Visibility of the Shafarevich-Tate Group at Higher Level

Dimitar P. Jetchev William A. Stein

Abstract. We study visibility of Shafarevich-Tate groups of modular abelian varieties in Jacobians of modular curves of higher level. We prove a theorem about the existence of visible elements at a specific higher level under hypotheses that can be verified explicitely. We also provide a table of examples of visible subgroups at higher level and state conjectures inspired by our data.

1 Introduction

1.1 Motivation

Mazur suggested that the Shafarevich-Tate group � (K, E) of an abelian variety A over a number field K could be studied via a collection of finite subgroups (the visible subgroups) corresponding to different embeddings of the variety into larger abelian varieties C over K (see [Maz99] and [CM00]). The advantage of this approach is that the isomorphism classes of principal homogeneous spaces, for which one has à priori little geometric information, can be given a much more explicit description as K-rational points on the quotient abelian variety C/A (the reason why they are called visible elements).

Agashe, Cremona, Klenke and the second author built upon the ideas of Mazur and developed a systematic theory of visibility of Shafarevich-Tate groups of abelian varieties over number fields (see [Aga99b, AS02, AS05, CM00, Kle01, Ste00]). More precisely, Agashe and Stein provided sufficient conditions for the existence of visible sugroups of certain order in the Shafarevich-Tate group and applied their general theory to the case of newform subvarieties A_{f}_{/ℚ} of the Jacobian J_{0}(N)_{/ℚ} of the modular curve X_{0}(N)_{/ℚ} (here, f is a newform of level N and weight 2 which is an eigenform for the Hecke operators acting on the space S_{2}(Γ_{0}(N)) of cuspforms of level N and weight 2). Unfortunately, there is no guarantee that a non-trivial element of � (ℚ, A_{f}) is visible for the embedding A_{f} ↪ J_{0}(N).

In this paper we consider the case of modular abelian varieties over ℚ and make use of the algebraic and arithmetic properties of the corresponding newforms to provide sufficient conditions for the existence of visible elements of � (ℚ, A_{f}) in modular Jacobians of level a multiple of the base level N. More

precisely, we consider morphism of the form A_{f} ↪ J_{0}(N) ⟶^{ϕ} J_{0}(MN), where ϕ is a suitable linear combination of degeneracy maps which makes the kernel of the composition morphism almost trivial (i.e., trivial away from the 2-part). For specific examples, the sufficient conditions can be verified explicitely. We also provide a table of examples where certain elements of � (ℚ, A_{f}) which are invisible in J_{0}(N) become visible at a suitably chosen higher level. At the end, we state some general conjectures inspired by our results.

1.2 Organization of the paper

Section 2 discusses the basic definitions and notation for modular abelian varieties, modular forms, Hecke algebras, the Shimura construction and modular degrees. Section 3 is a brief introduction to visibility theory for Shafarevich-Tate groups. In Section 4 we state and prove an equivariant version of a theorem of Agashe-Stein (see [AS05, Thm 3.1]) which guarantees existence of visible elements. The theorem is more general because it makes use of the action of the Hecke algebra on the modular Jacobian.

In Section 5 we introduce the notion of strong visibility which is relevant for visualizing cohomology classes in Jacobians of modular curves whose level is a multiple of the level of the original abelian variety. Theorem 5.1.3 guarantees existence of strongly visible elements of the Shafarevich-Tate group under some hypotheses on the component groups, a congruence condition between modular forms, and irreducibility of the Galois representation. In Section 5.4 we prove a variant of the same theorem (Theorem 5.4.2) with more stringent hypotheses that are easier to verify in specific cases.

Section 6 discusses in detail two computational examples for which strongly visible elements of certain order exist which provides evidence for the Birch and Swinnerton-Dyer conjecture. We state a general conjecture (Conjecture 7.1.1) in Section 7 according to which every element of the Shafarevich-Tate group of a modular abelian variety becomes visible at higher level. We provide evidence for the the conjecture in Section 7.2 and tables of computational data in Section 7.4.

Acknowledgement: The authors would like to thank David Helm, Ben Howard, Barry Mazur, Bjorn Poonen and Ken Ribet for discussions and comments on the paper.

2 Notation

Visibility of the Shafarevich-Tate Group at Higher Level 3

1. Abelian varieties. For a number field K, A_{/K} denotes an abelian variety over K. We denote the dual of A by A_{/K}^{∨}. If φ : A → B is an isogeny of degree n, we denote the complementary isogeny by φ′; this is the isogeny ϕ′ : B → A, such that φ ∘ φ′ = φ′ ∘ φ = [n], the multiplication-by-n map on A. Unless otherwise specified, Néron models of abelian varieties will be denoted by the corresponding caligraphic letters, e.g., A denotes the Néron model of A.

2. Galois cohomology. For a fixed algebraic closure \overline{K} of K, G_{K} will be the Galois group Gal(\overline{K}/K). If v is any non-archimedean place of K, K_{v} and k_{v} will always mean the completion and the residue field of K at v, respectively. By K_{v}^{ur} we always mean the maximal unramified extension of the completion K_{v}. Given a G_{K}-module M, we let H^{1}(K, M) denote the Galois cohomology group H^{1}(G_{K}, M).

3. Component groups. The component group of A at v is the finite group Φ_{A,v} = A_{k_{v}}/A_{k_{v}}^{0} which also has a structure of a finite group scheme over k_{v}. The Tamagawa number of A at v is c_{A,v} = #Φ_{A,v}(k_{v}), and the component group order of A at v is \overline{c}_{A,v} = #Φ_{A,v}(\overline{k}_{v}).

4. Modular abelian varieties. Let h = 0 or 1. A J_{h}-modular abelian variety is an abelian variety A_{/K} which is a quotient of J_{h}(N) for some N, i.e., there exists a surjective morphism J_{h}(N) ↠ A defined over K. We define the level of a modular abelian variety A to be the minimal N, such that A is a quotient of J_{h}(N). The modularity theorem of Wiles et al. (see [BCDT01]) implies that all elliptic curves over ℚ are modular. Serre’s modularity conjecture implies that the modular abelian varieties over ℚ are precisely the abelian varieties over ℚ of GL_{2}-type (see [Rib92, §4]).

5. Shimura construction. Let f = ∑_{n=1}^{∞} a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be a newform of level

N and weight 2 for Γ_{0}(N) which is an eigenform for all Hecke operators in the Hecke algebra 𝕋(N). Shimura (see [Shi94, Thm. 7.14]) associated to f an abelian subvariety A_{f}_{/ℚ} of J_{0}(N), simple over ℚ, of dimension d = [K : ℚ], where K = ℚ(…, a_{n}, …) is the Hecke eigenvalue field. More precisely, if I_{f} = Ann_{𝕋(N)}(f) then A_{f} is the connected component containing the identity of the I_{f}-torsion subgroup of J_{0}(N), i.e., A_{f} = J_{0}(N)[I_{f}]^{0} ⊂ J_{0}(N). The quotient 𝕋(N)/I_{f} of the Hecke algebra 𝕋(N) is a subalgebra of the endomorphism ring

End_{ℚ}(A_{/ℚ}). Also L(A_{f}, s) = ∏_{i=1}^{d} L(f_{i}, s), where the f_{i} are the G_{ℚ}-conjugates

of f. We also consider the dual abelian variety A_{f}^{∨} which is a quotient variety of J_{0}(N).

6. I-torsion submodules. If M is a module over a commutative ring R and I is an ideal of R, let

M[I] = {x ∈ M : mx = 0 all m ∈ I}

be the I-torsion submodule of M.

