is 1, and such that f does not occur at any level N′ | N with N′ ≠ N. If f is a newform, let I_{f} be the ideal in 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 a purely toric optimal quotient of dimension [K_{f} : Q].

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 which we view as a map α : H → Hom_{C}(S, C).

Theorem 4.14. 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 [Shi73].

Corollary 4.15. m_{A}^{2} = [α(H) : α(H[I_{f}])].

Proof. Recall that m_{A} is defined to be \sqrt{deg(θ_{A})}. The kernel of an isogeny of complex tori is isomorphic to the cokernel of the induced map on lattices. The corollary now follows from the diagram of Theorem 4.14 which indicates that the index [α(H) : α(H[I_{f}])] is the cokernel of the map H[I_{f}] → α(H).

Let Frob_{p} : X_{J} → X_{J} denote the map induced by Frobenius. One has 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 4.16.

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. Thus in this case, Φ(F_{p}) = Φ(\overline{F}_{p}). Next suppose w_{p} = 1. We have an exact sequence

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

Since W_{p} acts as +1 on f it acts as +1 on A, on X_{A}, on Hom(X_{A}, Z), and on Φ_{A}. Thus Frob_{p} = −W_{p} acts as −1 on Φ_{A}. The 2-torsion in a finite abelian group equals the fixed points under −1.

WARNING: When extending this result to the whole of J_{0}(N) be careful! The

action of Frob_{p} = T_{p} is not by ±1, even though it must be by an involution of order 2. For example, the component group of J_{0}(65)/F_{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 multipliction by −13 is the order 14 subgroup generated by 3.

Computation

Using the algorithms of Chapter 2, we can enumerate the optimal quotients A_{f}

and compute m_{A}. The method of graphs [Mes86] and [Koh98] can be used to compute X = X_{J_{0}(N)} with its T-action and the monodromy pairing. We can then compute

L = ⋂_{t∈I_{f}} ker(t|_{X}),

m_{X} := m_{L}, and Φ_{X} := Φ_{L}. By Theorem 4.11 we can now compute

|Φ_{A}| = |Φ_{X}| · \frac{m_{A}}{m_{X}} .

We have computed Φ_{A} in a number of cases. In the next subsection we discuss two conjectures suggested by our numerical computations.

4.5.1 Conjectures

Our numerical computations suggest the following conjectures. 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 on the submodule H_{new}[I_{f}] of elements annihilated by I_{f}. Integration defines a map α : H_{new} → Hom(S[I_{f}], C). Define the homology congruence modulus 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}.

Conjecture 4.17. m_{X} = m_{H}.

When N = p is prime we make the following conjecture.

Conjecture 4.18. 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 corresponding optimal quotients. Then #A_{i}(Q) = #Φ_{A_{i}} for each i and #Φ_{J_{0}(p)} = ∏_{i=1}^{d} #Φ_{A_{i}}.

Note, the natural map Φ_{J_{0}(113)} → ∏_{i=1}^{4} Φ_{A_{i}} is not an isomorphism because two of

the Φ_{A_{i}} have order two, so the product is not cyclic.

4.6 Tables

We computed several component groups of optimal quotients A_{f} of J_{0}(N) associ-

ated to newforms f. We denote such an optimal quotient by

N isogeny-class dimension

The dimension frequently determines the factor, so it is included in the notation.

4.6.1 Table 1: Some large component groups predicted by the Birch and Swinnerton-Dyer conjecture

Using the algorithm described in [AS99] we computed the special value L(A, 1)/Ω

(up to a Manin constant) for every optimal quotient A = A_{f} of level ≤ 1500. We found exactly five for which the numerator of L(A, 1)/Ω is nonzero and divisible by a prime number > 10^{9}. These are given below.

A  N  L(A, 1)/Ω · Manin constant
1154E20 2 · 577  2^{?} · 85495047371/17^{2}
1238G19 2 · 619  2^{?} · 7553329019/5 · 31
1322E21 2 · 661  2^{?} · 57851840099/331
1382D20 2 · 691  2^{?} · 37 · 1864449649/173
1478J20 2 · 739  2^{?} · 7 · 29 · 1183045463/5 · 37

The Birch and Swinnerton-Dyer conjecture predicts that these large prime divisors must divide either |Φ_{A}| or the Shafarevich-Tate group of A. We computed Φ_{A} and found that 58 CHAPTER 4. COMPONENT GROUPS this was the case.

