MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000–000 S 0025-5718(XX)0000-0

VISIBLE EVIDENCE FOR THE BIRCH AND

SWINNERTON-DYER CONJECTURE FOR MODULAR ABELIAN VARIETIES OF ANALYTIC RANK ZERO

(WITH AN APPENDIX BY J. CREMONA AND B. MAZUR)

AMOD AGASHE AND WILLIAM STEIN

Abstract. This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A_{f} that are optimal quotients of J_{0}(N) attached to newforms. We prove theorems about the ratio L(A_{f} , 1)/Ω_{A_{f}}, develop tools for computing with A_{f}, and gather data about certain arithmetic invariants of the nearly 20000 abelian varieties A_{f} of level ≤ 2333. Over half of these A_{f} have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of � (A_{f}). We find that there are at least 168 such that the Birch and Swinnerton-Dyer Conjecture implies that � (A_{f}) is divisible by an odd prime, and we prove for 39 of these that the odd part of the conjectural order of � (A_{f}) really divides #� (A_{f}) by constructing nontrivial elements of � (A_{f}) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.

Contents

1. Introduction 2 2. Background and Definitions 3 2.1. Modular Forms 3 2.2. Abelian Varieties Attached to Newforms 3 2.3. The Birch and Swinnerton-Dyer Conjecture 3 3. Explicit Approaches to Modular Abelian Varieties 4 3.1. Modular Symbols 5 3.2. Enumerating Newforms 5 3.3. The Modular Degree 6 3.4. Intersecting Complex Tori 6 3.5. Bounding the Torsion From Above 7 3.6. Bounding the Torsion From Below 9 3.7. Tamagawa Numbers 10 3.8. Visibility Theory 10 4. The Quotient L(A, 1)/Ω_{A} 12

Received by the editor March 8, 2004. 1991 Mathematics Subject Classification. Primary 11G40; Secondary 11F11, 11G10, 14K15,

14H25, 14H40.

Key words and phrases. Birch and Swinnerton-Dyer conjecture, modular abelian variety, visibility, Shafarevich-Tate groups.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

4.1. The Manin Constant 12 4.2. A Formula for L(A, 1)/Ω_{A} 12 4.3. The Denominator of L(A, 1)/Ω_{A} 14 5. Results and Conclusions 15 5.1. Example: Level 389 17 5.2. Invisible Elements of � (A) 17 5.3. The Part of � (A) That Must be a Perfect Square 17 6. Appendix by J. Cremona and B. Mazur:

“Explaining” Shafarevich-Tate via Mordell-Weil 23 References 30

1. Introduction

Let N be a positive integer, and f be a newform of weight 2 on Γ_{0}(N). A construction due to Shimura associates to f an abelian variety quotient A_{f} of J_{0}(N). We say that A_{f} has analytic rank zero if its L-function L(A_{f}, s) is nonzero at s = 1. In this paper we give evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A_{f} of arbitrary dimension. For such abelian varieties, the conjecture asserts that A_{f}(Q) is finite, and gives a formula for the order of the Shafarevich-Tate group � (A_{f}).

Kolyvagin and Logachev proved in [KL89, KL92] that if L(A_{f}, 1) ≠ 0, then A_{f}(Q) and � (A_{f}) are both finite. To the best of our knowledge, Birch and Swinnerton-Dyer’s formula for #� (A_{f}) has not been completely verified for a single abelian variety A_{f} of dimension greater than one. In [KL92, §1.6] Kolyvagin and Logachev remark that if one were able to compute the height of a certain Heegner point, their methods could be used to find an upper bound on #� (A_{f}), but we have not done this. Instead, in this paper we focus on computing nonzero subgroups of � (A_{f}) when the conjecture predicts that � (A_{f}) is nonzero.

Inspired by work of Cremona and Mazur (see [CM]), we had the idea to reverse their methods and prove, in some cases, that #� (A_{f}) is at least as big as predicted by the Birch and Swinnerton-Dyer conjecture. Instead of assuming that � (A_{f}) is as predicted by the conjecture and trying to understand whether or not it is visible in J_{0}(N), we prove a theorem (see [AS02]) that allows us to sometimes construct the odd part of � (A_{f}) without assuming any conjectures. After developing algorithms that allow us to compute the conjectural order of � (A_{f}) in most cases, we analyzed the 19608 abelian varieties A_{f} of level ≤ 2333, and constructed the tables of Section 5. This resulted in the first systematic experimental evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of dimension greater than 2 (see [FpS^{+}01] for dimension 2).

This paper is organized as follows. In Section 2 we review background about modular abelian varieties and state the Birch and Swinnerton-Dyer conjecture. Section 3 explains the basic facts about quotients A_{f} of J_{0}(N) that one needs to know in order to compute with them. In Section 4 we discuss a generalization of the Manin constant, derive a formula for the ratio L(A_{f}, 1)/Ω_{A_{f}}, and bound the denominator of this ratio, thus giving some theoretical evidence towards the Birch and Swinnerton-Dyer conjecture. Section 5 reports on our construction of a table of 168 rank 0 abelian varieties A_{f} of level ≤ 2333 such that the Birch and Swinnerton-Dyer conjecture predicts that #� (A_{f}) is divisible by an odd prime, and discusses what we computed to show that for 39 of the A_{f} there are at least as many elements of the odd part of #� (A_{f}) as predicted. The part of #� (A_{f}) that is coprime to the modular degree of A_{f} (which we define below) is a perfect square, and in the several cases where we could compute the odd part of the conjectured value of #� (A_{f}), we found the odd part to be a perfect square, which gives computational evidence for the conjecture. The appendix, written by Cremona and Mazur, fills in some gaps in the theoretical discussion in [CM].

Acknowledgment. It is a pleasure to thank Bryan Birch, Robert Coleman, Benedict Gross, Hednrik Lenstra, Dino Lorenzini, Loïc Merel, Bjorn Poonen, Ken Ribet, and John Tate for many helpful comments and discussions. Special thanks go to Barry Mazur for guiding our ideas on visibility and purchasing the second author a powerful computer, and to Allan Steel and David Kohel at Magma for their crucial computational support.

2. Background and Definitions

2.1. Modular Forms. Fix a positive integer N. The group

Γ_{0}(N) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(Z) : N | c\}

acts by linear fractional transformations on the extended complex upper halfplane 𝔥^{∗}. As a Riemann surface, X_{0}(N)(C) is the quotient Γ_{0}(N)\𝔥^{∗}. There is a standard model for X_{0}(N) over Q (see [Shi94, Ch. 6]), and the Jacobian J_{0}(N) of X_{0}(N) is an abelian variety over Q of dimension equal to the genus g of X_{0}(N), which is equipped with an action of the Hecke algebra T = Z[… T_{n} …]. The space S_{2}(Γ_{0}(N)) of cuspforms of weight 2 on Γ_{0}(N) is a module over T and S_{2}(Γ_{0}(N)) ≅ H^{0}(X_{0}(N), Ω_{X_{0}(N)}) as T-modules.

2.2. Abelian Varieties Attached to Newforms. A newform

f = ∑_{n≥1} a_{n}q^{n} ∈ S_{2}(Γ_{0}(N))

is an eigenvector for T that is normalized so that a_{1} = 1 and which lies in the orthogonal complement of the old subspace of S_{2}(Γ_{0}(N)). Let I_{f} denote the annihilator Ann_{T}(f) of f in T. Following Shimura [Shi73], attach to I_{f} the quotient

A_{f} = J_{0}(N)/I_{f}J_{0}(N),

which is an abelian variety over Q of dimension [Q(…, a_{n}, …) : Q], which is equipped with a faithful action of T/I_{f}. Moreover, A_{f} is an optimal quotient of J_{0}(N), in the sense that A_{f}^{∨} → J_{0}(N) is a closed immersion, or equivalently that the kernel of J_{0}(N) → A_{f} is connected (see [CS01, Prop. 3.3]).

Also, the complex torus A_{f}(C) fits into the exact sequence

H_{1}(X_{0}(N), Z) → Hom(S_{2}(Γ_{0}(N))[I_{f}], C) → A_{f}(C) → 0.

2.3. The Birch and Swinnerton-Dyer Conjecture. The conjecture of Birch and Swinnerton-Dyer makes sense for abelian varieties over fairly general global fields, but we only state a special case. This conjecture involves the L-function attached to A = A_{f}:

L(A, s) = ∏_{i=1}^{d} L(f^{(i)}, s) = ∏_{i=1}^{d} (∑_{n≥1} \frac{a_{n}^{(i)}}{n^{s}} ) ,

where f^{(i)} is the ith Galois conjugate of f and a_{n}^{(i)} is the ith Galois conjugate of a_{n}. It follows from work of Hecke that L(A, s) has an analytic continuation to the whole complex plane and satisfies a functional equation. Birch and Swinnerton-Dyer made the following conjecture, which relates the rank of A to the order of vanishing of L(A, s) at s = 1.

Conjecture 2.1 (Birch and Swinnerton-Dyer). The Mordell-Weil rank of A is equal to the order of vanishing of L(A, s) at s = 1, i.e.,

dim(A(Q) ⊗ Q) = ord_{s=1} L(A, s).

Birch and Swinnerton-Dyer also furnished a conjectural formula for the order of the Shafarevich-Tate group

� (A) := ker (H^{1}(Q, A) ⟶ ∏_{all places v} H^{1}(Q_{v}, A)) .

(They only made their conjecture for elliptic curves, but Tate [Tat66] reformulated it a functorial way which makes sense for abelian varieties. See also [Lan91, §III.5] for another formulation.) We now state their conjecture in the special case when L(A, 1) ≠ 0, where [KL89, KL92] implies that � (A) is finite. The conjecture involves the Tamagawa numbers c_{p} of A (see Section 3.7), and the canonical volume Ω_{A} of A(R) (see Section 4.2).

Conjecture 2.2 (Birch and Swinnerton-Dyer). Suppose L(A, 1) ≠ 0. Then

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

where A^{∨} is the abelian variety dual of A.

Remark 2.3. Since L(A, 1) ≠ 0, finiteness of � (A) and the existence of the Cassels-Tate pairing implies that #� (A) = #� (A^{∨}), so Conjecture 2.2 can also be viewed as a formula for #� (A^{∨}).

The algorithms outlined in this paper take advantage of the fact that A is attached to a newform in order to compute the conjectural order of � (A) away from certain bad primes.

3. Explicit Approaches to Modular Abelian Varieties

We use the algorithms of this section to enumerate the A_{f}, compute information about the invariants of A_{f} that appear in Conjecture 2.2, and to verify the hypothesis of Theorem 3.13 in order to construct nontrivial subgroups of � (A_{f}). The second author has implemented the algorithms discussed in this paper, and made many of them part of the Magma computer algebra system [BCP97].

In Section 3.1, we discuss modular symbols, which are the basic tool we use in many of the computations, and in Section 3.2 we discuss how we systematically enumerate modular abelian varieties. There is an analogue for A_{f} of the usual elliptic-curve modular degree, which we discuss in Section 3.3, and which we use to rule out the existence of visible elements of � (A_{f}) of a certain order. In Section 3.4 we describe how to compute the intersection of two abelian varieties, which will be needed to verify the hypothesis of Theorem 3.13. In Sections 3.5 and 3.6, we describe standard methods for bounding the torsion subgroup of an abelian variety above and below. Section 3.7 reviews an algorithm for computing the odd part of the Tamagawa number c_{p} when p || N, and discusses the Lenstra-Oort bound in the case when p^{2} | N.

Unless otherwise stated, f is a newform, I_{f} its annihilator, and A = A_{f} is the corresponding optimal quotient of J_{0}(N).

3.1. Modular Symbols. Modular symbols are crucial to many algorithms for computing with modular abelian varieties, because they can be used to construct a finite presentation for H_{1}(X_{0}(N), Z) in terms of paths between elements of P^{1}(Q) = Q ∪ {∞}. They were introduced by Birch [Bir71] and studied by Manin, Mazur, Merel, Cremona, and others.

Let 𝔐_{2} be the free abelian group with basis the set of all symbols {α, β}, with α, β ∈ P^{1}(Q), modulo the three-term relations

{α, β} + {β, γ} + {γ, α} = 0,

and modulo any torsion. The group GL_{2}(Q) acts on the left on 𝔐_{2} by

g{α, β} = {g(α), g(β)},

where g acts on α and β by a linear fractional transformation. The space 𝔐_{2}(Γ_{0}(N)) of modular symbols for Γ_{0}(N) is the quotient of 𝔐_{2} by the subgroup generated by all elements of the form x−g(x), for x ∈ 𝔐_{2} and g in Γ_{0}(N), modulo any torsion. A modular symbol for Γ_{0}(N) is an element of this space, and we frequently denote the equivalence class that defines a modular symbol by giving a representative element.