7. Hecke algebras. Let S_{2}(Γ) denote the space of cusp forms of weight 2 for any congruence subgroup Γ of SL_{2}(ℤ). Let

𝕋(N) = ℤ[…, T_{n}, …] ⊆ End_{ℚ}(J_{0}(N))

be the Hecke algebra, where T_{n} is the nth Hecke operator. 𝕋(N) also acts on S_{2}(Γ_{0}(N)) and the integral homology H_{1}(X_{0}(N), ℤ).

8. Modular degree. If A is an abelian subvariety of J_{0}(N), let

θ : A → J_{0}(N) ≅ J_{0}(N)^{∨} → A^{∨}

be the induced polarization. The modular degree of A is

m_{A} = \sqrt{# Ker(A ⟶^{θ} A^{∨})}.

See [AS02] for why m_{A} is an integer and for an algorithm to compute it.

3 Visible Subgroups of Shafarevich-Tate Groups

Let K be a number field and ι : A_{/K} ↪ C_{/K} be an embedding of an abelian variety into another abelian variety over K.

Definition 3.0.1. The visible subgroup of H^{1}(K, A) relative to ι is

Vis_{C} H^{1}(K, A) = Ker (ι_{∗} : H^{1}(K, A) → H^{1}(K, C)) .

The visible subgroup of � (K, A) relative to the embedding ι is

Vis_{C} � (K, A) = � (K, A) ∩ Vis_{C} H^{1}(K, A)
= Ker (� (K, A) → � (K, C))

Let Q be the abelian variety C/ι(A), which is defined over K. The long exact sequence of Galois cohomology corresponding to the short exact sequence 0 → A → C → Q → 0 gives rise to the following exact sequence

0 → A(K) → C(K) → Q(K) → Vis_{C} H^{1}(K, A) → 0.

The last map being surjective means that the cohomology classes of Vis_{C} H^{1}(K, A) are images of K-rational points on Q, which explains the meaning of the word visible in the definition. The group Vis_{C} H^{1}(K, A) is finite since it is torsion and since the Mordell-Weil group Q(K) is finitely generated. Remark 3.0.2. If A_{/K} is an abelian variety and c ∈ H^{1}(K, A) is any cohomology class, there exists an abelian variety C_{/K} and an embedding ι : A ↪ C defined over K, such that c ∈ Vis_{C} H^{1}(K, A), i.e., c is visible in C (see [AS02, Prop. 1.3]). The C of [AS02, Prop. 1.3] is the restriction of scalars of A_{L} = A ×_{K} L down to K, where L is any finite extension of K such that c has trivial image in H^{1}(L, A).

4 Equivariant Visibility

Let K be a number field, let A_{/K} and B_{/K} be abelian subvarieties of an abelian variety C_{/K}, such that C = A + B and A ∩ B is finite. Let Q_{/K} denotes the quotient C/B. Let N be a positive integer divisible by all primes of bad reduction for C.

Let ℓ be a prime such that B[ℓ] ⊂ A and e < ℓ − 1, where e is the largest ramification index of any prime of K lying over ℓ. Suppose that

ℓ ∤ N · #B(K)_{tor} · #Q(K)_{tor} · ∏_{v|N} c_{A,v}c_{B,v}.

Under those conditions, Agashe and Stein (see [AS02, Thm. 3.1]) construct a homomorphism B(K)/ℓB(K) → � (K, A)[ℓ] whose kernel has 𝔽_{ℓ}-dimension bounded by the Mordell-Weil rank of A(K).

In this paper, we refine [AS02, Prop. 1.3] by taking into account the algebraic structure coming from the endomorphism ring End_{K}(C). In particular, when we apply the theory to modular abelian varieties, we would like to use the additional structure coming from the Hecke algebra. There are numerous example (see [AS05]) where [AS02, Prop. 1.3] does not apply, but nevertheless, we can use our refinement to prove existence of visible elements of � (ℚ, A_{f}) at higher level (e.g., see Propositions 6.1.3 and 6.2.1 below).

4.1 The main theorem

Let A_{/K}, B_{/K}, C_{/K}, Q_{/K}, N and ℓ be as above. Let R be a commutative subring of End_{K}(C) that leaves A and B stable and let 𝔪 be a maximal ideal of R of residue characteristic ℓ. By the Néron mapping property, the subgroups Φ_{A,v}(k_{v}) and Φ_{B,v}(k_{v}) of k_{v}-points of the corresponding component groups can be viewed as R-modules.

Theorem 4.1.1 (Equivariant Visibility Theorem). Suppose that A(K) has rank zero and that the groups Q(K)[𝔪], B(K)[𝔪], Φ_{A,v}(k_{v})[𝔪] and Φ_{B,v}(k_{v})[ℓ] are all trivial for all nonarchimedean places v of K. Then there is an injective homomorphism of R/𝔪-vector spaces

(B(K)/ℓB(K))[𝔪] ↪ Vis_{C}(� (K, A))[𝔪].  (1)

Remark 4.1.2. Applying the above result for R = ℤ, we recover the result of Agashe and Stein in the case when A(K) has Mordell-Weil rank zero. We could relax the hypothesis that A(K) is finite and instead give a bound on the dimension of the kernel of (1) in terms of the rank of A(K) similar to the bound in [AS02, Thm. 3.1]. We will not need this stronger result in our paper.

4.2 Some commutative algebra

Before proving Theorem 4.1.1 we recall some well-known lemmas from commutative algebra. Let M be a module over a commutative ring R and let 𝔪 be a finitely generated prime ideal of R.

Lemma 4.2.1. If M_{𝔪} is Artinian, then M_{𝔪} ≠ 0 ⟺ M[𝔪] ≠ 0.

Proof. (⟸) We first prove that M_{𝔪} = 0 implies M[𝔪] = 0 by a slight modification of the proof of [AM69, Prop. I.3.8]. Suppose M_{𝔪} = 0, yet there is a nonzero x ∈ M[𝔪]. Let I = Ann_{R}(x). Then I ≠ (1) is an ideal that contains 𝔪, so I = 𝔪. Consider \frac{x}{1} ∈ M_{𝔪}. Since M_{𝔪} = 0, we have x/1 = 0, hence by definition of localization, x is killed by some element of R − 𝔪 (set-theoretic difference). But this is impossible since Ann_{R}(x) = 𝔪.

(⟹) Next we prove that M_{𝔪} ≠ 0 implies M[𝔪] ≠ 0. Since M_{𝔪} is an Artinian module over the (local) ring R_{𝔪}, by [AM69, Prop. 6.8], M_{𝔪} has a composition series:

M_{𝔪} = M_{0} ⊃ M_{1} ⊃ ⋯ ⊃ M_{n−1} ⊃ M_{n} = 0,

where by definition each quotient M_{i}/M_{i+1} is a simple R_{𝔪}-module. In particular, M_{n−1} is a simple R_{𝔪}-module. Suppose x ∈ M_{n−1} is nonzero, and let I = Ann_{R_{𝔪}}(x). Then

R_{𝔪}/I ≅ R_{𝔪} · x ⊂ M_{n−1},

so by simplicity R_{𝔪}/I ≅ M_{n−1} is simple. Thus I = 𝔪, otherwise R_{𝔪}/I would have 𝔪/I as a proper submodule. Thus x ∈ M_{n−1}[𝔪] is nonzero.

Write x = [y, a] with y ∈ M and a ∈ R − 𝔪, where [y/a] means the class of y/a in the localization (same as (y, a) on page 36 of [AM69]). Since a ∈ R − 𝔪, the element a acts as a unit on M_{𝔪}, hence ax = [y/1] ∈ M_{n−1} is nonzero and also still annihilated by 𝔪 (by commutativity).