A  p w |Φ_{X}|  m_{X}  |Φ_{A}|
1154E20 2 − 17^{2}  2^{24}  2^{?} · 17^{2} · 85495047371
577 + 1  2^{26} · 85495047371  2^{?}
1238G19 2 − 5 · 31  2^{26}  2^{?} · 5 · 31 · 7553329019
619 + 1  2^{28} · 7553329019  2^{?}
1322E21 2 − 331  2^{28}  2^{?} · 331 · 57851840099
661 + 1  2^{32} · 57851840099  2^{?}
1382D20 2 − 173  2^{29}  2^{?} · 37 · 173 · 1864449649
691 + 1  2^{31} · 37 · 1864449649  2^{?}
1478J20 2 − 5 · 37  2^{31}  2^{?} · 5 · 7 · 29 · 37 · 1183045463
739 + 1 2^{33} · 7 · 29 · 1183045463  2^{?}

4.6.2 Table 2: Some quotients of J_{0}(N)

In this table we give the invariants defined above for the optimal quotients of levels

65, 66, 68, and 69.

A  p w |Φ_{X}| m_{X} m_{H} m_{A} |Φ_{A}|
65A1 5 + 1  2  ?  2  1
13 + 1  2  ?  1
65B2 5 + 3  2^{2}  ?  2^{2}  3
13 − 3  2^{2}  ?  3
65C2 5 − 7  2^{2}  ?  2^{2}  7
13 + 1  2^{2}  ?  1
66A1 2 + 1  2  ?  2^{2}  2
3 − 3  2^{2}  ?  3
11 + 1  2^{2}  ?  1
66B1 2 − 2  2  ?  2^{2}  2^{2}
3 + 1  2^{2}  ?  1
11 + 1  2^{2}  ?  1
66C1 2 − 1  2  ? 2^{2} · 5 2 · 5
3 − 1  2^{2}  ?  5
11 − 1 2^{2} · 5 ?  1
68A2 17 + 2 2 · 3 ? 2 · 3  2
69A1 3 − 2  2  ?  2  2
23 + 1  2  ?  1
69B2 3 + 2  2  ? 2 · 11 2 · 11
23 − 2 2 · 11 ?  2

4.6.3 Table 3: Some quotients of J_{0}(p)

Using the method of graphs and modular symbols we computed the quantities

m_{A}, m_{L} and Φ_{L} for each abelian variety A = A_{f} associated to a newform of prime level p ≤ 757. The results were as follows:

1. In all cases m_{A} = m_{L}, 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}| = |Φ_{J}| Table 1 lists those A of level ≤ 631 for which w_{p} = −1, along with the order of the component group.

60 CHAPTER 4. COMPONENT GROUPS Table 3: Some quotients of J_{0}(p)

A  |Φ_{A}|
11A1  5
17A1  2^{2}
19A1  3
23A2  11
29A2  7
31A2  5
37B1  3
41A3  2 · 5
43B2  7
47A4  23
53B3  13
59A5  29
61B3  5
67A1  1
67C2  11
71A3  5
71B3  7
73A1  2
73C2  3
79B5  13
83B6  41
89B1  2
89C5  11
97B4  2^{3}
101B7  5^{2}
103B6  17
107B7  53
109A1  1
109C4  3^{2}
113A1  2
113B2  2
113D3  7
127B7 3 · 7
131B10 5 · 13
137B7 2 · 17
139A1  1
139C7  23
149B9  37
151B3  1
151C6  5^{2}

A  |Φ_{A}|
157B7  13
163C7  3^{3}
167B12 83
173B10 43
179A1  1
179C11 89
181B9  3 · 5
191B14 5 · 19
193C8  2^{4}
197C10 7^{2}
199A2  1
199C10 3 · 11
211A2  5
211D9  7
223C12 37
227B2  1
227C2  1
227E10 113
229C11 19
233A1  2
233C11 29
239B17 7 · 17
241B12 2^{2} · 5
251B17 5^{3}
257B14 2^{6}
263B17 131
269C16 67
271B16 3^{2} · 5
277B3  1
277D9  23
281B16 2 · 5 · 7
283B14 47
293B16 73
307A1  1
307B1  1
307C1  1
307D1  1
307E2  3
307F9  17
311B22 5 · 31