Let 𝔅_{2}(Γ_{0}(N)) be the free abelian group with basis the finite set Γ_{0}(N)\P^{1}(Q). The boundary map δ : 𝔐_{2}(Γ_{0}(N)) → 𝔅_{2}(Γ_{0}(N)) sends {α, β} to [β] − [α], where [β] denotes the basis element of 𝔅_{2}(Γ_{0}(N)) corresponding to β ∈ P^{1}(Q). The cuspidal modular symbols are the kernel 𝔖_{2}(Γ_{0}(N)) of δ, and the integral homology H_{1}(X_{0}(N), Z) is canonically isomorphic to 𝔖_{2}(Γ_{0}(N)).

Cremona’s book [Cre97, §2.2] contains a concrete description of how to compute 𝔐_{2}(Γ_{0}(N)) ⊗ Q using Manin symbols, which are a finite set of generators for 𝔐_{2}(Γ_{0}(N)). In general, the easiest way we have found to compute 𝔐_{2}(Γ_{0}(N)) is to compute 𝔐_{2}(Γ_{0}(N)) ⊗ Q, then compute the Z-submodule of 𝔐_{2}(Γ_{0}(N)) ⊗ Q generated by the Manin symbols.

3.2. Enumerating Newforms. Since X_{0}(N) is defined over Q it is defined over R, so complex conjugation acts on X_{0}(N)(C) hence on the homology H_{1}(X_{0}(N), Z). In terms of modular symbols, complex conjugation acts by sending {α, β} to {−α, −β}. Let H_{1}(X_{0}(N), Z)^{+} denote the +1-eigenspace for the action of the involution induced by complex conjugation, which we can compute using modular symbols. We list all newforms of a given level N by decomposing the new subspace of H_{1}(X_{0}(N), Q)^{+} under the action of the the Hecke operators and listing the corresponding systems of Hecke eigenvalues (see [Ste02a]). First we compute the characteristic polynomial of T_{2}, and use it to break up the new space. We apply this process recursively with T_{3}, T_{5}, … until either we have exceeded the bound coming from [Stu87] (see [LS02]), or we have found a Hecke operator T_{n} whose characteristic polynomial is irreducible.

We order the newforms in a way that extends the ordering in [Cre97]: First sort by dimension, with smallest dimension first; within each dimension, sort in binary by the signs of the Atkin-Lehner involutions, e.g., + + +, + + −, + − +, + − −, − + +, etc. When two forms have the same Atkin-Lehner sign sequence, order by | Tr(a_{p})| with ties broken by taking the positive trace first. We denote a Galoisconjugacy class of newforms by a bold symbol such as 389E, which consists of a level and isogeny class, where A denotes the first class, B the second, E the fifth, BB the 28th, etc. As discussed in [Cre97, pg. 5], for certain small levels the above ordering, when restricted to elliptic curves, does not agree with the ordering used in the tables of [Cre97]. For example, our 446B is Cremona’s 446D.

3.3. The Modular Degree. Since A_{f} is an optimal quotient, the dual map A_{f}^{∨} → J_{0}(N) is injective and the composite θ_{f} : A_{f}^{∨} → A_{f} has finite degree. The map θ_{f} is a polarization, so deg(θ_{f}) is a perfect square (see Lemma 3.14). The modular degree of A_{f} is the square root of the degree of θ_{f}:

moddeg(A_{f}) = \sqrt{deg(θ_{f})}.

When dim A_{f} = 1, moddeg(A_{f}) is the usual modular degree, i.e., the degree of X_{0}(N) → A_{f}.

If M is an abelian group, let M^{∗} = Hom_{Z}(M, Z). The Hecke algebra acts in a natural way on H_{1}(X_{0}(N), Z) and H_{1}(X_{0}(N), Z)^{∗}, and we have a natural restriction map

r_{f} : H_{1}(X_{0}(N), Z)^{∗}[I_{f}] → (H_{1}(X_{0}(N), Z)[I_{f}])^{∗}.

The following proposition leads to an algorithm for computing the modular degree.

Proposition 3.1. coker(r_{f}) ≅ ker(θ_{f}), so moddeg(A_{f}) = \sqrt{# coker(r_{f})}.

The proposition is proved in [KS00]. The proof makes use of the Abel-Jacobi theorem, which realizes the Jacobian J_{0}(N)(C) as a complex torus:

0 → H_{1}(X_{0}(N), Z) → Hom(S_{2}(Γ_{0}(N)), C) → J_{0}(N)(C) → 0,

where H_{1}(X_{0}(N), Z) is embedded as a lattice of full rank in the complex vector space Hom(S_{2}(Γ_{0}(N)), C) using the integration pairing, and this description of J_{0}(N)(C) is compatible with the action of Hecke operators.

3.4. Intersecting Complex Tori. Let V be a finite dimensional complex vector space and let Λ be a lattice in V, so that T = V/Λ is a complex torus. Suppose that V_{A} and V_{B} are subspaces of V such that Λ_{A} = V_{A} ∩ Λ and Λ_{B} = V_{B} ∩ Λ are lattices in V_{A} and V_{B}, respectively.

Proposition 3.2. If A ∩ B is finite, then there is an isomorphism

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

Proof. Extend the exact sequence

0 → A ∩ B → A ⊕ B ⟶^{(x,y)↦x−y} T

to the following diagram:

\xymatrix{ & Λ_{A} ⊕ Λ_{B} \ar[d] \ar[r] & Λ \ar[r] \ar[d] & Λ/(Λ_{A} + Λ_{B}) \ar[d] \\ 0 \ar[r] \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] & T \ar[r] & T/(A + B).}

The middle row is exact because A ∩ B is finite so V_{A} ∩ V_{B} = 0.

Using the snake lemma, which connects the kernel A ∩ B of A ⊕ B → T to the cokernel of Λ_{A} ⊕ Λ_{B} → Λ, we obtain an exact sequence

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

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

For abelian subvarieties of J_{0}(N) attached to newforms, we use the proposition above as follows. The complex vector space V = Hom(S_{2}(Γ_{0}(N)), C) is the tangent space of J_{0}(N)(C) at the identity. Setting Λ = H_{1}(X_{0}(N), Z) and considering Λ as a lattice in V via the integration pairing, we have J_{0}(N)(C) ≅ V/Λ. Suppose f and g are non-conjugate newforms, and let I_{f} and I_{g} be their annihilators in the Hecke algebra T, and let A = A_{f}^{∨} and B = A_{g}^{∨}. Then V_{A} = V [I_{f}] and V_{B} = V [I_{g}] are the tangent spaces to A and B at the identity, respectively. The above proposition shows that the group A ∩ B is canonically isomorphic to (Λ/(Λ_{A} + Λ_{B}))_{tor}. Here Λ_{A} = Λ[I_{f}] and Λ_{B} = Λ[I_{g}], because A_{f} and A_{g} are optimal quotients.

The following formula for the intersection of n subtori is obtained in a similar way to that of Proposition 3.2.

Proposition 3.3. For i = 1, …, n, with n ≥ 2, let A_{i} = V_{i}/Λ_{i} be a subtorus of T = V/Λ, and assume that each pairwise intersection A_{i} ∩ A_{j} is finite. Define a linear map

f : V_{1} × ⋯ × V_{n} ⟶ V^{⊕(n−1)}.

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

A_{1} ∩ ⋯ ∩ A_{n} ≅ (\frac{Λ^{⊕(n−1)}}{f(Λ_{1} ⊕ ⋯ ⊕ Λ_{n})})_{tor} .

3.5. Bounding the Torsion From Above. In this section we recall the standard upper bound on the order of #A(Q)_{tor}, and illustrate its usefulness.

Let f = ∑ a_{n}q^{n} be a weight 2 newform on Γ_{1}(N) with Nebentypus character ε : (Z/NZ)^{∗} → C^{∗} (recall that f is a form on Γ_{0}(N) if and only if ε = 1), and let A = A_{f} be the corresponding optimal quotient of J_{1}(N), as in [Shi73]. Shimura proved in [Shi94, Ch. 7] that the local Euler factor of A_{f} at p is

L_{p}(A_{f}, s) = ∏_{σ:K_{f} ↪\overline{Q}} \frac{1}{1 − σ(a_{p})p^{−s} + σ(ε(p))p^{1−2s}}

by showing that the characteristic polynomial F_{p} of Frobenius on any ℓ-adic Tate module of A_{F_{p}} (for ℓ ∤ pN) is

F_{p}(X) = ∏_{σ:K_{f} ↪\overline{Q}} X^{2} − σ(a_{p})X + σ(ε(p))p,

where K_{f} = Q(…, a_{n}, …). Let Q(ε) be the field generated by the values of ε (note that Q(ε) ⊂ K_{f}), and for any p ∤ N let G_{p}(X) ∈ Q(ε)[X] be the characteristic polynomial of left multiplication by a_{p} on the Q(ε)-vector space K_{f}, which is a polynomial of degree d′ = [K_{f} : Q(ε)]. Then

F_{p}(X) = Norm_{Q(ε)/Q} (X^{d′} · G_{p} (X + \frac{ε(p)p}{X})) ,

so

#A_{F_{p}}(F_{p}) = deg(1 − Frob_{p}) = | det(1 − Frob_{p})|
= |F_{p}(1)| = | Norm_{Q(ε)/Q}(G_{p}(1 + ε(p)p))|.

If p ∤ N is odd, standard facts about formal groups imply that the reduction map A(Q)_{tor} → A_{F_{p}}(F_{p}) is injective, so

#A(Q)_{tor} | gcd {#A_{F_{p}}(F_{p}) : primes p ∤ 2N} .

Likewise, since A^{∨} is isogenous to A, the same bound applies to A^{∨}(Q)_{tor}, since A^{∨} and A have the same L-series.

The upper bound is the same for every abelian variety isogenous to A, so it is not surprising that it is not sharp in general. For example, let E (resp., F) be the elliptic curve labeled 30A1 (resp. 30A2) in Cremona’s tables [Cre97]. Then E and F are isogenous, E(Q) ≈ Z/6Z, and F (Q) ≈ Z/12Z, so

12 | gcd {#E_{F_{p}}(F_{p}) : primes p ∤ 2N} .

(Incidentally, since #E(F_{5}) = 12, the gcd is 12.) For answers to some related deep questions about this gcd, see [Kat81].

Example 3.4. Let

f = q + (−1 + \sqrt{2})q^{2} + q^{3} + (−2\sqrt{2} + 1)q^{4} − 2\sqrt{2}q^{5} + ⋯ ∈ S_{2}(Γ_{0}(39))

be the form 39B. Then G_{5}(X) = X^{2} − 8, so

#A_{f}(Q)_{tor} | G_{5}(1 + 5) = 28.

We find in [FpS^{+}01] that A_{f} is isogenous to the Jacobian J of y^{2} + (x^{3} + 1)y = −5x^{4} − 2x^{3} + 16x^{2} − 12x + 2 and that #J(Q) = 28. However A_{f} is not isomorphic to J since, as reported in Table 2 of [FpS^{+}01], the Tamagawa numbers of J are c_{3} = 28, c_{13} = 1, whereas the methods of Section 3.7 below show that the Tamagawa numbers of A_{f} are c_{3} = 14, c_{13} = 2. The authors do not know for sure whether #A_{f}(Q) = 28, but in Example 3.6 below we show that 14 | #A_{f}(Q). (Also, using the computational techniques of this paper one sees that the Birch and Swinnerton-Dyer conjecture implies that #A_{f}(Q) = 28.)

Example 3.5. Let

f = q + (−ζ_{6} − 1)q^{2} + (2ζ_{6} − 2)q^{3} + ζ_{6}q^{4} + (−2ζ_{6} + 1)q^{5} + ⋯

be one of the two Galois-conjugate newforms in S_{2}(Γ_{1}(13)). This form has character ε : (Z/13Z)^{∗} → C^{∗} of order 6, and A_{f} = J_{1}(13). We have G_{3}(X) = X − 2ζ_{6} + 2 and ε(3) = −ζ_{6}, so

#J_{1}(13)(Q)_{tor} | #J_{1}(13)(F_{3}) = | Norm(G_{3}(1 − 3ζ_{6}))|
= | Norm(−5ζ_{6} + 3)| = 19.

In fact Ogg proved that J_{1}(13)(Q)_{tor} ≈ Z/19Z (see [Ogg73] and [MT74]).