To say that [y/1] is annihilated by 𝔪 means that for all α ∈ 𝔪 there exists t ∈ R − 𝔪 such that tαy = 0 in M. Since 𝔪 is finitely generated, we can write 𝔪 = (α_{1}, …, α_{n}) and for each α_{i} we get corresponding elements t_{1}, …, t_{n} and a product t = t_{1} ⋯ t_{n}. Also t ∉ 𝔪 since 𝔪 is a prime ideal and each t_{i} ∉ 𝔪. Let z = ty. Then for all α ∈ 𝔪 we have αz = tαy = 0. Also z ≠ 0 since t acts as a unit on M_{n−1}. Thus z ∈ M[𝔪], and is nonzero, which completes the proof of the lemma.

Lemma 4.2.2. Suppose 0 → M_{1} → N → M_{2} → 0 is an exact sequence of R-modules each of whose localization at 𝔪 is Artinian. Then N[𝔪] ≠ 0 ⟺ (M_{1} ⊕ M_{2})[𝔪] ≠ 0.

Proof. By Lemma 4.2.1 we have N[𝔪] ≠ 0 if and only if N_{𝔪} ≠ 0. By Proposition 3.3 on page 39 of [AM69], the localized sequence

0 → (M_{1})_{𝔪} → N_{𝔪} → (M_{2})_{𝔪} → 0

is exact. Thus N_{𝔪} ≠ 0 if and only if at least one of (M_{1})_{𝔪} or (M_{2})_{𝔪} is nonzero. Again by Lemma 4.2.1, at least one of (M_{1})_{𝔪} or (M_{2})_{𝔪} is nonzero if and only if at least one of M_{1}[𝔪] or M_{2}[𝔪] is nonzero. The latter is the case if and only if (M_{1} ⊕ M_{2})[𝔪] ≠ 0.

Remark 4.2.3. One could also prove the lemmas using the isomorphism M[𝔪] ≅ Hom_{R}(R/𝔪, M) and exactness properties of Hom, but even with this approach many of the details in Lemma 4.2.1 still have to be checked.

Remark 4.2.4. In Theorem 4.1.1, we have R ⊂ End(C), hence R is finitely generated as a ℤ-module, so R is noetherian.

Lemma 4.2.5. Let G be a finite cyclic group, M be a finite G-module that is also a module over a commutative ring R such that the action of G and R commute (i.e., M is an R[G]-module). Suppose 𝔭 is a finitely-generated prime ideal of R, and H^{0}(G, M)[𝔭] = 0. Then H^{1}(G, M)[𝔭] = 0.

Proof. Argue as in [Se79, Prop. VIII.4.8], but noting that all modules are modules over R and maps are morphisms of R-modules.

4.3 Proof of Theorem 4.1.1

Proof of Theorem 4.1.1. We argue as in the proof of [AS02, Thm. 3.1]. The construction of the map (1) is similar to the one in the proof of [AS02, Lem. 3.6]. We have the commutative diagram

\xymatrix{0 \ar[r] & B[ℓ] \ar[r] \ar[d] & B \ar[d] \ar[r]^{ℓ} \ar[dr]^{ψ} & B \ar[d]^{π} \ar[r] & 0 \\ 0 \ar[r] & A \ar[r] & C \ar[r] & Q \ar[r] & 0,}

where ψ : B → Q is the composition of the inclusion B ↪ C with the quotient map C → Q, and the existence of the morphism π : B → Q follows from the inclusion B[ℓ] ⊂ Ker(ψ) = A ∩ B. By naturality for the long exact sequence of Galois cohomology we obtain the following commutative diagram with exact rows and columns

\xymatrix{ & M_{0} \ar[d] & M_{1} \ar[d] & M_{2} \ar[d] \\ 0 \ar[r] & B(K)/(B(K)[ℓ]) \ar[r]^{ℓ} \ar[d] & B(K) \ar[dr]^{π} \ar[r] \ar[d] & B(K)/ℓB(K) \ar[r] \ar[d]^{φ} & 0 \\ 0 \ar[r] & C(K)/A(K) \ar[r] \ar[d] & Q(K) \ar[r] & Vis_{C}(H^{1}(K, A)) \ar[r] & 0 \\  & M_{3}.}

Here, M_{0}, M_{1} and M_{2} denote the kernels of the corresponding vertical maps and M_{3} denotes the cokernel of the first map. Since R preserves A, B, and B[ℓ], all objects in the diagram are R-module and the morphisms of abelian varieties are also R-module homomorphisms.

The snake lemma yields an exact sequence

0 → M_{0} → M_{1} → M_{2} → M_{3}.

By hypothesis, B(K)[𝔪] = 0, so N_{0} = Ker(B(K) → C(K)/A(K)) has no 𝔪 torsion. Noting that B(K)[ℓ] ⊂ N_{0}, it follows that M_{0} = N_{0}/(B(K)[ℓ]) has no 𝔪 torsion either, by Lemma 4.2.2. Also, M_{1}[𝔪] = 0 again since B(K)[𝔪] = 0.

By the long exact sequence on Galois cohomology, the quotient C(K)/B(K) is isomorphic to a subgroup of Q(K) and by hypothesis Q(K)[𝔪] = 0, so (C(K)/B(K))[𝔪] = 0. Since Q is isogenous to A and A(K) is finite and C(K)/B(K) ↪ Q(K), we see that C(K)/B(K) is finite. Thus M_{3} is a quotient of the finite R-module C(K)/B(K), which has no 𝔪-torsion, so Lemma 4.2.2 implies that M_{3}[𝔪] = 0. The same lemma implies that M_{1}/M_{0} has no 𝔪torsion, since it is a quotient of the finite module M_{1}, which has no 𝔪-torsion. Thus, we have an exact sequence

0 → M_{1}/M_{0} → M_{2} → M_{3} → 0,

and both of M_{1}/M_{0} and M_{3} have trivial 𝔪-torsion. It follows by Lemma 4.2.2, that M_{2}[𝔪] = 0. Therefore, we have an injective morphism of R/𝔪-vector spaces

φ : (B(K)/ℓB(K))[𝔪] ↪ Vis_{C}(H^{1}(K, A))[𝔪].

It remains to show that for any x ∈ B(K), we have φ(x) ∈ Vis_{C}(� (K, A)), i.e., that φ(x) is locally trivial.

We proceed exactly as in Section 3.5 of [AS05]. In both cases char(v) ≠ ℓ and char(v) = ℓ we arrive at the conclusion that the restriction of φ(x) to H^{1}(K_{v}, A) is an element c ∈ H^{1}(K_{v}^{ur}/K_{v}, A(K_{v}^{ur})). (Note that in the case char(v) ≠ ℓ the proof uses our hypothesis that ℓ ∤ #Φ_{B,v}(k_{v}).) By [Mil86, Prop I.3.8], there is an isomorphism

H^{1}(K_{v}^{ur}/K_{v}, A(K_{v}^{ur})) ≅ H^{1}(\overline{k}_{v}/k_{v}, Φ_{A,v}(\overline{k}_{v})).  (2)

We will use our hypothesis that

Φ_{A,v}(k_{v})[𝔪] = Φ_{B,v}(k_{v})[ℓ] = 0

for all v of bad reduction to deduce that the image of φ lies in Vis_{C}(� (K, A))[𝔪]. Let d denote the image of c in H^{1}(\overline{k}_{v}/k_{v}, Φ_{A,v}(\overline{k}_{v})). The construction of d is compatible with the action of R on Galois cohomology, since (as is explained in the proof of [Mil86, Prop. I.3.8]) the isomorphism (2) is induced from the exact sequence of Gal(K_{v}^{ur}/K_{v})-modules

0 → A^{0}(K_{v}^{ur}) → A(K_{v}^{ur}) → Φ_{A,v}(\overline{k}_{v}) → 0,

where A is the Néron model of A and A^{0} is the subgroup scheme whose generic fiber is A and whose closed fiber is the identity component of A_{k_{v}}. Since φ(x) ∈ H^{1}(K, A)[𝔪], it follows that

d ∈ H^{1}(\overline{k}_{v}/k_{v}, Φ_{A,v}(\overline{k}_{v}))[𝔪].