A  |Φ_{A}|
313A2  1
313C12 2 · 13
317B15 79
331D16 5 · 11
337B15 2^{2} · 7
347D19 173
349B17 29
353A1  2
353B3  2
353D14 2 · 11
359D24 179
367B19 61
373C17 31
379B18 3^{2} · 7
383C24 191
389A1  1
389E20 97
397B2  1
397C5  11
397D10  3
401B21 2^{2} · 5^{2}
409B20 2 · 17
419B26 11 · 19
421B19 5 · 7
431B1  1
431D3  1
431F24 5 · 43
433A1  1
433B3  1
433D16 2^{2} · 3^{2}
439C25 73
443C1  1
443E22 13 · 17
449B23 2^{4} · 7
457C20 2 · 19
461D26 5 · 23
463B22 7 · 11
467C26 233
479B32 239
487A2  1

A  |Φ_{A}|
487B2  3
487C3  1
487D16  3^{3}
491C29 5 · 7^{2}
499C23  83
503B1  1
503C1  1
503D3  1
503F26 251
509B28 127
521B29 2 · 5 · 13
523C26 3 · 29
541B24 3^{2} · 5
547C25 7 · 13
557B1  1
557D26 139
563A1  1
563E31 281
569B31 2 · 71
571A1  1
571B1  1
571C2  1
571D2  1
571F4  1
571I18 5 · 19
577A2  3
577B2  1
577C3  1
577D18  2^{4}
587C31 293
593B1  2
593C2  1
593E27 2 · 37
599C37 13 · 23
601B29 2 · 5^{2}
607D31 101
613C27 3 · 17
617B28 2 · 7 · 11
619B30 103
631B32 3 · 5 · 7

Chapter 5

Visibility of Shafarevich-Tate groups

In this chapter, we study visibility of Shafarevich-Tate group of abelian varieties. Consider an abelian variety A and an injection ι : A → J of A into an abelian

variety J. Mazur (see [CM98]) introduced the subgroup H^{1}(Q, A)^{∘} of elements of H^{1}(Q, A) visible inside of J. This is the kernel of the map H^{1}(Q, A) → H^{1}(Q, J) induced by ι. Of particular interest is the visible subgroup � (A/Q)^{∘} consisting of elements of � (A/Q) that lie in H^{1}(Q, A)^{∘}.

In [CM98], Cremona and Mazur asked the following question: if E ⊂ J_{0}(N) is a

elliptic curve, how much of � (E/Q) is visible in an abelian surface inside J_{0}(N)? In their numerical experiment, visibility appears to occur surprisingly often. However, they warn that the conductors considered (N ≤ 5500) are quite small, and the situation is likely to change as the conductors get larger.

In [Maz99], Mazur proved that if E is an elliptic curve over a number field K, then

every element of order 3 in � (E/K) is visible in some abelian surface.

In this paper, we consider questions of visibility for abelian varieties A of dimension

possibly greater than one. This extension is important for several reasons. The numbers become large at much smaller conductor, so behavior that is not observed when considering elliptic curves becomes apparent. Certain moduli surfaces are of general type, so one does not expect visibility, in abelian surfaces, of large order Shafarevich-Tate group of elliptic curves; it is thus essential to consider visibility in higher dimensional abelian varieties. J. de Jong proved in [CM98, §3] that every element of � (E/Q) becomes visible in some abelian variety.

5.1 The Birch and Swinnerton-Dyer conjecture

In [Tat95], Tate stated a generalization of the conjecture of Birch and Swinnerton-

Dyer to abelian varieties over number fields. We recall a special case of the conjecture. Consider an abelian variety A over Q that is a quotient of J_{0}(N) for some N, and let A/Z be its Néron model. For each prime p, let C_{A,p} denote the component group of the closed fiber A_{F_{p}}. A basis h_{1}, …, h_{d} for H^{0}(A, Ω_{A/Z}) defines a measure μ on A(R). Under the further assumption that L(A, 1) ≠ 0, it is proved in [KL89] that A(Q) and � (A/Q) are both finite groups. Birch, Swinnerton-Dyer, and Tate gave a conjectural relationship between all of these finite groups that expresses the special value at 1 of L(A, s) in terms of the volumes of all of these groups of rational points.