3.6. Bounding the Torsion From Below. A cusp α ∈ Γ_{0}(N)\P^{1}(Q) ⊂ X_{0}(N) defines a point (α) − (∞) ∈ J_{0}(N)(\overline{Q})_{tor}. The rational cuspidal subgroup C of J_{0}(N)(Q)_{tor} generated by Q-rational cusps is of interest because the order of the image of C in A_{f}(Q)_{tor} provides a lower bound on #A_{f}(Q)_{tor}. Stevens [Ste82, §1.3] computed the action of Gal(\overline{Q}/Q) on the subgroup of J_{0}(N)(\overline{Q}) generated by all cusps (and for other congruence subgroups besides Γ_{0}(N)). He found that Gal(\overline{Q}/Q) acts on the cusps through Gal(Q(ζ_{N})/Q) ≅ (Z/NZ)^{∗}, and that d ∈ (Z/NZ)^{∗} acts by x/y ↦ x/(d′y), where dd′ ≡ 1 (mod N). Thus, e.g., (0) − (∞) ∈ J_{0}(N)(Q)_{tor}, and if N is square-free then all cusps are rational.

To compute the image of C in A_{f}(Q)_{tor}, first make a list of inequivalent cusps using, e.g., the method described in [Cre97, §2.2, pg. 17]. Keep only the Q-rational cusps, which can be determined using the result of Stevens above and [Cre97, Prop. 2.2.3] (when N is squarefree all cusps are rational). Next compute the subgroup C of 𝔐_{2}(Γ_{0}(N)) generated by modular symbols {α, ∞}, where α is a Q-rational cusp. The image of C in A_{f}(Q)_{tor} is isomorphic to the image of C in

P = Φ_{f}(𝔐_{2}(Γ_{0}(N)))/Φ_{f}(𝔖_{2}(Γ_{0}(N))),

where Φ_{f} : 𝔐_{2}(Γ_{0}(N)) → Hom(S_{2}(Γ_{0}(N))[I_{f}], C) is defined by the integration pairing. To keep everything rational, note that P can be computed using any map with the same kernel as Φ_{f}; for example, such a map can be constructed by finding a basis for Hom(𝔐_{2}(Γ_{0}(N)), Q)[I_{f}] as described at the end of Section 4.2).

Example 3.6. Let the notation be as in Example 3.4. The cusps on X_{0}(39) are represented by 0, ∞, −1/9, and −4/13, and since N = 39 is squarefree, these cusps are all rational. Using Magma we find that the image of C in A_{f}(Q)_{tor} is isomorphic to Z/14Z. Thus A_{f}(Q)_{tor} is isomorphic to one of Z/14Z, Z/28Z, or Z/14Z × Z/2Z, but we do not know which.

Example 3.7. Let

f = q + \frac{1 + \sqrt{5}}{2}q^{2} + \frac{1 − \sqrt{5}}{2}q^{3} + \frac{5 + \sqrt{5}}{2}q^{4} + ⋯ ∈ S_{2}(Γ_{0}(175))

be the form 175D. The cusps of X_{0}(175) are represented by

0, ∞, \frac{1}{25}, \frac{1}{28}, \frac{1}{30}, \frac{1}{35}, \frac{1}{45}, \frac{1}{60}, \frac{1}{65}, \frac{1}{70}, \frac{1}{105}, \frac{1}{140}.

The Q-rational cusps in this list are 0, ∞, \frac{1}{25}, \frac{1}{28}, and these generate a subgroup of A_{f}(Q)_{tor} of order 2. (Incidentally, the group generated by all cusps, both rational and not, is isomorphic to Z/32Z.) Using a_{p} for p ≤ 17 and the method of the previous section, we see that #A_{f}(Q)_{tor} | 4. The authors do not know if the cardinality is 2 or 4.

Example 3.8. The form 209C is

f = q + αq^{2} + (1/2α^{4} − α^{3} − 5/2α^{2} + 4α + 1)q^{3} + (α^{2} − 2)q^{4} + ⋯,

where α^{5} − 2α^{4} − 6α^{3} + 10α^{2} + 5α − 4 = 0. As above, we find that #A_{f}(Q)_{tor} divides 5. The image of the (rational) cuspidal subgroup in A_{f}(\overline{Q})_{tor} is isomorphic to Z/5Z, so A_{f}(Q)_{tor} ≈ Z/5Z.

3.7. Tamagawa Numbers. Suppose p | N and let Φ_{A,p} denote the component group of A at p, which is defined by the following exact sequence:

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

where A_{F_{p}} is the closed fiber of the Néron model of A over Z_{p} and A_{F_{p}}^{0} is the component of A_{F_{p}} that contains the identity.

Definition 3.9. The Tamagawa number of A at p is

c_{p} = c_{A,p} = #Φ_{A,p}(F_{p}).

When p || N, the second author found a computable formula for #Φ_{A,p}(\overline{F}_{p}) and (sometimes only up to a power of 2) for #Φ_{A,p}(F_{p}). There is a discussion about how to compute this number in [KS00] and [CS01] contains a proof of the formula. Note also that in this case the Tamagawa number of A at p is the same as the Tamagawa number of A^{∨} at p.

When p^{2} | N the authors do not know an algorithm to compute c_{p}. However, in this case Lenstra and Oort (see [LO85]) proved that

∑_{ℓ≠p}(ℓ − 1) ord_{ℓ}(#Φ_{A,p}(\overline{F}_{p})) ≤ 2 dim(A_{f}),

so if ℓ | #Φ_{A,p}(\overline{F}_{p}) then ℓ ≤ 2 · dim(A_{f}) + 1 or ℓ = p. (Here ord_{ℓ}(x) denotes the exponent of the largest power of ℓ that divides x.)

Example 3.10. Let f be 39B as in Example 3.4. Running the algorithm of [KS00], we find that c_{3} = 14 and c_{13} = 2.

Example 3.11. Let f be 175D as in Example 3.7. Running the algorithm of [KS00], we find that c_{7} = 1, and the Lenstra-Oort bound implies that the only possible prime divisors of c_{5} are 2, 3, and 5.

3.8. Visibility Theory. We briefly recall visibility theory, which we will use to construct elements of Shafarevich-Tate groups. Section 6 contains another approach to the results reported in this section, but in the special case of elliptic curves.

Definition 3.12. Let ι : A ↪ J be an embedding of abelian varities over Q. The visible subgroup of � (A) with respect to the embedding ι is

Vis_{J}(� (A)) = Ker(� (A) → � (J)).

The following is a special case of Theorem 3.1 of [AS02].

Theorem 3.13. Let A and B be abelian subvarieties of an abelian variety J over Q such that A(\overline{Q}) ∩ B(\overline{Q}) is finite. Let N be an integer divisible by the residue characteristics of primes of bad reduction for J (e.g., the conductor of J). Suppose p is a prime such that

p ∤ 2 · N · #(J/B)(Q)_{tor} · #B(Q)_{tor} · ∏_{ℓ} c_{A,ℓ} · c_{B,ℓ},

where c_{A,ℓ} = #Φ_{A,ℓ}(F_{ℓ}) (resp., c_{B,ℓ}) is the Tamagawa number of A (resp., B) at ℓ. Suppose furthermore that B[p](\overline{Q}) ⊂ A(\overline{Q}) as subgroups of J(\overline{Q}). Then there is a natural map

φ : B(Q)/pB(Q) → Vis_{J}(� (A))

such that dim_{F_{p}} ker(φ) ≤ dim_{Q} A(Q) ⊗ Q.

We return to the situation where A = A_{f} is an optimal quotient of J_{0}(N) attached to a newform. In Proposition 3.15 below we show that Vis_{J_{0}(N)}(� (A^{∨})) is annihilated by multiplication by moddeg(A) (see also [CM, p.19]). We first state a lemma; the outline of the proof was indicated to us by B. Poonen.

Lemma 3.14. Let A be an abelian variety over k, where k is a field, and let λ : A → A^{∨} be a polarization. Suppose either that k has characteristic 0 or that its characteristic does not divide the degree of λ. Then there is a finite abelian group H such that ker(λ) ≈ H × H as groups.

Proof. We work in the setting of Section 16 of [Mil86], using the notation used there. Consider the pairing

e^{λ} : Ker(λ) × Ker(λ) → μ_{m} ⊆ \overline{k}^{∗},

as in [Mil86, p. 135], where m is an integer that kills Ker(λ). We will show that this pairing is nondegenerate.

Suppose a ∈ Ker(λ) is such that e^{λ}(a, a′) = 1 for all a′ ∈ Ker(λ). Let a′′ ∈ A^{∨}[m]. There exists an isogeny λ′ : A^{∨} → A such that λ′ ∘ λ is multiplication by m on A and λ ∘ λ′ is multiplication by m on A^{∨} (to construct λ′, note that λ′ is the quotient map A^{∨} → A^{∨}/λ(A[m])). Pick an element b ∈ A(\overline{k}) such that λb = a′′. Then mb = λ′(λb) = λ′(a′′). So \overline{e}_{m}(a, a′′) = \overline{e}_{m}(a, λb) = e^{λ}(a, λ′a′′) = 0 (note that λ(λ′a′′) = ma′′ = 0, so that λ′a′′ ∈ Ker(λ)). This is true for all a′′ ∈ A^{∨}[m], so the non-degeneracy of \overline{e}_{m} ([Mil86, p. 131]) implies that a = 0.

Similarly, suppose a′ ∈ Ker(λ) is such that e^{λ}(a, a′) = 1 for all a ∈ Ker(λ). Since e^{λ} is skew-symmetric ([Mil86, p. 135]), this implies that e^{λ}(a′, a) = 1 for all a ∈ Ker(λ). Then by the previous paragraph, a′ = 0. This finishes the proof of non-degeneracy.

As mentioned before, the pairing e^{λ} is skew-symmetric. It is alternating because it extends to pairings on Tate modules (denoted by e_{ℓ}^{λ} in [Mil86, p. 132]), and the latter take values in a torsion-free group, so there is no distinction between skew-symmetric and alternating.

Now the lemma follows from the fact that if G is a finite abelian group with an alternating nondegenerate pairing, then there is a finite abelian group H such that G ≈ H × H as groups (e.g., see [Del01, Prop. 2]). □

Proposition 3.15. Let m_{A} = moddeg(A). We have

Vis_{J_{0}(N)}(� (A^{∨})) ⊂ � (A^{∨})[m_{A}].

Proof. The polarization θ_{f} (from Section 3.3) is the composite map A^{∨} → J_{0}(N) → A. Let e_{A} be the exponent of the finite group ker(θ_{f}). By Lemma 3.14, multiplication by m_{A} kills ker(θ_{f}), so e_{A} | m_{A}. Also θ_{f} factors through multiplication by e_{A}, so there is a map θ′_{f} : A → A^{∨} such that θ′_{f} ∘ θ_{f} is multiplication by e_{A}. If ϕ is a map of abelian varieties (over Q), let ϕ_{∗} denote the corresponding map on Shafarevich-Tate groups. Since Vis_{J_{0}(N)}(� (A^{∨})) is contained in ker((θ_{f})_{∗}), it is also contained in

ker((δ′ ∘ δ)_{∗}) = � (A^{∨})[e_{A}] ⊂ � (A^{∨})[m_{A}].
□

Since � (A^{∨})[n] is finite for any n, we obtain the following corollary.

Corollary 3.16. Vis_{J_{0}(N)}(� (A^{∨})) is finite.

4. The Quotient L(A, 1)/Ω_{A}

Fix a newform f ∈ S_{2}(Γ_{0}(N)), let I_{f} be the annihilator of f in T, and A = A_{f} = J_{0}(N)/I_{f}J_{0}(N) the corresponding optimal quotient. Suppose for the rest of this section that L(A, 1) ≠ 0.

4.1. The Manin Constant. When trying to compute the conjectural order of � (A), we try to compute the quotient L(A, 1)/Ω_{A}, but find that it is easier to compute c_{A} · L(A, 1)/Ω_{A} where c_{A} is the Manin constant of A, which is defined as follows:

Definition 4.1 (Manin constant). The Manin constant of A is

c_{A} = # (\frac{S_{2}(Γ_{0}(N), Z)[I_{f}]}{H^{0}(A, Ω_{A/Z})}) ∈ Z,

where we consider H^{0}(A, Ω_{A/Z}) as a submodule of S_{2}(Γ_{0}(N), Q) using

H^{0}(A, Ω_{A/Z}) → H^{0}(J , Ω_{J /Z})[I_{f}] → H^{0}(J, Ω_{J/Q})[I_{f}] → S_{2}(Γ_{0}(N), Q)[I_{f}],

where A and J are the Néron models of A and J, respectively. (See [AS04] for a discussion of why the image of H^{0}(A, Ω_{A/Z}) is contained in S_{2}(Γ_{0}(N), Z).)