Lemma 4.2.5, our hypothesis that Φ_{A,v}(k_{v})[𝔪] = 0, and that

H^{1}(\overline{k}_{v}/k_{v}, Φ_{A,v}(\overline{k}_{v})) = \underrightarrow{lim} H^{1}(Gal(k′_{v}/k_{v}), Φ_{A,v}(k′_{v}))),

together imply that H^{1}(\overline{k}_{v}/k_{v}, Φ_{A,v}(\overline{k}_{v}))[𝔪] = 0, hence d = 0. Thus c = 0, so φ(x) is locally trivial, which completes the proof.

5 Strong Visibility at Higher Level

5.1 Strongly visible subgroups

Let A_{/ℚ} be an abelian subvariety of J_{0}(N)_{/ℚ} and let p ∤ N be a prime. Let

φ = δ_{1}^{∗} + δ_{p}^{∗} : J_{0}(N) → J_{0}(pN),  (3)

where δ_{1}^{∗} and δ_{p}^{∗} are the pullback maps on equivalence classes of degree-zero divisors of the degeneracy maps δ_{1}, δ_{p} : X_{0}(pN) → X_{0}(N). Let H^{1}(ℚ, A)^{odd} be the prime-to-2-part of the group H^{1}(ℚ, A).

Definition 5.1.1 (Strongly Visibility). The strongly visible subgroup of H^{1}(ℚ, A) for J_{0}(pN) is

Vis_{pN} H^{1}(ℚ, A) = Ker (H^{1}(ℚ, A)^{odd} ⟶^{φ_{∗}} H^{1}(ℚ, J_{0}(pN))) ⊂ H^{1}(ℚ, A).

Also,

Vis_{pN} � (ℚ, A) = � (ℚ, A) ∩ Vis_{pN} H^{1}(ℚ, A).

The reason we replace H^{1}(ℚ, A) by H^{1}(ℚ, A)^{odd} is that the kernel of φ is a 2-group (see [Rib90b]).

Remark 5.1.2. We could obtain more visible subgroups by considering the map δ_{1}^{∗} − δ_{p}^{∗} in Definition 5.1.1. However, the methods of this paper do not apply to this map.

For a positive integer N, let

ν(N) = \frac{1}{6} · ∏_{q^{r}∥N}(q^{r} + q^{r−1}) = \frac{1}{6} · [SL_{2}(ℤ) : Γ_{0}(N)].

We call the number ν(N) the Sturm bound (see [Stu87]).

Theorem 5.1.3. Let A_{/ℚ} = A_{f} be a newform abelian subvariety of J_{0}(N) for which L(A_{/ℚ}, 1) ≠ 0 and let p ∤ N be a prime. Suppose that there is a maximal ideal λ ⊂ 𝕋(N) and an elliptic curve E_{/ℚ} of conductor pN such that:

1. [Nondivisibility] The residue characteristic ℓ of λ satisfies

ℓ ∤ 2 · N · p · ∏_{q|N} c_{E,q}.

2. [Component Groups] For each prime q | N,

Φ_{A,q}(𝔽_{q})[λ] = 0.

3. [Fourier Coefficients] Let a_{n}(E) be the n-th Fourier coefficient of the modular form attached to E, and a_{n}(f) the n-th Fourier coefficient of f. Assume that a_{p}(E) = −1,

a_{p}(f) ≡ −(p + 1) (mod λ) and a_{q}(f) ≡ a_{q}(E) (mod λ),

for all primes q ≠ p with q ≤ ν(pN).

4. [Irreducibility] The mod ℓ representation \overline{ρ}_{E,ℓ} is irreducible.

Then there is an injective homomorphism

E(ℚ)/ℓE(ℚ) ↪ Vis_{pN}(� (ℚ, A_{f}))[λ].

Remark 5.1.4. In fact, we have

E(ℚ)/ℓE(ℚ) ↪ Ker(� (ℚ, A_{f}) → � (ℚ, C))[λ] ⊂ Vis_{pN}(� (ℚ, A_{f}))[λ],

where C ⊂ J_{0}(pN) is isogenous to A_{f} × E.

5.2 Some auxiliary lemmas

We will use the following lemmas in the proof of Theorem 5.1.3. The notation is as in the previous section. In addition, if f ∈ S_{2}(Γ_{0}(N)), we denote by a_{n}(f) the n-th Fourier coefficient of f and by K_{f} and O_{f} the Hecke eigenvalue field and its ring of integers, respectively.

Lemma 5.2.1. Suppose A_{f} ⊂ J_{0}(N) and A_{g} ⊂ J_{0}(pN) are attached to newforms f and g of level N and pN, respectively, with p ∤ N. Suppose that there is a prime ideal λ of residue characteristic ℓ ∤ 2pN in an integrally closed subring O of \overline{ℚ} that contains the ring of integers of the composite field K = K_{f}K_{g} such that for q ≤ ν(pN),

a_{q}(f) ≡ \begin{cases}a_{q}(g) (mod λ) & if q ≠ p, \\ (p + 1)a_{p}(g) (mod λ) & if q = p. \end{cases}

Assume that a_{p}(g) = −1. Let λ_{f} = O_{f} ∩ λ and λ_{g} = O_{g} ∩ λ and assume that A_{f}[λ_{f}] is an irreducible G_{ℚ}-module. Then we have an equality

φ(A_{f}[λ_{f}]) = A_{g}[λ_{g}]

of subgroups of J_{0}(pN), where φ is as in (3).

Proof. Our hypothesis that a_{p}(f) ≡ −(p + 1) (mod λ_{f}) implies, by the proofs in [Rib90b], that

φ(A_{f}[λ_{f}]) ⊂ φ(A_{f}) ∩ J_{0}(pN)_{p-new},

where J_{0}(pN)_{p-new} is the p-new abelian subvariety of J_{0}(N).

By [Rib90b, Lem. 1], the operator U_{p} = T_{p} on J_{0}(pN) acts as −1 on φ(A_{f}[λ_{f}]). Consider the action of U_{p} on the 2-dimensional vector space spanned by {f(q), f(q^{p})}. The matrix of U_{p} with respect to this basis is

U_{p} = (\begin{matrix}a_{p}(f) & p \\ −1 & 0 \end{matrix}) .

In particular, neither of f(q) and f(q^{p}) is an eigenvector for U_{p}. The characteristic polynomial of U_{p} acting on the span of f(q) and f(q^{p}) is x^{2} − a_{p}(f)x + p. Using our hypothesis on a_{p}(f) again, we have

x^{2} − a_{p}(f)x + p ≡ x^{2} + (p + 1)x + p ≡ (x + 1)(x + p) (mod λ).

Thus we can choose an algebraic integer α such that

f_{1}(q) = f(q) + αf(q^{p})

is an eigenvector of U_{p} with eigenvalue congruent to −1 modulo λ. (It does not matter for our purposes whether x^{2} +a_{p}(f)x+p has distinct roots; nonetheless, since p ∤ N, [CV92, Thm. 2.1] implies that it does have distinct roots.) The cusp form f_{1} has the same prime-indexed Fourier coefficients as f at primes other than p. Enlarge O if necessary so that α ∈ O. The p-th coefficient of f_{1} is congruent modulo λ to −1 and f_{1} is an eigenvector for the full Hecke algebra. It follows from the recurrence relation for coefficients of the eigenforms that

a_{n}(g) ≡ a_{n}(f_{1}) (mod λ)

for all integers n ≤ ν(pN).