Conjecture 5.1. If A is a modular abelian variety of analytic rank 0, then

L(A, 1) = \frac{∏_{p|N} #C_{A,p}(F_{p})}{#A(Q) · #A^{∨}(Q)} · #� (A/Q) · μ(A(R)).

As evidence for this conjecture, Tate proved that both the left and right hand

sides are invariant under isogeny. Recent work of Kato gives bounds on #� (A/Q) that are consistent with this conjecture. [[Reference the blue book with the article on this by Scholl.]]

In what follows we will frequently write c_{p} = #C_{A,p}(F_{p}) and Ω_{A} = μ(A(R)).

5.2 Tables

This section contains tables computed using the above algorithms as implemented

in the author’s program HECKE (a C++ program using LiDIA and NTL), David Kohel’s Magma software, and PARI. Each factor A_{f} of J_{0}(N) is denoted as follows:

N isogeny-class dimension

The dimension frequently determines the factor, so it is included in the notation. We consider only the odd part of � so we only computed the odd parts of the arithmetic invariants of A_{f}. Thus at this point we make the

WARNING: ONLY ODD PARTS OF INVARIANTS ARE GIVEN!

Tables 1-3: New Visible � Let n_{f} be the largest odd square dividing the numerator of L(A_{f}, 1)/Ω_{f}. Table 1 lists those A_{f} such that, for p|n_{f} there exists a new factor B_{g} of J_{0}(N), of positive analytic rank, and such that (A_{f}^{∨} ∩ B_{g}^{∨})[p] ≠ 0. This is necessary (and usually sufficient) for the p-torsion in the new visible part of � to be nonzero. In many cases it could be seen that there were no other appropriate new factors by looking at the signs of the Atkin-Lehner involutions. Up to level 1001 our search was systematic. The two examples after level 1001 were not found by systematic search (i.e., there may be a gap). In those cases for which 4|N, we put c_{2} = a, as we don’t know how to compute c_{2} exactly when the reduction is additive. Table 2 contains further arithmetic information about each explanatory factor.

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

of level ≤ 1028. By [Bru95], the explanatory factor at level 1061 is the first surface of rank 4 (and prime level). Table 4: Component groups Table 4 gives the quantities involved in the formula for Tamagawa numbers, for each of the A_{f} from table 1.

Table 5: Odd square numerator In order to find the A_{f}, we first enumerated those A_{f} for which the numerator of L(A_{f}, 1)/Ω_{f} is divisible by an odd square n_{f}. For N < 1000, these are given in table 5. Any odd visible � coprime to primes dividing torsion and c_{p} must show up as a divisor of the numerator, and given BSD, it must show up as a square divisor because the Mordell-Weil rank of the explanatory factor is even. It would be interesting to compute the conjectural order of � for each abelian variety in this table, but not in table 1, and show (when possible) that the visible � is old. Large Sha Upon seeing an early draft of this paper, Armand Brumer provided us with the first quotients of J_{0}(p) having positive rank and dimension 3 and 4. Computing further I’ve found the following spectacular table. The listed orders of � really is the numerator of the L-function as a long computation showed.

A_{f}  divisior of |� | B_{g}
389E20  5  389A1
1061D46  151  1061B2
1693B72  1301  1693A3
2593B109  67 · 2213  2593A4

For 1693 the odd part of the special value is 1301^{2}/141. For 2593 it is 67^{2}·2213^{2}/3^{3}.

5.3 Prime conductor

Using [REFS] we computed the odd part � _{an} of the analytic order of the Sha-

farevich and Tate group of every rank zero optimal quotient of J_{0}(p) for p ≤ 2593. Those for which � _{an} > 1 are laid out in the first column of Table ??. The fourth column lists the explanatory factor, if there is one at the same level. If � is not visible we were sometimes able to find an explanatory factor at higher level; this is discussed in the next section.