Theorem 4.2. If ℓ | c_{A} is a prime then ℓ^{2} | 4N.

Proof. Mazur proved this when dim A = 1 in [Maz78, §4], and we generalized his proof in [AS04]. □

When dim A = 1, Edixhoven [Edi91] obtained strong results towards the folklore conjecture that c_{A} = 1, and when A has arbitrary dimension the authors have made the following conjecture (see [AS04] for evidence):

Conjecture 4.3. c_{A} = 1.

4.2. A Formula for L(A, 1)/Ω_{A}. If L and M are lattices in a real vector space V, then the lattice index [L : M] is the absolute value of the determinant of a linear transformation of V taking L onto M. The lattice index satisfies the usual properties suggested by the notation, e.g., [L : M] · [M : N] = [L : N].

The real volume Ω_{A} is defined as follows. If L^{∗} is a lattice in the cotangent space

T^{∗} = H^{0}(A_{R}, Ω_{A_{R}}) = S_{2}(Γ_{0}(N), R)[I_{f}]

of A_{R}, then L^{∗} determines a lattice L = Hom(L^{∗}, Z) in the tangent space T = Hom(T^{∗}, R), and hence a measure on T by declaring that the quotient T/L has measure 1. Let A(R)^{0} denote the identity component of A(R). Then A(R)^{0} inherits a measure by virtue of being viewed as T/H_{1}(A(R)^{0}, Z), and we have

μ_{L}(A(R)^{0}) = [L : H_{1}(A(R)^{0}, Z)].

We also set

μ_{L}(A(R)) = μ_{L}(A(R)^{0}) · c_{∞},

where c_{∞} = #(A(R)/A(R)^{0}). Let A be the Néron model of A (see [BLR90]). The Néron differentials H^{0}(A, Ω_{A/Z}) define a lattice Λ^{∗} in T^{∗}, and we define Ω_{A} = μ_{Λ}(A(R)).

Lemma 4.4. H_{1}(A(R)^{0}, Z) ≅ H_{1}(A(C), Z)^{+}.

Proof. This lemma is well known, but we give a proof for the reader’s convenience (which was suggested by H. Lenstra and B. Poonen). We have the commutative diagram

\xymatrix{0 \ar[r] & H_{1}(A(R)^{0}, Z) \ar[r] \ar[d]^{ψ} & H_{1}(A(R)^{0}, R) \ar[r] \ar[d]^{≅} & A(R)^{0} \ar[r] \ar@{^{(}->}[d]^{i} & 0 \\ 0 \ar[r] & H_{1}(A(C), Z)^{+} \ar[r] & H_{1}(A(C), R)^{+} \ar[r]^{π} & A(C)^{+}}

where the upper horizontal sequences is exact (we view the real torus A(R)^{0} as the quotient of the tangent space at the identity by the first integral homology), and the lower horizontal sequence is exact because it is the beginning of the long exact sequence of Gal(R/C)-cohomology that arises from

0 → H_{1}(A(C), Z) → H_{1}(A(C), R) → A(C) → 0.

The middle vertical map is an isomorphism because if it were not then its kernel would be an uncountable set that maps to 0 in A(R)^{0}. The snake lemma then yields an exact sequence

0 → ker(ψ) → 0 → 0 → coker(ψ) → 0,

which implies that ψ is an isomorphism. □

Let

Φ : H_{1}(X_{0}(N), Q) → Hom(S_{2}(Γ_{0}(N))[I_{f}], C)

be the map induced by integration, scaled so that

Φ({0, ∞})(f) = L(f, 1)

(that {0, ∞} ∈ H_{1}(X_{0}(N), Q) is the Manin-Drinfeld theorem, and that ∫_{0}^{∞} f is a multiple of L(f, 1) follows from the definition of L(f, s) as a Mellin transform).

Theorem 4.5. Recall that A is an abelian variety attached to a newform f ∈ S_{2}(Γ_{0}(N)), that c_{∞} is the number of connected components of A(R), that c_{A} is the Manin constant of A, that Ω_{A} is the Néron canonical volume of A(R), and that Φ is the period mapping on homology induced by integrating homology classes on X_{0}(N) against the C-vector space spanned by the Gal(\overline{Q}/Q)-conjugates of f. Then we have the following equation:

c_{∞} · c_{A} · \frac{L(A, 1)}{Ω_{A}} = [Φ(H_{1}(X_{0}(N), Z))^{+} : Φ(T\{0, ∞\})] ∈ Q,

where the lattice index on the right hand side should be interpreted as 0 if Φ(T{0, ∞}) has rank less than the dimension of A.

Proof. It is easier to compute with Λ̃^{∗} = S_{2}(Γ_{0}(N), Z)[I_{f}] than with Λ^{∗}, so let Ω̃_{A} = μΛ̃(A(R)). Note that Ω̃_{A} · c_{A} = Ω_{A}, where c_{A} is the Manin constant. By Lemma 4.4 and Section 2.2,

Ω̃_{A} = c_{∞} · [Λ̃ : H_{1}(A(R)^{0}, Z)]
= c_{∞} · [Hom(S_{2}(Γ_{0}(N), Z)[I_{f}], Z) : Φ(H_{1}(X_{0}(N), Z))^{+}].

For any ring R the pairing

T_{R} × S_{2}(Γ_{0}(N), R) → R

given by ⟨T_{n}, f⟩ = a_{1}(T_{n}f) is perfect, so (T/I_{f}) ⊗ R ≅ Hom(S_{2}(Γ_{0}(N), R)[I_{f}], R). Using this pairing, we may view Φ as a map

Φ : H_{1}(X_{0}(N), Q) → (T/I_{f}) ⊗ C,

so that

Ω̃_{A} = c_{∞} · [T/I_{f} : Φ(H_{1}(X_{0}(N), Z))^{+}].

Note that (T/I_{f}) ⊗ C is isomorphic as a ring to a product of copies of C, with one copy corresponding to each Galois conjugate f^{(i)} of f. Let π_{i} ∈ (T/I_{f}) ⊗ C be the projector onto the subspace of (T/I_{f}) ⊗ C corresponding to f^{(i)}. Then Φ({0, ∞}) · π_{i} = L(f^{(i)}, 1) · π_{i}. Since the π_{i} form a basis for the complex vector space (T/I_{f}) ⊗ C, we see that

det(Φ({0, ∞})) = ∏_{i} L(f^{(i)}, 1) = L(A, 1).

Letting H = H_{1}(X_{0}(N), Z), we have

[Φ(H)^{+} : Φ(T{0, ∞})] = [Φ(H)^{+} : (T/I_{f}) · Φ({0, ∞})]

= [Φ(H)^{+} : T/I_{f}] · [T/I_{f} : T/I_{f} · Φ({0, ∞})]

= \frac{c_{∞}}{Ω̃_{A}} · det(Φ(\{0, ∞\}))

= \frac{c_{∞}c_{A}}{Ω_{A}} · L(A, 1),

which proves the theorem. □

Theorem 4.5 was inspired by the case when A is an elliptic curve (see [Cre97, §II.2.8]) or the winding quotient of J_{0}(p) (see [Aga99]), and it generalizes to forms of weight > 2 (see [Ste00]).

Theorem 4.5 is true with Φ replaced by any linear map with the same kernel as Φ. One way to find such a linear map with image in a Q-vector space is to compute a basis φ_{1}, … φ_{d} for Hom(H_{1}(X_{0}(N), Q), Q)[I_{f}] and let Φ = φ_{1} × ⋯ × φ_{d}. Also, since H_{1}(X_{0}(N), Z)^{+} and T{0, ∞} are contained in H_{1}(X_{0}(N), Q)^{+}, Theorem 4.5 implies that L(A, 1)/Ω_{A} ∈ Q, a fact well known to the experts (see [Gro94, Prop. 2.7] for the statement, but without proof).

4.3. The Denominator of L(A, 1)/Ω_{A}. In this section, we prove a result about the denominator of the rational number L(A, 1)/Ω_{A} and compare it to what is predicted by the Birch and Swinnerton-Dyer conjecture.

Proposition 4.6. Let z be the point in J_{0}(N)(Q) defined by the degree 0 divisor (0) − (∞) on X_{0}(N), and let n = n_{f} be the order of the image of z in A(Q). Then the denominator of c_{∞} · c_{A} · L(A, 1)/Ω_{A} divides n.

Proof. Let x be the image of z in A(Q), and let I = Ann_{T}(x) be the ideal of elements of T that annihilate x. Since f is a newform, the Hecke operators T_{p}, for p | N, act as 0 or ±1 on A(Q) (see, e.g., [DI95, §6]). If p ∤ N, then a standard calculation (see, e.g., [Cre97, §2.8]) shows that T_{p}(x) = (p + 1)x.

Let C be the cyclic subgroup of A(Q) of order n generated by x. Consider the map T → C given by T_{p} ↦ T_{p}(x). The kernel of this map is I, and the map is surjective because its image is an additive group that contains x, and C is the smallest such group. Thus the map induces an isomorphism T/I ⟶^{≅} C. □

Conjecture 2.2 predicts that

#A(Q) · #A^{∨}(Q) · \frac{L(A, 1)}{Ω_{A}} = #� (A) · ∏ c_{p} ∈ Z,

and since n | #A(Q), Proposition 4.6 implies that

c_{∞} · c_{A} · #A(Q) · \frac{L(A, 1)}{Ω_{A}} ∈ Z.

Since c_{∞} is a power of 2, and c_{A} is conjecturally 1 (if N is prime, then by Theorem 4.2 it is a power of 2), Proposition 4.6 provides theoretical evidence for Conjecture 2.2, and also reflects a surprising amount of cancellation between ∏ c_{p} and #A^{∨}(Q).

5. Results and Conclusions

We computed all 19608 abelian varieties A = A_{f} attached to newforms of level N ≤ 2333. Interesting data about some of these abelian varieties is summarized in Tables 1–4, which use the notation described in this section.

Suppose that A is one of the 10360 of these for which L(A, 1) ≠ 0, so Conjecture 2.2 asserts that � (A) has order

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

(See Section 4.2 for the definition of Ω̃_{A} and c_{A}.)

For any rational number x, let x^{odd} be the odd part of x. If a and b are rational numbers with a ≠ 0, we say that a | b if b/a is an integer.

Define integers S_{l} and S_{u} such that

S_{l} | numer(#� _{?}^{odd}) | S_{u}

as follows:

S_{u} The upper bound S_{u} is the odd part of the numerator of

\frac{L(A, 1)}{Ω̃_{A}} · \frac{T^{2}}{∏_{p||N} c_{p}} ,

where T is the upper bound on #A(Q) and #A^{∨}(Q) computed using Section 3.5 using a_{p} for p ≤ 17. Since the Manin constant and the Tamagawa numbers are integers, S_{u} is an upper bound on the odd part of #� _{?}.

S_{l} The lower bound S_{l} is defined as follows: Let S_{l,1} be the odd part of the rational number

\frac{L(A, 1)}{Ω̃_{A}} · \frac{#C · #D}{∏_{p||N} c_{p}} ,

where C ⊂ A(Q)_{tor} and D is the part of C coprime to the modular degree of A. Usually C is the group generated by the image of (0)−(∞), and in all cases it contains this subgroup. More precisely, when A is an elliptic curve, we instead let C and D be the full torsion subgroup A(Q)_{tor}, because it is easy to calculate. When A is not an elliptic curve it would be better to let C be the subgroup generated by all rational cusps, but the authors only realized this after completing the calculations, so we did not do this.

If N is square free, we let S_{l} = S_{l,1}. Otherwise, let S_{l,2} be the largest part of S_{l,1} coprime to all primes whose square divides N. This takes care of the Manin constant, which only involves primes whose square divides N. To take care of Tamagawa numbers, remove all primes p ≤ 2 dim(A) + 1 from S_{l,2} to obtain S_{l}.

Remark 5.1. When N is square free we have

S_{l} | #� _{?}^{odd} | S_{u}

since c_{A} is a power of 2 and no Tamagawa numbers have been omitted from the formulas for S_{l} and S_{u}. For every N ≤ 2333 we found that S_{l} is an integer, so when N ≤ 2333 is squarefree, #� _{?}^{odd} is an integer. Since Conjecture 2.2 asserts that #� _{?} is the order of a group, hence an integer, our data gives evidence for Conjecture 2.2.