By [Stu87], we have g ≡ f_{1} (mod λ), so a_{q}(g) ≡ a_{q}(f) (mod λ) for all primes q ≠ p. Thus by the Brauer-Nesbitt theorem [CR62], the 2-dimensional G_{ℚ}-representations φ(A_{f}[λ_{f}]) and A_{g}[λ_{g}] are isomorphic.

Let 𝔪 be a maximal ideal of the Hecke algebra 𝕋(pN) that annihilates the module A_{g}[λ_{g}]. Note that A_{g}[𝔪] = A_{g}[λ_{g}] since A_{g}[𝔪] ⊂ A_{g}[λ_{g}] and A_{g}[λ_{g}] ≅ φ(A_{f}[λ_{f}]) is irreducible as a G_{ℚ}-module. The maximal ideal 𝔪 gives rise to a Galois representation \overline{ρ}_{𝔪} : G_{ℚ} → GL_{2}(𝕋(pN)/𝔪) isomorphic to A_{g}[λ_{g}], which is irreducible since the Galois module A_{f}[λ_{f}] is irreducible. Finally, we apply [Wil95, Thm. 2.1(i)] for H = (ℤ/Nℤ)^{×} (i.e., J_{H} = J_{0}(N)) to conclude that J_{0}(N)(\overline{ℚ})[𝔪] ≅ (𝕋(pN)/𝔪)^{2}, i.e., the representation \overline{ρ}_{𝔪} occurs with multiplicity one in J_{0}(pN). Thus

A_{g}[λ_{g}] = φ(A_{f}[λ_{f}]).

Lemma 5.2.2. Suppose φ : A → B and ψ : B → C are homomorphisms of abelian varieties over a number field K, with φ an isogeny and ψ injective. Suppose n is an integer that is relatively prime to the degree of φ. If G = Vis_{C}(� (ℚ, B))[n^{∞}], then there is some injective homomorphism

f : G ↪ Ker {(ψ ∘ φ)_{∗} : � (ℚ, A) ⟶ � (ℚ, C)} ,

such that φ_{∗}(f(G)) = G.

Proof. Let m be the degree of the isogeny φ : A → B. Consider the complementary isogeny φ′ : B → A, which satisfies φ ∘ φ′ = φ′ ∘ φ = [m]. By hypothesis m is coprime to n, so gcd(m, #G) = gcd(m, n^{∞}) = 1, hence

φ_{∗}(φ′_{∗}(G)) = [m]G = G.

Thus φ′_{∗}(G) maps, via φ_{∗}, to G ⊂ � (ℚ, B), which in turn maps to 0 in � (ℚ, C).

Lemma 5.2.3. Let M be an odd integer coprime to N and let R be the subring of 𝕋(N) generated by all Hecke operators T_{n} with gcd(n, M) = 1. Then R = 𝕋(N).

Proof. See the lemma on page 491 of [Wil95]. (The condition that M is odd is necessary, as there is a counterexample when N = 23 and M = 2.)

Lemma 5.2.4. Suppose λ is a maximal ideal of 𝕋(N) with generators a prime ℓ and T_{n} − a_{n} (for all n ∈ ℤ), with a_{n} ∈ ℤ. For each integer p ∤ N, let λ_{p} be the ideal in 𝕋(N) generated by ℓ and all T_{n} − a_{n}, where n varies over integers coprime to p. Then λ = λ_{p}.

Proof. Since λ_{p} ⊂ λ and λ is maximal, it suffices to prove that λ_{p} is maximal. Let R be the subring of 𝕋(N) generated by Hecke operators T_{n} with p ∤ n. The quotient R/λ_{p} is a quotient of ℤ since each generator T_{n} is equivalent to an integer. Also, ℓ ∈ λ_{p}, so R/λ_{p} = 𝔽_{ℓ}. But by Lemma 5.2.3, R = 𝕋(N), so 𝕋(N)/λ_{p} = 𝔽_{ℓ}, hence λ_{p} is a maximal ideal.

Lemma 5.2.5. Suppose that A is an abelian variety over a field K. Let R be a commutative subring of End(A) and I an ideal of R. Then

(A/A[I])[I] ≅ A[I^{2}]/A[I],

where the isomorphism is an isomorphism of R[G_{K}]-modules.

Proof. Let a + A[I], for some a ∈ A, be an I-torsion element of A/A[I]. Then by definition, xa ∈ A[I] for each x ∈ I. Therefore, a ∈ A[I^{2}]. Thus a + A[I] ↦ a + A[I] determines a well-defined homomorphism of R[G_{K}]-modules

φ : (A/A[I])[I] → A[I^{2}]/A[I].

Clearly this homomorphism is injective. It is also surjective as every element a + A[I] ∈ A[I^{2}]/A[I] is I-torsion as an element of A/A[I], as Ia ∈ A[I]. Therefore, φ is an isomorphism of R[G_{K}]-modules.

Lemma 5.2.6. Let ℓ be a prime and let ϕ : E → E′ be an isogeny of elliptic curves of degree coprime to ℓ defined over a number field K. If v is any place of K then ℓ | c_{E,v} if and only if ℓ | c_{E′,v}.

Proof. Consider the complementary isogeny ϕ′ : E′ → E. Both ϕ and ϕ′ induce homomorphisms ϕ : Φ_{E,v}(k_{v}) → Φ_{E′,v}(k_{v}) and ϕ′ : Φ_{E′,v}(k_{v}) → Φ_{E,v}(k_{v}) and ϕ ∘ ϕ′ and ϕ′ ∘ ϕ are multiplication-by-n maps. Since (n, ℓ) = 1 then # ker ϕ and # ker ϕ′ must be coprime to ℓ which implies the statement.

5.3 Proof of Theorem 5.1.3

Proof of Theorem 5.1.3. By [BCDT01] E is modular, so there is a rational newform f ∈ S_{2}^{new}(pN) which is an eigenform for the Hecke operators and an isogeny E → E_{f} defined over ℚ, which by Hypothesis 4 can be chosen to have degree coprime to ℓ. Indeed, every cyclic rational isogeny is a composition of rational isogenies of prime degree, and E admits no rational ℓ-isogeny since \overline{ρ}_{E,ℓ} is irreducible.

By Hypothesis 1 the Tamagawa numbers of E are coprime to ℓ. Since E and E_{f} are related by an isogeny of degree coprime to ℓ, the Tamagawa numbers of E_{f} are also not divisible by ℓ by Lemma 5.2.6. Moreover, note that

E(ℚ) ⊗ 𝔽_{ℓ} ≅ E_{f}(ℚ) ⊗ 𝔽_{ℓ}.

Let 𝔪 be the ideal of 𝕋(pN) generated by ℓ and T_{n} − a_{n}(E) for all integers n coprime to p. Note that 𝔪 is maximal by Lemma 5.2.4.

Let φ be as in (3), and let A = φ(A_{f}). Note that if T_{n} ∈ 𝕋(pN) then T_{n}(E_{f}) ⊂ E_{f} since E_{f} is attached to a newform, and if, moreover p ∤ n, then T_{n}(A) ⊂ A since the Hecke operators with index coprime to p commute with the degeneracy maps. Lemma 5.2.1 implies that

E_{f}[ℓ] = E_{f}[𝔪] = φ(A_{f}[λ]) ⊂ A,

so Ψ = E_{f}[ℓ] is a subgroup of A as a G_{ℚ}-module. Let

C = (A × E_{f})/Ψ,

where we embed Ψ in A × E_{f} anti-diagonally, i.e., by the map x ↦ (x, −x). The antidiagonal map Ψ → A × E_{f} commutes with the Hecke operators T_{n} for p ∤ n, so (A × E_{f})/Ψ is preserved by the T_{n} with p ∤ n. Let R be the subring of End(C) generated by the action of all Hecke operators T_{n}, with p ∤ n. Also note that T_{p} ∈ End(J_{0}(pN)) acts by Hypothesis 3 as −1 on E_{f}, but T_{p} need not preserve A.