Table ?? has many interesting properties.^{1} There are 23 examples in which � is

(almost certainly) visible and 18 examples in which � is invisible. The largest visible � found occurs at level 2333 and has order at least 83341^{2} (83341 is prime). Believe it or not, the largest invisible � occurs in a 112 dimensional abelian variety at level 2111 and has order at least 211^{2}. Furthermore,

1. The Shafarevich-Tate group is never Eisenstein in these examples [[recheck, as I’ve added new examples!]]

2. 1283C. This example demonstrates that � _{an} can divide the modular degree, yet still be invisible. The other factor 1283A (of dimension 2) which satisfies a 5-congruence with 1283C has analytic rank (and hence algebraic rank) 0.

1Examples 1091C and 1429B were discoverd in [Aga99] and 1913B in [CM98].

64 CHAPTER 5. VISIBILITY OF SHAFAREVICH-TATE GROUPS

Table 5.1: New visible Shafarevich-Tate

A_{f}  n_{f}  w_{q}  c_{p}  T T L(1)/Ω_{f}  δ_{A}  B_{g}
389E20  5^{2}  −  97  97  5^{2}  5  389A1
433D16  7^{2}  −  3^{2}  3^{2}  7^{2}  3 · 7 · 37  433A1
446F8  11^{2}  +− 1, 3  3  11^{2}  11 · 359353  446B1
563E31  13^{2}  −  281 281  13^{2}  13  563A1
571D2  3^{2}  −  1  1  3^{2}  3^{2} · 127  571B1
655D13  3^{4}  +− 1, 1  1  3^{4}  3^{2} · 19 · 515741 655A1
664F8  5^{2}  −+ a, 1  1  5^{2}  5  664A1
681B1  3^{2}  +− 1, 1  1  3^{2}  3 · 5^{3}  681C1
707G15  13^{2}  +− 1, 1  1  13^{2}  13 · 800077  707A1
709C30  11^{2}  −  59  59  11^{2}  11  709A1
718F7  7^{2}  +− 1, 1  1  7^{2}  7 · 151 · 35573  718B1
794G14  11^{2}  +− 3, 1  3  11^{2}  3 · 7 · 11 · 47 · 35447 794A1
817E15  7^{2}  +− 1, 5  5  7^{2}  7 · 79  817A1
916G9  11^{2}  −+ a, 1  1  11^{2}  3^{9} · 11 · 17 · 239 916C1
944O6  7^{2}  +− a, 1  1  7^{2}  7  944E1
997H42  3^{4}  −  83  83  3^{4}  3^{2}  997B1, C1
1001L7  7^{2}  + − + 1, 1, 1 1  7^{2}  7 · 19 · 47 · 2273 1001C1
1028E14  3^{2} · 11^{2}  −+ a, 1  3  3^{4} · 11^{2}  3^{13} · 11  1028A1
1061D46  151^{2}  − 5 · 53 5 · 53  151^{2}  61 · 151 · 179  1061B2
2593B109 67^{2} · 2213^{2} −  ?  ?  ?  ?  2593A4

5.3. PRIME CONDUCTOR 65

Table 5.2: Explanatory factors

B_{g} rank w_{q} c_{p} T δ_{B} Comments

389A1

2  −  1 1 5  first curve of rank 2
433A1  2  −  1 1 7

446B1

2  +− 1, 1 1 11  this is 446D in [Cre97a]
563A1  2  −  1 1 13
571B1  2  −  1 1 3
655A1  2  +− 1, 1 1 3^{2}
664A1  2  −+ 1, 1 1 5
681C1  2  +− 1, 1 1 3
707A1  2  +− 1, 1 1 13
709A1  2  −  1 1 11
718B1  2  +− 1, 1 1 7
794A1  2  +− 1, 1 1 11
817A1  2  +− 1, 1 1 7
916C1  2  −+ 3, 1 1 3 · 11
944E1  2  +− 1, 1 1 7
997B1  2  −  1 1 3
997C1  2  −  1 1 3
1001C1 2 + − + 1, 3, 1 1 3^{2} · 7

1028A1

2  −+ 3, 1 1 3 · 11  intersects 1028E mod 11

1061B2

4  −  1 1 151 first prime level surface of pos. even rank [Bru95]