Tables 1–4 list every A of level N ≤ 2333 such that S_{l} > 1. The A column contains the label of A (see Section 3.2), and the next column (labeled dim) contains dim A. A star next to the label for A indicates that we have proved that the odd part of #� (A) is at least as large as conjectured by the Birch and Swinnerton-Dyer conjecture. This is the case for 39 of the 168 examples. The columns labeled S_{l} contains the number S_{l} defined above. If S_{l} = S_{u} then the column labeled S_{u} contains an = sign, and otherwise, it contains S_{u} (there are only 13 cases in which S_{u} ≠ S_{l}). The column labeled moddeg(A)^{odd} contains the odd part m of the modular degree of A, written as a product gcd(m, S_{u}) · m/ gcd(S_{u}, m), where m/ gcd(S_{u}, m) is shrunk to save space. The only non-square-free levels of A_{f} for which S_{l} > 1 are 1058, 1664, 2224, and 2264.

The column labeled B contains all B such that L(B, 1) = 0 and

gcd(S_{l}, #(A^{∨} ∩ B̃^{∨})) > 1.

(In retrospect, it would probably have been more interesting to list those B such that gcd(S_{u}, #(A^{∨} ∩ B̃^{∨})) > 1.) Here if B = A_{g} for some newform g of level dividing N, and B̃^{∨} is the abelian subvariety of J_{0}(N) generated by all images of B^{∨} under the degeneracy maps. Thus, e.g., when B^{∨} is of level N, B̃^{∨} = B^{∨}. The next column, labeled dim, contains the dimension of B.

The final two columns contain information about the relationship between A and B. The one labeled A^{∨} ∩ B̃^{∨} contains the abelian group structure of the indicated abelian group, where e.g., [a^{b}c^{d}] means the abelian group (Z/aZ)^{b} × (Z/cZ)^{d}. The column labeled Vis contains a divisor of the order of Vis_{C}(� (A^{∨})), where C = A^{∨} + B̃^{∨} (note that Vis_{C}(� (A^{∨})) ⊂ Vis_{J_{0}(N)}(� (A^{∨}))).

The table is divided into three vertical regions, where the columns in the first region are about A only, the columns of the second region are about B only, and the third column is about the relationship between A and B.

5.1. Example: Level 389. We illustrate what is involved in computing the first line of Table 1. Using the method sketched in Section 3.2, we find that S_{2}(Γ_{0}(389)) contains exactly five Galois-conjugacy classes of newforms, and these are defined over extensions of Q of degrees 1, 2, 3, 6, and 20. Thus J = J_{0}(389) decomposes, up to isogeny, as a product A_{1} × A_{2} × A_{3} × A_{6} × A_{20} of abelian varieties, where dim A_{d} = d and A_{d} is the optimal quotient corresponding to the appropriate Galoisconjugacy class of newforms.

Next we consider the arithmetic of the A_{d}. Using Theorem 4.5 we find that

L(A_{1}, 1) = L(A_{2}, 1) = L(A_{3}, 1) = L(A_{6}, 1) = 0,

and

\frac{L(A_{20}, 1)}{Ω_{A_{20}}} = \frac{5^{2} · 2^{11}}{97 · c_{A}} ,

where c_{A} is the Manin constant attached to A_{20}, which, by Theorem 4.2, is of the form 2^{n} with n ≥ 0. Using the algorithms of Sections 3.5, 3.6, 3.7, we find that #A_{20}(Q) = c_{389} = 97. Thus Conjecture 2.2 predicts that #� (A_{20}) = 5^{2} · 2^{11}/c_{A}. The following proposition provides support for this conjecture.

Proposition 5.2. There is a natural inclusion

(Z/5Z)^{2} ≅ A_{1}(Q)/5A_{1}(Q) ↪ Vis_{J_{0}(389)}(� (A_{20}^{∨})).

Proof. Let A = A_{20}^{∨}, B = A_{1}^{∨} and J = A + B ⊂ J_{0}(389). Using Proposition 3.3, we find that A ∩ B ≅ (Z/4)^{2} × (Z/5Z)^{2}, so B[5] ⊂ A. Since 5 does not divide the numerator of (389 − 1)/12, it does not divide the Tamagawa numbers or the orders of the torsion groups, so Theorem 3.13 yields the asserted injection. To see that (Z/5Z)^{2} ≅ A_{1}(Q)/5A_{1}(Q) use the standard elliptic curves algorithms [Cre97]. □

5.2. Invisible Elements of � (A). Tables 1–4 suggest that much of � (A^{∨}) is invisible in J_{0}(N). This is because Proposition 3.15 implies that if a prime divides #� (A^{∨}) but not moddeg(A^{∨}) then � (A^{∨}) contains an element of order p that is invisible. We find many examples in the table where p divides the conjectural order of � (A^{∨}), but p ∤ moddeg(A^{∨}).

Invisible elements might become visible at higher level (see [AS02, §4.3] for a discussion and example).

5.3. The Part of � (A) That Must be a Perfect Square. When dim A = 1, properties of the Cassels-Tate pairing imply that if � (A) is finite then #� (A) is a perfect square, and the fact that one finds in examples (see [Cre97]) that #� _{?} is a perfect square is computational evidence for Conjecture 2.2.

In contrast, when the dimension is greater than one, Poonen and Stoll [PS99] discovered Jacobians J such that � (J) has order twice a square, and the second author found for each prime p < 25000 an abelian variety A of dimension p−1 such that #� (A) = pn^{2} for some integer n (see [Ste02b]).

Proposition 5.3. Let A = A_{f} be a quotient of J_{0}(N) and ℓ be a prime that does not divide the modular degree of A. Suppose that � (A)[ℓ^{∞}] is finite. Then #� (A)[ℓ^{∞}] is a perfect square.

Proof. The Cassels-Tate pairing (see [Tat63, §3]) induces a pairing

ϕ : � (A)[ℓ^{∞}] × � (A^{∨})[ℓ^{∞}] → Q/Z.

Since � (A)[ℓ^{∞}] is finite, it follows from [Tat63, Thm. 3.2] that � (A^{∨})[ℓ^{∞}] is also finite and ϕ is non-degenerate. In particular, #� (A^{∨})[ℓ^{∞}] = #� (A)[ℓ^{∞}].

Since J_{0}(N) is a Jacobian, it possesses a canonical polarization arising from the theta divisor; this divisor is rational over Q, since X_{0}(N) always has a point over Q (the cusp ∞ is rational). This polarization induces a polarization θ : A^{∨} → A, which also comes from a divisor that is rational over Q. Hence, by [Tat63, Thm. 3.3] (see also [PS99, Thm. 5]), the pairing

ϕ′ : � (A^{∨})[ℓ^{∞}] × � (A^{∨})[ℓ^{∞}] → Q/Z

obtained by composing θ with the pairing ϕ above is alternating.

Since ℓ does not divide the modular degree of A, it does not divide the degree of the isogeny θ. Hence θ induces an isomorphism � (A^{∨})[ℓ^{∞}] →^{≅} � (A)[ℓ^{∞}]. Thus by the non-degeneracy of the pairing ϕ, the pairing ϕ′ is also non-degenerate. Since ϕ′ is also alternating, it follows from arguments similar to those in [Cas62, p. 260] that #� (A^{∨})[ℓ^{∞}] is a perfect square. Since #� (A)[ℓ^{∞}] = #� (A^{∨})[ℓ^{∞}], we see that #� (A)[ℓ^{∞}] is also a perfect square.

□

For the entries in Tables 1–4, � (A) is finite, so if ℓ ∤ moddeg(A) then the ℓ-power part of #� (A) must be a perfect square. When S_{l} = S_{u} and the level is square free, then S_{l} is the odd part of the conjectural order of � (A). We found that S_{l} is a perfect square whenever S_{l} = S_{u}, which provides evidence for Conjecture 2.2.

MODULAR ABELIAN VARIETIES 19

Table 1. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l}  S_{u} moddeg(A)^{odd}  B dim A^{∨} ∩ B̃^{∨} Vis
389E∗ 20 5^{2}  =  5  389A 1  [20^{2}]  5^{2}
433D∗ 16 7^{2}  =  7·111  433A 1  [14^{2}]  7^{2}
446F∗ 8 11^{2}  =  11·359353  446B 1  [11^{2}]  11^{2}
551H 18 3^{2}  =  169  NONE
563E∗ 31 13^{2}  =  13  563A 1  [26^{2}]  13^{2}
571D∗ 2  3^{2}  =  3^{2}·127  571B 1  [3^{2}]  3^{2}
655D∗ 13 3^{4}  = 3^{2}·9799079  655A 1  [36^{2}]  3^{4}
681B 1  3^{2}  =  3·125  681C 1  [3^{2}]  −
707G∗ 15 13^{2}  =  13·800077  707A 1  [13^{2}]  13^{2}
709C∗ 30 11^{2}  =  11  709A 1  [22^{2}]  11^{2}
718F∗ 7  7^{2}  =  7·5371523  718B 1  [7^{2}]  7^{2}
767F 23 3^{2}  =  1  NONE
794G∗ 12 11^{2}  = 11·34986189 794A 1  [11^{2}]  −
817E∗ 15 7^{2}  =  7·79  817A 1  [7^{2}]  −
959D 24 3^{2}  =  583673  NONE
997H∗ 42 3^{4}  =  3^{2}  997B 1  [12^{2}]  3^{2}
997C 1  [24^{2}]  3^{2}
1001F 3  3^{2}  =  3^{2}·1269  1001C 1  [3^{2}]  −
91A 1  [3^{2}]  −
1001L 7  7^{2}  =  7·2029789  1001C 1  [7^{2}]  −
1041E 4  5^{2}  =  5^{2}·13589  1041B 2  [5^{2}]  −
1041J 13 5^{4}  = 5^{3}·21120929983 1041B 2  [5^{4}]  −
1058D 1  5^{2}  =  5·483  1058C 1  [5^{2}]  −
1061D 46 151^{2} =  151·10919  1061B 2 [2^{2}302^{2}] −
1070M 7 3·5^{2} 3^{2}·5^{2} 3·5·1720261 1070A 1  [15^{2}]  −
1077J 15 3^{4}  = 3^{2}·1227767047943 1077A 1  [9^{2}]  −
1091C 62 7^{2}  =  1  NONE
1094F∗ 13 11^{2}  = 11^{2}·172446773 1094A 1  [11^{2}]  11^{2}
1102K 4  3^{2}  =  3^{2}·31009  1102A 1  [3^{2}]  −
1126F∗ 11 11^{2}  = 11·13990352759 1126A 1  [11^{2}]  11^{2}
1137C 14 3^{4}  = 3^{2}·64082807  1137A 1  [9^{2}]  −
1141I 22 7^{2}  =  7·528921  1141A 1  [14^{2}]  −
1147H 23 5^{2}  =  5·729  1147A 1  [10^{2}]  −
1171D∗ 53 11^{2}  =  11·81  1171A 1  [44^{2}]  11^{2}
1246B 1  5^{2}  =  5·81  1246C 1  [5^{2}]  −
1247D 32 3^{2}  =  3^{2}·2399  43A 1  [36^{2}]  −
1283C 62 5^{2}  =  5·2419  NONE
1337E 33 3^{2}  =  71  NONE
1339G 30 3^{2}  =  5776049  NONE
1355E 28  3  3^{2} 3^{2}·2224523985405 NONE
1363F 25 31^{2}  =  31·34889  1363B 2 [2^{2}62^{2}] −
1429B 64 5^{2}  =  1  NONE
1443G 5  7^{2}  =  7^{2}·18525  1443C 1 [7^{1}14^{1}] −
1446N 7  3^{2}  = 3·17459029  1446A 1  [12^{2}]  −