Suppose for the moment that we have verified that the hypothesis of Theorem 4.1.1 are satisfied with A, B = E_{f}, C, Q = C/B, R as above and K = ℚ. Then we obtain an injective homomorphism

E(ℚ)/ℓE(ℚ) ≅ E_{f}(ℚ)/ℓE_{f}(ℚ) ↪ Ker(� (ℚ, A) → � (ℚ, C))[𝔪].

We then apply Lemma 5.2.2 with n = ℓ, A_{f}, A, and C, respectively, to see that

E_{f}(ℚ)/ℓE_{f}(ℚ) ⊂ Ker(� (ℚ, A_{f}) → � (ℚ, C))[λ].

That E_{f}(ℚ)/ℓE_{f}(ℚ) lands in the λ-torsion is because the subgroup of Vis_{C}(� (ℚ, E_{f})) that we constructed is 𝔪-torsion.

Finally, consider A × E_{f} → J_{0}(pN) given by (x, y) ↦ x + y. Note that Ψ maps to 0, since (x, −x) ↦ 0 and the elements of Ψ are of the form (x, −x). We have a (not-exact!) sequence of maps

� (ℚ, A_{f}) → � (ℚ, C) → � (ℚ, J_{0}(pN)),

hence inclusions

E_{f}(ℚ)/ℓE_{f}(ℚ) ⊆ Ker(� (ℚ, A_{f}) → � (ℚ, C))
⊆ Ker(� (ℚ, A_{f}) → � (ℚ, J_{0}(pN))),

which gives the conclusion of the theorem.

It remains to verify the hypotheses of Theorem 4.1.1. That C = A + B is clear from the definition of C. Also, A ∩ E_{f} = E_{f}[ℓ], which is finite. We explained above when defining R that each of A and E_{f} is preserved by R. Since K = ℚ and ℓ is odd the condition 1 = e < ℓ − 1 is satisfied. That A(ℚ) is finite follows from our hypothesis that L(A_{f}, 1) ≠ 0 (by [KL89]).

It remains is to verify that the groups

Q(ℚ)[𝔪], E_{f}(ℚ)[𝔪], Φ_{A,q}(𝔽_{q})[𝔪], and Φ_{E_{f} ,q}(𝔽_{q})[ℓ],

are 0 for all primes q | pN. Since ℓ ∈ 𝔪, we have by Hypothesis 4 that

E_{f}(ℚ)[𝔪] = E_{f}(ℚ)[ℓ] = 0.

We will now verify that Q(ℚ)[𝔪] = 0. From the definition of C and Ψ we have Q ≅ A/Ψ. Let λ_{p} be as in Lemma 5.2.4 with a_{n} = a_{n}(E). The map φ induces an isogeny of 2-power degree

A_{f}/(A_{f}[λ]) → A/Ψ.

Thus there is λ_{p}-torsion in (A_{f}/(A_{f}[λ]))(ℚ) if and only if there is 𝔪-torsion in (A/Ψ)(ℚ). Thus it suffices to prove that (A_{f}/A_{f}[λ])(ℚ)[λ_{p}] = 0.

By Lemma 5.2.4, we have λ_{p} = λ, and by Lemma 5.2.5,

(A_{f}/A_{f}[λ])[λ] ≅ A_{f}[λ^{2}]/A_{f}[λ].

By [Maz77, §II.14], the quotient A_{f}[λ^{2}]/A_{f}[λ] injects into a direct sum of copies of A_{f}[λ] as Galois modules. But A_{f}[λ] ≅ E[ℓ] is irreducible, so (A_{f}[λ^{2}]/A_{f}[λ])(ℚ) = 0, as required.

By Hypothesis 2, we have Φ_{A_{f} ,q}(𝔽_{q})[λ] = 0 for each prime divisor q | N.Since A is 2-power isogenous to A_{f} and ℓ is odd, this verifies the Tamagawa number hypothesis for A. Our hypothesis that a_{p}(E) = −1 implies that Frob_{p} acts on Φ_{E_{f} ,p}(\overline{𝔽}_{p}) as −1. Thus Φ_{E_{f} ,p}(𝔽_{p})[ℓ] = 0 since ℓ is odd. This completes the proof.

Remark 5.3.1. An essential ingrediant in the proof of the above theorem is the multiplicity one result used in the paper of Wiles (see [Wil95, Thm. 2.1.]). Since this result holds for Jacobians J_{H} of the curves X_{H}(N) that are intermediate covers for the covering X_{1}(N) → X_{0}(N) corresponding to subgroups H ⊆ (ℤ/Nℤ)^{×} (i.e., the Galois group of X_{1}(N) → X_{H} is H), one should be able to give a generalization of Theorem 5.1.3 which holds for newform subvarieties of J_{H}. This requires generalizing some results from [Rib90b] to arbitrary H.

5.4 A Variant of Theorem 5.1.3 with Simpler Hypothesis

Proposition 5.4.1. Suppose A = A_{f} ⊂ J_{0}(N) is a newform abelian variety and q is a prime that exactly divides N. Suppose 𝔪 ⊂ 𝕋(N) is a non-Eisenstein maximal ideal of residue characteristic ℓ and that ℓ ∤ m_{A}, where m_{A} is the modular degree of A. Then Φ_{A,q}(\overline{𝔽}_{q})[𝔪] = 0.

Proof. The component group of Φ_{J_{0}(N),q}(\overline{𝔽}_{q}) is Eisenstein by [Rib87], so

Φ_{J_{0}(N),q}(\overline{𝔽}_{q})[𝔪] = 0.

By Lemma 4.2.2, the image of Φ_{J_{0}(N),q}(\overline{𝔽}_{q}) in Φ_{A^{∨},q}(\overline{𝔽}_{q}) has no 𝔪 torsion. By the main theorem of [CS01], the cokernel Φ_{J_{0}(N),q}(\overline{𝔽}_{q}) in Φ_{A^{∨},q}(\overline{𝔽}_{q}) has order that divides m_{A}. Since ℓ ∤ m_{A}, it follows that the cokernel also has no 𝔪 torsion. Thus Lemma 4.2.2 implies that Φ_{A^{∨},q}(\overline{𝔽}_{q})[𝔪] = 0. Finally, the modular polarization A → A^{∨} has degree m_{A}, which is coprime to ℓ, so the induced map Φ_{A,q}(\overline{𝔽}_{q}) → Φ_{A^{∨},q}(\overline{𝔽}_{q}) is an isomorphism on ℓ primary parts. In particular, that Φ_{A^{∨},q}(\overline{𝔽}_{q})[𝔪] = 0 implies that Φ_{A,q}(\overline{𝔽}_{q})[𝔪] = 0.

If E is a semistable elliptic curve over ℚ with discriminant Δ, then we see using Tate curves that \overline{c}_{p} = ord_{p}(Δ).

Theorem 5.4.2. Suppose A = A_{f} ⊂ J_{0}(N) is a newform abelian variety with L(A_{/ℚ}, 1) ≠ 0 and N square free, and let ℓ be a prime. Suppose that p ∤ N is a prime, and that there is an elliptic curve E of conductor pN such that:

1. [Rank] The Mordell-Weil rank of E(ℚ) is positive.

2. [Divisibility] We have a_{p}(E) = −1, ℓ | \overline{c}_{E,p}, and

ℓ ∤ 2 · N · p · c_{E,p} · ∏_{q|N} \overline{c}_{E,q}.