Table 5.3: 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}

66 CHAPTER 5. VISIBILITY OF SHAFAREVICH-TATE GROUPS

Table 5.4: Component groups

A_{f}  p w_{p} | coker |  | disc(X[I_{f}])|  |Φ(\overline{F}_{p})|
389E20 389 −  97  5 · 97  97
433D16 433 −  3^{2}  3^{3} · 7 · 37  3^{2}
446F8  223 −  3  3 · 11 · 359353  3
2 +  3  3 · 11  3 · 359353
563E31 563 − 281  13 · 281  281
571D2  571 −  1  3^{2} · 127  1
655D13 131 −  1  3^{2} · 19 · 515741  1
5 +  1  3^{2}  19 · 515741
664F8  83 +  1  5  1
681B1  227 −  1  3 · 5^{3}  1
3 +  1  3 · 5^{2}  5
707G15 101 −  1  13 · 800077  1
7 +  1  13  800077
709C30 709 −  59  11 · 59  59
718F7  359 −  1  7 · 151 · 35573  1
2 +  1  7  151 · 35573
794G14 397 −  3  3^{2} · 7 · 11 · 47 · 35447  3
2 +  3  3 · 11  3^{2} · 7 · 47 · 35447
817E15  43 −  5  5 · 7 · 79  5
19 +  1  7  79
916G9  229 +  1  3^{9} · 11 · 17 · 239  1
944O6  59 −  1  7  1
997H42 997 −  83  3^{2} · 83  83
1001L7  13 +  1  7 · 19 · 47 · 2273  1
11 −  1  7 · 19 · 47 · 2273  1
7 +  1  7 · 19 · 47  2273
1028E14 257 +  1  3^{13} · 11  1
1061D46 1061 − 5 · 53 5 · 53 · 61 · 151 · 179  5 · 53

5.3. PRIME CONDUCTOR 67

Table 5.5: Odd square numerator

305D7 : 3  309D8 : 5  335E11 : 3^{2}  389E20 : 5 394A2 : 5  399G5 : 3^{4}
433D16 : 7 435G2 : 3  436C4 : 3  446E7 : 3  446F8 : 11 455D4 : 3
473F9 : 3  500C4 : 3  502E6 : 11  506I4 : 5  524D4 : 3  530G4 : 7
538E7 : 3  551H18 : 3 553D13 : 3  555E2 : 3  556C7 : 3  563E31 : 13
564C3 : 3  571D2 : 3  579G13 : 3 · 5 597E14 : 19 602G3 : 3  604C6 : 3
615F6 : 5  615G8 : 7  620D3 : 3  620E4 : 3  626F12 : 5 629G15 : 3
642D2 : 3  644C5 : 3  644D5 : 3  655D13 : 3^{2} 660F2 : 3  662E10 : 43
664F8 : 5  668B5 : 3  678I2 : 3  681B1 : 3  681I10 : 3 682I6 : 11
707G15 : 13 709C30 : 11 718F7 : 7  721F14 : 3^{2} 724C8 : 3  756G2 : 3
764A8 : 3  765M4 : 3 766B4 : 3  772C9 : 3  790H6 : 3  794G12 : 11
794H14 : 5^{2} 796C8 : 3  817E15 : 7  820C4 : 3  825E2 : 3  844C10 : 3^{2}
855M4 : 3 860D4 : 3  868E5 : 3  876E5 : 3  878C2 : 3  884D6 : 3
885L9 : 3^{2} 894H2 : 3  902I5 : 3  913G17 : 3 916G9 : 11 918O2 : 5
918P2 : 3  925K7 : 3  932B13 : 3^{2}  933E14 : 19 934I12 : 7 944O6 : 7
946K7 : 3  949B2 : 3  951D19 : 3  959D24 : 3 964C12 : 3^{2} 966J1 : 3
970I5 : 3  980F1 : 3  980J2 : 3  986J7 : 5  989E22 : 5 993B3 : 3^{2}
996E4 : 3  997H42 : 3^{2} 998A2 : 3  998H9 : 3  999J10 : 3

68 CHAPTER 5. VISIBILITY OF SHAFAREVICH-TATE GROUPS

Table 5.6: Odd parts of Shafarevich-Tate groups at prime level

A  dim � _{an}  mod deg  B  dim mod deg
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 1  3 · 5^{2}
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

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