20 AMOD AGASHE AND WILLIAM STEIN

Table 2. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim S_{l} S_{u} moddeg(A)^{odd}  B dim A^{∨} ∩ B̃^{∨}  Vis
1466H∗ 23  13^{2} = 13·25631993723  1466B 1  [26^{2}]  13^{2}
1477C∗ 24  13^{2} = 13·57037637  1477A 1  [13^{2}]  13^{2}
1481C 71  13^{2} =  70825  NONE
1483D∗ 67 3^{2}·5^{2} =  3·5  1483A 1  [60^{2}]  3^{2}·5^{2}
1513F 31  3  3^{4}  3·759709  NONE
1529D 36  5^{2}  =  535641763  NONE
1531D 73  3  3^{2}  3  1531A 1  [48^{2}]  −
1534J 6  3  3^{2}  3^{2}·635931  1534B 1  [6^{2}]  −
1551G 13  3^{2}  = 3·110659885  141A 1  [15^{2}]  −
1559B 90  11^{2} =  1  NONE
1567D 69 7^{2}·41^{2} =  7·41  1567B 3 [4^{4}1148^{2}] −
1570J∗ 6  11^{2} = 11·228651397  1570B 1  [11^{2}]  11^{2}
1577E 36  3  3^{2}  3^{2}·15  83A 1  [6^{2}]  −
1589D 35  3^{2}  = 6005292627343  NONE
1591F∗ 35  31^{2} =  31·2401  1591A 1  [31^{2}]  31^{2}
1594J 17  3^{2}  = 3·259338050025131 1594A 1  [12^{2}]  −
1613D∗ 75  5^{2}  =  5·19  1613A 1  [20^{2}]  5^{2}
1615J 13  3^{4}  = 3^{2}·13317421  1615A 1 [9^{1}18^{1}]  −
1621C∗ 70  17^{2} =  17  1621A 1  [34^{2}]  17^{2}
1627C∗ 73  3^{4}  =  3^{2}  1627A 1  [36^{2}]  3^{4}
1631C 37  5^{2}  =  6354841131  NONE
1633D 27 3^{6}·7^{2} = 3^{5}·7·31375  1633A 3 [6^{4}42^{2}]  −
1634K 12  3^{2}  = 3·3311565989  817A 1  [3^{2}]  −
1639G∗ 34  17^{2} =  17·82355  1639B 1  [34^{2}]  17^{2}
1641J∗ 24  23^{2} = 23·1491344147471 1641B 1  [23^{2}]  23^{2}
1642D∗ 14  7^{2}  = 7·123398360851  1642A 1  [7^{2}]  7^{2}
1662K 7  11^{2} = 11·16610917393  1662A 1  [11^{2}]  −
1664K 1  5^{2}  =  5·7  1664N 1  [5^{2}]  −
1679C 45  11^{2} =  6489  NONE
1689E 28  3^{2}  = 3·172707180029157365 563A 1  [3^{2}]  −
1693C 72 1301^{2} =  1301  1693A 3 [2^{4}2602^{2}] −
1717H∗ 34  13^{2} =  13·345  1717B 1  [26^{2}]  13^{2}
1727E 39  3^{2}  =  118242943  NONE
1739F 43 659^{2} = 659·151291281  1739C 2 [2^{2}1318^{2}] −
1745K 33  5^{2}  = 5·1971380677489 1745D 1  [20^{2}]  −
1751C 45  5^{2}  =  5·707  103A 2  [505^{2}]  −
1781D 44  3^{2}  =  61541  NONE
1793G∗ 36  23^{2} = 23·8846589  1793B 1  [23^{2}]  23^{2}
1799D 44  5^{2}  =  201449  NONE
1811D 98  31^{2} =  1  NONE
1829E 44  13^{2} =  3595  NONE
1843F 40  3^{2}  =  8389  NONE
1847B 98  3^{6}  =  1  NONE
1871C 98  19^{2} =  14699  NONE

MODULAR ABELIAN VARIETIES 21

Table 3. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim  S_{l}  S_{u}  moddeg(A)^{odd}  B dim A^{∨} ∩ B̃^{∨}  Vis
1877B 86  7^{2}  =  1  NONE
1887J 12  5^{2}  =  5·10825598693  1887A 1  [20^{2}]  −
1891H 40  7^{4}  =  7^{2}·44082137  1891C 2  [4^{2}196^{2}]  −
1907D∗ 90  7^{2}  =  7·165  1907A 1  [56^{2}]  7^{2}
1909D∗ 38  3^{4}  =  3^{2}·9317  1909A 1  [18^{2}]  3^{4}
1913B∗ 1  3^{2}  =  3·103  1913A 1  [3^{2}]  3^{2}
1913E 84 5^{4}·61^{2}  =  5^{2}·61·103  1913A 1  [10^{2}]  −
1913C 2  [2^{2}610^{2}]  −
1919D 52  23^{2}  =  675  NONE
1927E 45  3^{2}  3^{4}  52667  NONE
1933C 83  3^{2}·7 3^{2}·7^{2}  3·7  1933A 1  [42^{2}]  3^{2}
1943E 46  13^{2}  =  62931125  NONE
1945E∗ 34  3^{2}  =  3·571255479184807  389A 1  [3^{2}]  3^{2}
1957E∗ 37 7^{2}·11^{2}  =  7·11·3481  1957A 1  [22^{2}]  11^{2}
1957B 1  [14^{2}]  7^{2}
1979C 104  19^{2}  =  55  NONE
1991C 49  7^{2}  =  1634403663  NONE
1994D 26  3  3^{2}  3^{2}·46197281414642501  997B 1  [3^{2}]  −
1997C 93  17^{2}  =  1  NONE
2001L 11  3^{2}  =  3^{2}·44513447  NONE
2006E 1  3^{2}  =  3·805  2006D 1  [3^{2}]  −
2014L 12  3^{2}  =  3^{2}·126381129003  106A 1  [9^{2}]  −
2021E 50  5^{6}  =  5^{2}·729  2021A 1  [100^{2}]  5^{4}
2027C∗ 94  29^{2}  =  29  2027A 1  [58^{2}]  29^{2}
2029C 90 5^{2}·269^{2} =  5·269  2029A 2  [2^{2}2690^{2}]  −
2031H∗ 36  11^{2}  =  11·1014875952355  2031C 1  [44^{2}]  11^{2}
2035K 16  11^{2}  =  11·218702421  2035C 1  [11^{1}22^{1}]  −
2038F 25  5  5^{2}  5^{2}·92198576587  2038A 1  [20^{2}]  −
1019B 1  [5^{2}]  −
2039F 99 3^{4}·5^{2}  =  13741381043009  NONE
2041C 43  3^{4}  =  61889617  NONE
2045I 39  3^{4}  =  3^{3}·3123399893  2045C 1  [18^{2}]  −
409A 13 [9370199679^{2}] −
2049D 31  3^{2}  =  29174705448000469937  NONE
2051D 45  7^{2}  =  7·674652424406369  2051A 1  [56^{2}]  −
2059E 45  5·7^{2} 5^{2}·7^{2}  5^{2}·7·167359757  2059A 1  [70^{2}]  −
2063C 106  13^{2}  =  8479  NONE
2071F 48  13^{2}  =  36348745  NONE
2099B 106  3^{2}  =  1  NONE
2101F 46  5^{2}  =  5·11521429  191A 2  [155^{2}]  −
2103E 37 3^{2}·11^{2}  = 3^{2}·11·874412923071571792611 2103B 1  [33^{2}]  11^{2}
2111B 112 211^{2}  =  1  NONE
2113B 91  7^{2}  =  1  NONE
2117E∗ 45  19^{2}  =  19·1078389  2117A 1  [38^{2}]  19^{2}

22 AMOD AGASHE AND WILLIAM STEIN

Table 4. Visibility of Nontrivial Odd Parts of Shafarevich-Tate Groups

A dim  S_{l}  S_{u} moddeg(A)^{odd}  B dim  A^{∨} ∩ B̃^{∨}  Vis
2119C 48  7^{2}  =  89746579  NONE
2127D 34  3^{2}  = 3·18740561792121901 709A 1  [3^{2}]  −
2129B 102  3^{2}  =  1  NONE
2130Y 4  7^{2}  =  7·83927  2130B 1  [14^{2}]  −
2131B 101 17^{2}  =  1  NONE
2134J 11  3^{2}  =  1710248025389  NONE
2146J 10  7^{2}  =  7·1672443  2146A 1  [7^{2}]  −
2159E 57  13^{2}  =  31154538351  NONE
2159D 56  3^{4}  =  233801  NONE
2161C 98  23^{2}  =  1  NONE
2162H 14  3  3^{2}  3·6578391763  NONE
2171E 54  13^{2}  =  271  NONE
2173H 44 199^{2}  =  199·3581  2173D 2  [398^{2}]  −
2173F 43  19^{2} 3^{2}·19^{2} 3^{2}·19·229341  2173A 1  [38^{2}]  19^{2}
2174F 31  5^{2}  = 5·21555702093188316107 NONE
2181E 27  7^{2}  = 7·7217996450474835 2181A 1  [28^{2}]  −
2193K 17  3^{2}  =  3·15096035814223  129A 1  [21^{2}]  −
2199C 36  7^{2}  = 7^{2}·13033437060276603 NONE
2213C 101  3^{4}  =  19  NONE
2215F 46  13^{2}  =  13·1182141633  2215A 1  [52^{2}]  −
2224R 11  79^{2}  =  79  2224G 2  [79^{2}]  −
2227E 51  11^{2}  =  259  NONE
2231D 60  47^{2}  =  91109  NONE
2239B 110 11^{4}  =  1  NONE
2251E∗ 99  37^{2}  =  37  2251A 1  [74^{2}]  37^{2}
2253C∗ 27  13^{2}  = 13·14987929400988647 2253A 1  [26^{2}]  13^{2}
2255J 23  7^{2}  =  15666366543129  NONE
2257H 46 3^{6}·29^{2}  =  3^{3}·29·175  2257A 1  [9^{2}]  −
2257D 2  [2^{2}174^{2}]  −
2264J 22  73^{2}  =  73  2264B 2  [146^{2}]  −
2265U 14  7^{2}  = 7^{2}·73023816368925 2265B 1  [7^{2}]  −
2271I∗ 43  23^{2}  = 23·392918345997771783 2271C 1  [46^{2}]  23^{2}
2273C 105  7^{2}  =  7^{2}  NONE
2279D 61  13^{2}  =  96991  NONE
2279C 58  5^{2}  =  1777847  NONE
2285E 45 151^{2}  = 151·138908751161 2285A 2  [2^{2}302^{2}]  −
2287B 109 71^{2}  =  1  NONE
2291C 52  3^{2}  =  427943  NONE
2293C 96 479^{2}  =  479  2293A 2  [2^{2}958^{2}]  −
2294F 15  3^{2}  =  3·6289390462793  1147A 1  [3^{2}]  −
2311B 110  5^{2}  =  1  NONE
2315I 51  3^{2}  =  3·4475437589723  463A 16 [13426312769169^{2}] −
2333C 101 83341^{2}  =  83341  2333A 4  [2^{6}166682^{2}]  −

6. Appendix by J. Cremona and B. Mazur: “Explaining” Shafarevich-Tate via Mordell-Weil

Introduction. In our article [CM] we discussed the notion of visibility and offered some tables of examples of that phenomenon. We gave, however, very little theoretical discussion in that article. Here we wish to take the opportunity to correct some gaps in our commentary on our tables and to offer the details of the proof of a general criterion that is sometimes useful to test visibility. Regarding Table 1 of [CM] we said that for each pair (E, p) that occurs there and for which there is a corresponding “F ” on the table of the same conductor of E, the Shafarevich-Tate group of E is explained by the Mordell-Weil group of F, in the technical sense that we gave to the word explained in that article. Now this is indeed the case for all entries of our table such that E has semistable reduction at p and it is also the case for those entries where the conductor of F properly divides the conductor of E. We will review why this is so, below. It is also true that for each of the remaining 7 entries (E = 2601H, 2718D, 2900D, 3555E, 3879E, 3933A, 5499E) a nontrivial subgroup of the Shafarevich-Tate group of E is explained by the Mordell-Weil group of the corresponding F, but we wish to notify our readers that we have not yet checked whether or not all of the “� ” of these 7 elliptic curves is so explained. These 7 cases deserve to be looked at (the issue being local at the prime 3 for all but 2900D, where it is local at the prime 5). Regarding Table 2 of [CM], although our commentary in [CM] does not say this clearly, for all the entries E of that table for which there is a corresponding F of the same conductor we only have checked that E[2] = F [2] in J_{0}(N) and nothing more, except, of course, for those entries we particularly signal to have shown something less; namely, in the language of our article, that they “seem to satisfy a 2-congruence.” In these latter cases where we signal that we have shown something less, W. Stein has checked that in fact E[2] ≠ F [2] in J_{0}(N).