3. [Irreducibility] The mod ℓ representation \overline{ρ}_{E,ℓ} is irreducible.

4. [Noncongruence] The representation \overline{ρ}_{E,ℓ} is not isomorphic to any representation \overline{ρ}_{g,λ} where g ∈ S_{2}(Γ_{0}(N)) is a newform of level dividing N that is not conjugate to f.

Then there is an element of order ℓ in � (ℚ, A_{f}) that is not visible in J_{0}(N) but is strongly visible in J_{0}(pN). More precisely, there is an inclusion

E(ℚ)/ℓE(ℚ) ↪ Ker(� (ℚ, A_{f}) → � (ℚ, C))[λ] ⊂ Vis_{pN}(� (ℚ, A_{f}))[λ],

where C ⊂ J_{0}(pN) is isogenous to A_{f} × E, the homomorphism A_{f} → C has degree a power of 2, and λ is the maximal ideal of 𝕋(N) corresponding to \overline{ρ}_{E,ℓ}.

Proof. The divisibility assumptions of Hypothesis 2 on the \overline{c}_{E,q} imply that the Serre level of \overline{ρ}_{E,ℓ} is N and since ℓ ∤ N, the Serre weight is 2 (see [RS01, Thm. 2.10]). Since ℓ is odd, Ribet’s level lowering theorem [Rib91] implies that there is some newform h = ∑ b_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) and a maximal ideal λ over ℓ such that a_{q}(E) ≡ b_{q} (mod λ) for all primes q ≠ p. By our non-congruence hypothesis, the only possibility is that h is a G_{ℚ}-conjugate of f. Since we can replace f by any Galois conjugate of f without changing A_{f}, we may assume that f = h. Also a_{p}(f) ≡ −(p + 1) (mod λ), as explained in [Rib83, pg. 506].

Hypothesis 3 implies that λ is not Eisenstein, and by assumption ℓ ∤ m_{A}, so Proposition 5.4.1 implies that Φ_{A,q}(\overline{𝔽}_{q})[λ] = 0 for each q | N.

The theorem now follows from Theorem 5.1.3.

Remark 5.4.3. The condition a_{p}(E) = −1 is redundant. Indeed, we have \overline{c}_{E,p} ≠ c_{E,p} since \overline{c}_{E,p} is divisible by ℓ and c_{E,p} is not. By studying the action of Frobenius on the component group at p one can show that this implies that E has nonsplit multiplicative reduction, so a_{p}(E) = −1.

Remark 5.4.4. The non-congruence hypothesis of Theorem 5.4.2 can be verified using modular symbols as follows. Let W ⊂ H_{1}(X_{0}(N), ℤ)_{new} be the saturated submodule of H_{1}(X_{0}(N), ℤ) that corresponds to all newforms in S_{2}(Γ_{0}(N)) that are not Galois conjugate to f. Let \overline{W} = W ⊗ 𝔽_{ℓ}. We require that the intersection of the kernels of T_{q}|\overline{W} − a_{q}(E), for q ≠ p, has dimension 0.

6 Computational Examples

In this section we give examples that illustrate how to use Theorem 5.4.2 to prove existence of elements of the Shafarevich-Tate group of a newform subvariety of J_{0}(N) (for 767 and 959) which are invisible at the base level, but become visible in a modular Jacobian of higher level.

Hypothesis 6.0.5. The statements in this section all make the hypothesis that certain commands of the computer algebra system Magma [BCP97] produce correct output.

6.1 Level 767

Consider the modular Jacobian J_{0}(767). Using the modular symbols package in Magma, one decomposes J_{0}(767) (up to isogeny) into a product of six optimal quotients of dimensions 2, 3, 4, 10, 17 and 23. The duals of these quotients are subvarieties A_{2}, A_{3}, A_{4}, A_{10}, A_{17} and A_{23} defined over ℚ, where A_{d} has dimension d. Consider the subvariety A_{23}.

We first show that the Birch and Swinnerton-Dyer conjectural formula predicts that the orders of the groups � (ℚ, A_{23}) and � (ℚ, A_{23}^{∨}) are both divisible by 9.

Proposition 6.1.1. Assume [AS05, Conj. 2.2]. Then

3^{2} | #� (ℚ, A_{23}) and 3^{2} | #� (ℚ, A_{23}^{∨}).

Proof. Let A = A_{23}^{∨}. We use [AS05, §3.5 and §3.6] (see also [Ka81]) to compute a multiple of the order of the torsion subgroup A(ℚ)_{tor}. This multiple is obtained by injecting the torsion subgroup into the group of 𝔽_{p}-rational points on the reduction of A for odd primes p of good reduction and then computing the order of that group. Hence, the multiple is an isogeny invariant, so one gets the same multiple for A^{∨}(ℚ)_{tor}. For producing a divisor of #A(ℚ)_{tor}, we use the injection of the subgroup of rational cuspidal divisor classes of degree 0 into A(ℚ)_{tor}. Using the implementation in Magma we obtain 120 | #A(ℚ)_{tor} | 240. To compute a divisor of A^{∨}(ℚ)_{tor}, we use the algorithm described in [AS05, §3.3] to find that the modular degree m_{A} = 2^{34}, which is not divisible by any odd primes, hence 15 | #A^{∨}(ℚ)_{tor} | 240.

Next, we use [AS05, §4] to compute the ratio of the special value of the L-function of A_{/ℚ} at 1 over the real Néron period Ω_{A}. We obtain

\frac{L(A_{/ℚ}, 1)}{Ω_{A}} = c_{A} · \frac{2^{9} · 3}{5}, where c_{A} ∈ ℤ is the Manin constant. Since c_{A} | 2^{dim(A)}

by [ARS06] then

\frac{L(A_{/ℚ}, 1)}{Ω_{A}} = \frac{2^{n+2} · 3}{5},

for some 0 ≤ n ≤ 23. In particular, the modular abelian variety A_{/ℚ} has rank zero over ℚ.

Next, using the algorithms from [CS01, KS00] we compute the Tamagawa number c_{A,13} = 1920 = 2^{3} · 3 · 5. We also find that 2 | c_{A,59} is a power of 2 because W_{59} acts as 1 on A, and on the component group Frob_{59} = −W_{59}, so the fixed subgroup Φ_{A,59}(𝔽_{59}) of Frobenius is a 2-group (for more details, see [Rib90a, Prop. 3.7–8]).

Finally, the Birch and Swinnerton-Dyer conjectural formula for abelian varieties of Mordell-Weil rank zero (see [AS05, Conj. 2.2]) asserts that

\frac{L(A_{/ℚ}, 1)}{Ω_{A}} = \frac{#� (ℚ, A) · c_{A,13} · c_{A,59}}{#A(ℚ)_{tor} · #A^{∨}(ℚ)_{tor}} .

By substituting what we computed above, we obtain 3^{2} | #� (ℚ, A). Since L(A_{/ℚ}, 1) ≠ 0, [KL89] implies that � (ℚ, A) is finite. By the nondegeneracy of the Cassels-Tate pairing, #� (ℚ, A) = #� (A^{∨}/ℚ). Thus, if the BSD conjectural formula is true then 3^{2} | #� (ℚ, A) = #� (ℚ, A^{∨}).

We next observe that there are no visible elements of odd order for the embedding A_{23}_{/ℚ} ↪ J_{0}(767)_{/ℚ}.

Lemma 6.1.2. Any element of � (ℚ, A_{23}) which is visible in J_{0}(767) has order a power of 2.

Proof. Since m_{A_{23}} = 2^{34}, [AS05, Prop. 3.15] implies that any element of � (ℚ, A_{23}) that is visible in J_{0}(767) has order a power of 2.