Let p be an odd prime number. If E is an (optimal) elliptic curve over Q of conductor N then E may be unambiguously identified (up to sign) with a subabelian variety of the modular jacobian J_{0}(N) (over Q). If (E, F, p) is an entry of Table 1 of [CM] such that E and F are of the same conductor N we checked that we have equality of the finite group schemes E[p]_{/Q} = F [p]_{/Q} in J_{0}(N)_{/Q}. For the remaining three entries we checked that there is an isomorphism of finite group schemes ι : E[p]_{/Q} ≅ F [p]_{/Q}. In both cases, identifying the two finite group schemes let H denote the common cohomology group,

H := H^{1}(G_{Q}, E[p]_{/Q}) = H^{1}(G_{Q}, F [p]_{/Q}),

and S_{E} ⊂ H, and S_{F} ⊂ H the p-Selmer groups of, respectively, E and F. What we will show is that

Proposition 6.1. For each of the entries (E, p) in Table 1 of [CM] such that p is a prime of semistable reduction for E and for which there is a “corresponding” F, we have

S_{E} = S_{F} ⊂ H.

To discuss this, we need some notation.

Let X := Spec(Z), Y := Spec(Z[1/p]) = X − Spec(F_{p}), and η := Spec(Q). Let E_{η} := E be our elliptic curve over Q of conductor N, E_{/X} the Néron model over X of E_{η} and E_{/X}^{o} ⊂ E_{/X} the “connected component” of Néron (meaning the open subgroup scheme every fiber of which is connected). We have, of course, similar notation for the corresponding elliptic curve F. Let E[p]_{/X} denote the closed subgroup scheme given as the kernel of multiplication by p in the Néron model: E[p]_{/X} ⊂ E_{/X}. We have, in general, that the restriction E[p]_{/Y} of E[p]_{/X} to the base Y is an étale quasi-finite flat group scheme; and if p^{2} doesn’t divide N we have that the group scheme E[p]_{/X} is a quasi-finite flat group scheme [Gro72, Prop. 3.1(d), pg. 343]. The étale quasi-finite flat group scheme E[p]_{/Y} can be characterized by the following features:

(i) Its generic fiber is the group scheme E[p]_{/η} ⊂ J_{0}(N)_{/η}, and (one has a choice here) either:

(ii) E[p]_{/Y} ⊂ J_{0}(N)_{/Y} is a closed étale quasi-finite flat) subgroup scheme, or:

(ii’) E[p]_{/Y} enjoys the Néronian property over the base Y. Similar statements hold for F [p]_{/X}.

Let Φ be the (punctual) sheaf of abelian groups for the flat topology over X which fits into the exact sequence (of abelian sheaves over X)

(1)  0 → E^{o} → E → Φ → 0.

We will use the same notation to indicate the corresponding exact sequence of sheaves for the étale topology over X. Since E^{0} and E are smooth group schemes, the long exact sequences of cohomology derived from the short exact sequence (1), viewed either as sheaves of abelian groups for the flat or étale topology, coincide; cf Section 11 Appendice: Un théorème de comparaison de la cohomologie étale et de la cohomologie fppf in [Gro68]. Thinking now of Φ as a sheaf for the étale topology, denote by Φ_{ℓ} its stalk at the prime ℓ. So Φ_{ℓ} is representable as a finite étale group scheme over the field F_{ℓ}. We have that

Φ = ⨁_{ℓ | N}(i_{ℓ})_{∗}Φ_{ℓ},

where i_{ℓ} : SpecF_{ℓ} ↪ X is the natural closed immersion. We have an exact sequence

(2) 0 → E^{o}(X) → E(Q) → H^{0}(X, Φ) → H^{1}(X, E^{o}) → H^{1}(X, E) → H^{1}(X, Φ),

where cohomology is computed for the étale topology. We have, for either topology,

H^{i}(X, Φ) = ⨁_{ℓ | N} H^{i}(Spec(F_{ℓ}), Φ_{ℓ}).

Viewing (1) as an exact sequence of sheaves for the flat topology, and passing to the associated cohomology sequence we see that (2) may be thought of, ambiguously as computed for either the étale or the flat topology.

If p is an odd prime number, the p-primary component of the Shafarevich-Tate group of E is the p-primary component of the image of H^{1}(X, E^{o}) → H^{1}(X, E) (see the appendix to [Maz72]), or equivalently the intersection of the kernels of

H^{1}(X, E) → H^{1}(Spec(F_{ℓ}), Φ_{ℓ}).

Let p be an odd prime number. Let E′ ⊂ E be the open subgroup scheme of E which is the inverse image of pΦ ⊂ Φ, so that we have an exact sequence of sheaves for the flat (or étale) topology:

(3)  0 → E′ → E → Φ/pΦ → 0,

and if p is a prime of semistable reduction for E (equivalently: p^{2} doesn’t divide N) we have an exact sequence of flat group schemes

(4)  0 → E[p] → E → E′ → 0.

Put

E[p]_{/X}^{o} := E[p]_{/X} ⋂ E_{/X}^{o}.

Then E[p]_{/X}^{o} is an open (quasi-finite) subgroup scheme of E[p]_{/X}. Let Ẽ[p]_{/X} be any “intermediate” open (quasi-finite) subgroup scheme

E[p]_{/X}^{o} ⊂ Ẽ[p]_{/X} ⊂ E[p]_{/X}

so that we have the exact sequence of sheaves for the finite flat topology

(5)  0 → Ẽ[p]_{/X} → E[p]_{/X} → Ψ → 0,

with Ψ a subquotient of Φ.

Consider the following hypothesis: A(E, p, ℓ): The Galois module Φ_{ℓ}/pΦ_{ℓ} is either trivial, or else is a non-

constant cyclic Galois module over F_{ℓ}.

Let A(E, p) denote the conjunction of Hypotheses A(E, p, ℓ) for all prime numbers ℓ, or equivalently, for all ℓ dividing N.

Lemma 6.2. These are equivalent formulations of Hypothesis A(E, p). (a) Φ/pΦ is cohomologically trivial; that is, H^{0}(X, Φ/pΦ) = H^{1}(X, Φ/pΦ) = 0.

(b) If Ψ is any subquotient of Φ, Ψ is “p-cohomologically trivial” in the sense

that the p-primary components of H^{i}(X, Ψ) vanish for all i. Moreover, if p ≥ 5, or if p = 3 and E has no Néron fibers of type IV or IV*, the above conditions are equivalent to:

(c) For every ℓ at which E has split multiplicative reduction, p does not divide the order of the group of connected components of the Néron fiber of E at ℓ.

Proof. The equivalence of Hypothesis A(E, p) with (a) and with (b) is straightforward using standard exact sequences plus the fact that the p-primary components of the (underlying abelian group of) Φ_{ℓ} is cyclic since p > 2; and noting that a (finite) G-module of prime order with nontrivial G-action has trivial cohomology. For (c) we are using that if p > 2 the p-primary component of Φ_{ℓ} vanishes for all primes ℓ of additive reduction for E except when p = 3 and the Néron fiber type of E at ℓ is IV or IV*. □

A morphism G_{1} → G_{2} of flat (commutative, finite type) groups schemes over X will be said to induce an isomorphism on p-cohomology if the induced mappings

H^{i}(X, G_{1}) ⊗ Z_{p} → H^{i}(X, G_{2}) ⊗ Z_{p}

are isomorphisms for all i ≥ 0, where cohomology is computed for the flat topology.

Lemma 6.3. Let p be an odd prime number for which A(E, p) holds. We have that the natural morphisms

E_{/X}^{o} → E′_{/X}  and  E′_{/X} → E_{/X}

induce isomorphisms on p-cohomology. If p is of semistable reduction for E, we also have that

Ẽ[p]_{/X} → E[p]_{/X}

induces isomorphisms on p-cohomology, for any of the open subgroup schemes Ẽ[p]_{/X} in E[p]_{/X} described above.

Proof. These all sit in short exact sequences of sheaves of abelian groups for the flat topology over X where the third sheaf is p-cohomologically trivial. □

Corollary 6.4. If p > 2 and A(E, p) holds we have natural isomorphisms

H^{0}(X, E^{o}) ⊗ Z_{p} ≅ H^{0}(X, E′) ⊗ Z_{p} ≅ E(Q) ⊗ Z_{p}

and

� (E) ⊗ Z_{p} ≅ H^{1}(X, E^{o}) ⊗ Z_{p} ≅ H^{1}(X, E′) ⊗ Z_{p} ≅ H^{1}(X, E) ⊗ Z_{p}.

Corollary 6.5. Let p be an odd prime number, semistable for E, and suppose that A(E, p) holds.

(i) The image of the natural (injective) coboundary mapping

0 → E(Q)/pE(Q) ↪ H^{1}(G_{Q}, E[p])

attached to the Kummer sequence is contained in the image of the natural injection

H^{1}(X, E[p]^{o}) ↪ H^{1}(G_{Q}, E[p]).

(ii) We have an exact sequence

0 → E(Q)/pE(Q) → H^{1}(X, Ẽ[p]) → � (E)[p] → 0

for any of the open subgroup schemes Ẽ[p]_{/X} ⊂ E[p]_{/X} defined above. (iii) The image of H^{1}(X, Ẽ[p]) ↪ H^{1}(G_{Q}, E[p]) is equal to the p-Selmer sub-

group,

S_{p}(E) ⊂ H^{1}(G_{Q}, E[p]).

Proof. All this follows from straightforward calculations using the cohomological exact sequences associated to the exact sequences (1)–(5) in the light of the previous discussion. □

To set things up for our application, let us record the following:

Corollary 6.6. Let E_{/Q} and F_{/Q} be elliptic curves over Q. Let p be an odd prime number of semistable reduction for E and F, and for which A(E, p) and A(F, p) both hold. Define Ẽ[p]_{/X} ⊂ E[p]_{/X} to be the open quasi-finite subgroup scheme whose restriction to Y is equal to E[p]_{/Y} and whose fiber at F_{p} is equal to E[p]_{/F_{p}}^{o} = (E[p] ⋂ E^{o})_{/F_{p}}. Define F̃[p]_{/X} similarly. Suppose, finally, that we have an isomorphism of G_{Q}-modules ι : F [p]_{/Q} ≅ E[p]_{/Q} which extends to an injection of quasi-finite flat group schemes

F̃[p]_{/X} ↪ Ẽ[p]_{/X}.

Letting

H := H^{1}(G_{Q}, E[p]) = H^{1}(G_{Q}, F [p])

(making the identification via ι) we have that the p-Selmer groups S_{p}(E) ⊂ H and S_{p}(F) ⊂ H are the same.

Proposition 6.7. Let (E, F, p) be a triple which is an entry of Table 1 of [CM]. Suppose further that p is of semistable reduction for E and for F. Then, with the notation of the previous corollary, the p-Selmer groups S_{p}(E) ⊂ H and S_{p}(F) ⊂ H are the same. In the terminology of [CM] the Shafarevich-Tate group of E is explained by the Mordell-Weil group of F.

Proof. As mentioned above, we have checked that E[p]_{/Q} = F [p]_{/Q} ⊂ J_{0}(N) whenever the pair E and F (appearing as entry of Table 1 of [CM]) have the same conductor. We have checked that E[p]_{/Q} ≅ F [p]_{/Q} for the three entries where E and F have different conductor (E = 2932A, 3306B, and 5136B). We have checked that Hypothesis A(E, p, ℓ) and A(F, p, ℓ) hold for all quadruples (E, F, p, ℓ) such that (E, F, p) occurs as an entry in Table 1 of [CM] (even when p is not semistable for E and F) with the exception of the entry (E, F, p, ℓ) = (2366D, 2366E, 3, 13).

Sublemma 6.8. Under the hypotheses of our proposition, the isomorphism of G_{Q}-modules ι : E[p]_{/Q} ≅ F [p]_{/Q} extends to an injection of quasi-finite flat group schemes

Ẽ[p]_{/X} ↪ F̃[p]_{/X}

which is an isomorphism except in two instances (E = 3306B, and 5136B).

Proof. First, since Ẽ[p]_{/Y} = E[p]_{/Y}, F̃[p]_{/Y} = F [p]_{/Y}, and, as we mentioned at the beginning, both of these quasi-finite, flat (étale) group schemes F [p]_{/Y} and E[p]_{/Y} enjoy the Néronian property, the isomorphism ι extends to an isomorphism Ẽ[p]_{/Y} ≅ F̃[p]_{/Y}. The remaining question is then local about p. If p is of good reduction for E, then Ẽ[p]_{/X_{p}} and F̃[p]_{/X_{p}} are both finite flat group schemes of odd order, so by Fontaine’s Theorem [Fon75], the isomorphism between their generic fibers extends to an isomorphism over X_{p}. (Compare: Theorem I.1.4 in [Maz77].) A standard result allows us to patch the isomorphism extending ι over Y with the isomorphism (“extending ι”) over X_{p} to get the extension of ι to an isomorphism of group schemes over X, Ẽ[p]_{/X} ≅ F̃[p]_{/X}. Now consider the case where p is of bad reduction. By the assumptions of our proposition, p is then of multiplicative reduction for E, and hence the fiber of E over F_{p} is a finite multiplicative type group scheme of order p. We therefore have that Ẽ[p]_{/X_{p}} sits in an exact sequence

(6)  0 → C_{/X_{p}} → Ẽ[p]_{/X_{p}} → E_{/X_{p}} → 0

where C_{/X_{p}} is a finite flat group scheme of order p (and with fiber of multiplicative type in characteristic p) and where E_{/X_{p}} is an étale quasi-finite group scheme, with trivial fiber in characteristic p.

Let us take a moment to recall (see [Maz78, Lem. 1.1]) the construction of such an exact sequence (6): working in the category of formal schemes, let X̂_{p} := Spf(Z_{p}), and let Ĉ_{/X̂_{p}} be the formal completion of the zero-section in Ẽ[p]_{/X_{p}}. One checks

that Ĉ_{/X̂_{p}} may be identified with a finite flat formal group scheme over X̂_{p} which

admits a closed immersion into the formal group scheme over X̂_{p} associated to Ẽ[p]_{/X_{p}}. A standard algebrization argument establishes that there is a (unique) finite flat subgroup scheme C_{/X_{p}} ⊂ Ẽ[p]_{/X_{p}} whose associated formal group scheme over X̂_{p} is Ĉ_{/X̂_{p}}. The exact sequence (6) is then obtained by letting E_{/X_{p}} be the evident quotient (quasi-finite flat) group scheme, and noting that, by construction, its special fiber is trivial.

Now let us return to the proof of the sublemma. Since the restriction of ι to C_{/X_{p}} (a finite flat multiplicative type group scheme of order p) is injective over the generic point, it follows (by elementary considerations, or by Fontaine’s Theorem cited above) that ι restricted to C_{/X_{p}} is an injection over X_{p}. Since E_{/X_{p}} has trivial fiber in characteristic p, ι is an injection as was to be proved. In all cases under consideration, then,

ι : Ẽ[p]_{/X} ↪ F̃[p]_{/X}

is an injection. If E is of good reduction at p, or if F is of bad reduction at p, ι is therefore an isomorphism. The cases remaining are when E is of bad reduction at p and F is of good reduction (i.e., E = 3306B, and 5136B) in which case we can only assert that ι is an injection. □

Returning to our proposition, suppose that F̃[p]_{/X_{p}} is finite flat (which happens in the two cases signalled above: E = 3306B, and 5136B). Then the isomorphism induced by ι on generic fibers

Ẽ[p]_{/Q_{p}} ≅ F̃[p]_{/Q_{p}}

restricted to the G_{Q_{p}}-stable subgroup C_{/Q_{p}} ⊂ Ẽ[p]_{/Q_{p}} extends to a morphism of the finite flat group scheme C_{/X_{p}} into F̃[p]_{/X_{p}}. This extended morphism j : C_{/X_{p}} → F̃[p]_{X_{p}} is necessarily a closed immersion since C_{/X_{p}} is a multiplicative type finite flat group scheme. Since E_{/X_{p}} has trivial fiber in characteristic p an application of the standard patching argument (as used in the previous case) allows us to put together the isomorphism of group schemes over Y extending ι with the closed immersion j over X_{p} to get a closed immersion

Ẽ[p]_{/X} ↪ F̃[p]_{/X}.

Finally suppose that both E and F have multiplicative reduction at p. We then have exact sequences (6) for each of our quasi-finite flat group schemes Ẽ[p]_{/X_{p}} and F̃[p]_{/X_{p}}. Let V denote their common generic fiber (identified via ι) considered as two-dimensional F_{p}-vector space with G_{Q_{p}}-action. Let C(E) ⊂ V and C(F) ⊂ V denote the one-dimensional subspaces given by the generic fibers of the finite flat subgroup schemes C_{/X_{p}} corresponding to the exact sequence (6) for for E and for F respectively. Suppose, first, that these one-dimensional F_{p}-subspaces C(E) and C(F) are different. It then follows that the G_{Q_{p}}-representation V splits as the direct sum of C(E) and C(F), both F_{p}-subspaces being isomorphic, as I_{Q_{p}}-modules to μ_{p}, where I_{Q_{p}} ⊂ G_{Q_{p}} is the inertia subgroup of G_{Q_{p}}. But this contradicts the fact that V is self-Cartier dual (under the Weil pairing). Consequently, C(E) = C(F) ⊂ V. From the above discussion it follows that we can extend ι to an isomorphism Ẽ[p]_{/X_{p}} ≅ F̃[p]_{/X_{p}}.

Our proposition then follows (from Corollary 6.6) for all entries in Table 1 of [CM] where p is of semistable reduction for E once we produce special arguments to cover the three special cases E = 3306B, 5136B and 2366D. The first two of these cases are “special” because we only have an injection Ẽ[p]_{/X_{p}} ↪ F̃[p]_{/X_{p}} and not an isomorphism. However, the cokernel of this morphism restricted to the fiber in characteristic 3 is, in both of these cases, a cyclic group with nontrivial G_{Q_{3}}- action and hence is 3-cohomologically trivial. In particular, the injection Ẽ[p]_{/X_{p}} ↪ F̃[p]_{/X_{p}} induces an isomorphism on flat cohomology over X, and the argument for these two cases proceeds as before. This leaves (E, F, p) = (2366D, 2366E, 3) which is the only example of an entry (E, F, p) in our table, where E has a Qrational point of order p, and (this is no accident) where Hypothesis A(E, p) and Hypothesis A(F, p) fail. (Indeed there are no other failures of Hypothesis A(F, p) for any of the (E, F, p)’s occurring in Table 1 of [CM] and only one other failure of Hypothesis A(E, p), which is for (E, p, ℓ) = (2932A, 3, 2).)

Let us now deal with the case (E, F, p) = (2366D, 2366E, 3). The subgroup C of Q-rational points of order 3 on E specialize in characteristic 13 to yield an isomorphism

C ≅ Φ_{13}

and the same for the subgroup of Q-rational points of order 3 on F. We make use of this information to cut down the group schemes Ẽ[3]_{/X} and F̃[3]_{/X} and define open

subgroup schemes: ˜ Ẽ[3]_{/X} ⊂ Ẽ[3]_{/X} and ˜ Ẽ[3]_{/X} ⊂ Ẽ[3]_{/X} by requiring that these closed immersions of subgroup schemes be isomorphisms outside characteristic 13, and that the “double-tilded” group schemes each have trivial fiber in characteristic 13. We get via the above argument an isomorphism of group schemes ˜ Ẽ[3]_{/X} ≅

˜F̃ [3]_{/X} extending ι, and an identification of the 3-Selmer groups of E and F with

H^{1}(X, ˜ Ẽ[3]) and H^{1}(X, ˜ F̃ [3]) respectively. Our proposition is proved. □

It remains to say a few words about why, in the 7 cases of entries (E, F, p) in our Table 1 of [CM] for which p is a prime of additive reduction for E some nontrivial elements of the Shafarevich-Tate group of E are explained by the Mordell-Weil group of F. Briefly, the reason is as follows. By the inflated p-Selmer group of E (and of F) let us mean the subgroup of H obtained by insisting upon all the local Selmer conditions at primes different from p, but putting no condition at p. The p-Selmer group of E (and of F) are, in all 7 instances, F_{p}-vector spaces of dimension 2 and therefore, the inflated p-Selmer groups are of dimensions either 2 or 3. Working over Y rather than over X, the above argument applied to these 7 remaining cases gives us an identification of the inflated p-Selmer groups of E and of F in H. But the true p-Selmer groups (vector spaces of dimension 2) being subspaces in a vector space of dimension ≤ 3 must have a nontrivial intersection.

References

[Aga99] A. Agashe, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369–374. MR 2000e:11083

[AS02] A. Agashe and W. A. Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties, to appear in J. of Number Theory (2002).

[AS04] A. Agashe and W. A. Stein, The manin constant, congruence primes, and the modular degree, Preprint,

http://modular.fas.harvard.edu/papers/manin-agashe/ (2004).

[BCP97] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265, Computational algebra and number theory (London, 1993). MR 1 484 478

[Bir71] B. J. Birch, Elliptic curves over Q: A progress report, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp. 396–400. MR 47:3395

[BLR90] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990. MR 91i:14034

[Cas62] J. W. S. Cassels, Arithmetic on curves of genus 1. III. The Tate-Šafarevič and Selmer groups, Proc. London Math. Soc. (3) 12 (1962), 259–296. MR 29 #1212

[Cre97] J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997. MR 99e:11068

[CM] J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13–28. MR 1 758 797

[CS01] B. Conrad and W. A. Stein, Component groups of purely toric quotients, Math. Res. Lett. 8 (2001), no. 5-6, 745–766. MR 2003f:11087

[Del01] C. Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over Q, Experiment. Math. 10 (2001), no. 2, 191–196.

[DI95] F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem, Providence, RI, 1995, pp. 39–133. MR 97g:11044

[Edi91] B. Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989), Birkhäuser Boston, Boston, MA, 1991, pp. 25–39. MR 92a:11066

[Eme01] M. Emerton, Optimal quotients of modular Jacobians. Preprint. [FpS^{+}01] E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell,

Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675–1697. MR 1 836 926

[Fon75] J-M. Fontaine, Groupes finis commutatifs sur les vecteurs de Witt, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Ai, A1423–A1425. MR 51:10353

[Gro94] B. H. Gross, L-functions at the central critical point, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 527–535. MR 95a:11060

[Gro68] A. Grothendieck, Le groupe de Brauer. III. Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88–188. MR 39:5586c

[Gro72] A. Grothendieck, Modèles de Néron et monodromie in Groupes de monodromie en géométrie algébrique. I, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Vol. 288. MR 50:7134

[GZ86] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 87j:11057

[Kat81] N. M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481–502. MR 82d:14025

[KL89] V. A. Kolyvagin and D. Y. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196. MR 91c:11032

[KL92] V. A. Kolyvagin and D. Y. Logachev, Finiteness of � over totally real fields, Math. USSR Izvestiya 39 (1992), no. 1, 829–853. MR 93d:11063 [KS00] D. R. Kohel and W. A. Stein, Component Groups of Quotients of J_{0}(N), Proceedings

of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, July 2–7, 2000 (Berlin), Springer, 2000. MR 1 850 621

[Lan91] S. Lang, Number theory. III, Springer-Verlag, Berlin, 1991, Diophantine geometry. MR 93a:11048

[LO85] H. W. Lenstra, Jr. and F. Oort, Abelian varieties having purely additive reduction, J. Pure Appl. Algebra 36 (1985), no. 3, 281–298. MR 86e:14020

[LS02] Joan-C. Lario and René Schoof, Some computations with Hecke rings and deformation rings, Experiment. Math. 11 (2002), no. 2, 303–311, With an appendix by Amod Agashe and William Stein. MR 2004b:11072

[Maz72] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 56:3020

[Maz77] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33–186 (1978). MR 80c:14015

[Maz78] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 80h:14022

[MT74] B. Mazur and J. Tate, Points of order 13 on elliptic curves, Invent. Math. 22 (1973/74), 41–49. MR 50:327

[Mil86] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103–150.

[Ogg73] A. P. Ogg, Rational points on certain elliptic modular curves, Analytic number theory (Proc. Sympos. Pure Math., Vol XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 221–231. MR 49:2743

[PS99] B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109–1149. MR 2000m:11048

[Shi73] G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), no. 3, 523–544. MR 47:6709

[Shi94] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1. MR 95e11048

[Ste82] G. Stevens, Arithmetic on modular curves, Birkhäuser Boston Inc., Boston, Mass., 1982. MR 87b:11050

[Ste00] W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000).

[Ste02a] W. A. Stein, An introduction to computing modular forms using modular symbols, to appear in an MSRI Proceedings (2002).

[Ste02b] W. A. Stein, Shafarevich-tate groups of nonsquare order, Proceedings of MCAV 2002, Progress of Mathematics (to appear).

[Stu87] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Springer, Berlin, 1987, pp. 275–280. MR 88h:11031

[Tat63] J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 288–295. MR 31:168

[Tat66] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1966 (reprinted in 1995), pp. Exp. No. 306, 415–440. MR 1 610 977

Department of Mathematics, U niversity of Texas, Austin, Texas 78712

E-mail address: agashe@math.utexas.edu

Department of Mathematics, Harvard University, Cambridge, Massachussetts 02138

E-mail address: was@math.harvard.edu