Finally, we use Theorem 5.4.2 to prove the existence of non-trivial elements of order 3 in � (ℚ, A_{23}) which are invisible at level 767, but become visible at higher level. In particular, we prove unconditionally that 3 | #� (ℚ, A_{23}) which provides evidence for the Birch and Swinnerton-Dyer conjectural formula.

Proposition 6.1.3. There is an element of order 3 in � (ℚ, A_{23}) which is not visible in J_{0}(767) but is strongly visible in J_{0}(2 · 767).

Proof. Let A = A_{23}, and note that A has rank 0, since L(A_{/ℚ}, 1) ≠ 0. Using Cremona’s database [Cre] we find that the elliptic curve

E : y^{2} + xy = x^{3} − x^{2} + 5x + 37

has conductor 2 · 767 and Mordell-Weil group E(ℚ) = ℤ ⊕ ℤ. Also

c_{2} = 2, c_{13} = 2, c_{59} = 1, \overline{c}_{2} = 6, \overline{c}_{13} = 2, \overline{c}_{59} = 1.

We apply Theorem 5.4.2 with ℓ = 3 and p = 2. Since E does not admit any rational 3-isogeny (by [Cre]), Hypothesis 3 is satisfied. The level is square free and the modular degree of A is a power of 2, so Hypothesis 2 is satisfied.

We have a_{3}(E) = −3. Using Magma we find

det(T_{3}|\overline{W} − (−3)) ≡ 1 (mod 3),

which verifies the noncongruence hypothesis and completes the proof.

6.2 Level 959

We do similar computations for a 24-dimensional abelian subvariety of J_{0}(959). We have 959 = 7 · 137, which is square free. There are five newform abelian subvarieties of the Jacobian, A_{2}, A_{7}, A_{10}, A_{24} and A_{26}, whose dimensions are the corresponding subscripts. Let A_{f} = A_{24} be the 24-dimensional newform abelian subvariety.

Proposition 6.2.1. There is an element of order 3 in � (A_{f}/ℚ) which is not visible in J_{0}(959) but is strongly visible in J_{0}(2 · 959).

Proof. Using Magma we find that m_{A} = 2^{32} · 583673, which is coprime to 3. Thus we apply Theorem 5.4.2 with ℓ = 3 and p = 2. Consulting [Cre] we find the curve E=1918C1, with Weierstrass equation

y^{2} + xy + y = x^{3} − 22x − 24,

with Mordell-Weil group E(ℚ) ≅ ℤ ⊕ ℤ ⊕ (ℤ/2ℤ), and

c_{2} = 2, c_{7} = 2, c_{137} = 1, \overline{c}_{2} = 6, \overline{c}_{7} = 2, \overline{c}_{137} = 1.

Using [Cre] we find that E has no rational 3-isogeny. The modular form attached to E is

g = q − q^{2} − 2q^{3} + q^{4} − 2q^{5} + ⋯,

and we have

det(T_{2}|\overline{W} − (−2)) = 2177734400 ≡ 2 (mod 3),

which completes the verification.

7 Conjecture, evidence and more computational data

We state several conjectures, provide some evidence and finally, provide a table that we computed using similar techniques to those in Section 6

7.1 The conjecture

The two examples computed in Section 6 show that for an abelian subvariety A of J_{0}(N) an invisible element of � (ℚ, A) at the base level N might become visible at a multiple level NM. We state a general conjecture according to which any element of � (ℚ, A) should have such a property.

Conjecture 7.1.1. Let h = 0 or 1. Suppose A is a J_{h}-modular abelian variety and c ∈ � (ℚ, A). Then there is a J_{h}-modular abelian variety C and an inclusion ι : A → C such that ι_{∗}c = 0.

Remark 7.1.2. For any prime ℓ, the Jacobian J_{h}(N) comes equipped with two morphisms α^{∗}, β^{∗} : J_{h}(N) → J_{h}(Nℓ) induced by the two degeneracy maps α, β : X_{h}(ℓN) → X_{h}(N) between the modular curves of levels ℓN and N, and it is natural to consider visibility of � (ℚ, A) in J_{h}(Nℓ) via morphisms ι constructed from these degeneracy maps.

Remark 7.1.3. It would be interesting to understand the set of all levels N of J_{h}-modular abelian varieties C that satisfy the conclusion of the conjecture.

7.2 Theoretical Evidence for the Conjectures

The first piece of theoretical evidence for Conjecture 7.1.1 is Remark 3.0.2, according to which any cohomology class c ∈ H^{1}(K, A) is visible in some abelian variety C_{/K}.

The next proposition gives evidence for elements of � (ℚ, E) for an elliptic curve E and elements of order 2 or 3.

Proposition 7.2.1. Suppose E is an elliptic curve over ℚ. Then Conjecture 7.1.1 for h = 0 is true for all elements of order 2 and 3 in � (ℚ, E).

Proof. We first show that there is an abelian variety C of dimension 2 and an injective homomorphism i : E ↪ C such that c ∈ Vis_{C}(� (ℚ, E)). If c has order 2, this follows from [AS02, Prop. 2.4] or [Kle01], and if c has order 3, this follows from [Maz99, Cor. pg. 224]. The quotient C/E is an elliptic curve, so C is isogenous to a product of two elliptic curves. Thus by [BCDT01], C is a quotient of J_{0}(N), for some N.

We also prove that Conjecture 7.1.1 is true with h = 1 for all elements of � (ℚ, A) which split over abelian extensions.

Proposition 7.2.2. Suppose A_{/ℚ} is a J_{1}-modular abelian variety over ℚ and c ∈ � (ℚ, A) splits over an abelian extension of ℚ. Then Conjecture 7.1.1 is true for c with h = 1.

Proof. Suppose K is an abelian extension such that res_{K}(c) = 0 and let C = Res_{K/ℚ}(A_{K}). Then c is visible in C (see Section 3.0.2). It remains to verify that C is modular. As discussed in [Mil72, pg. 178], for any abelian variety B over K, we have an isomorphism of Tate modules

Tate_{ℓ}(Res_{K/ℚ}(B_{K})) ≅ Ind_{G_{K}}^{G_{ℚ}} Tate_{ℓ}(B_{K}),

and by Faltings’s isogeny theorem [Fal86], the Tate module determines an abelian variety up to isogeny. Thus if B = A_{f} is an abelian variety attached to a newform, then Res_{K/ℚ}(B_{K}) is isogenous to a product of abelian varieties A_{f^{χ}}, where χ runs through Dirichlet characters attached to the abelian extension K/ℚ. Since A is isogenous to a product of abelian varieties of the form A_{f} (for various f), it follows that the restriction of scalars C is modular.

Remark 7.2.3. Suppose that E is an elliptic curve and c ∈ � (ℚ, E). Is there an abelian extension K/ℚ such that res_{K}(c) = 0? The answer is “yes” if and only if there is a K-rational point (with K-abelian) on the locally trivial principal homogeneous space corresponding to c (this homogenous space is a genus one curve). Recently, M. Ciperiani and A. Wiles proved that any genus one curve over ℚ which has local points everywhere and whose Jacobian is a semistable elliptic curve admits a point over a solvable extension of ℚ (see [CW06]). Unfortunately, this paper does not answer our question about the existence of abelian points.

Remark 7.2.4. As explained in [Ste04], if K/ℚ is an abelian extension of prime degree then there is an exact sequence

0 → A → Res_{K/ℚ}(E_{K}) ⟶^{Tr} E → 0,

where A is an abelian variety with L(A_{/ℚ}, s) = ∏ L(f_{i}, s) (here, the f_{i}’s are the G_{ℚ}-conjugates of the twist of the newform f_{E} attached to E by the Dirichlet character associated to K/ℚ). Thus one could approach the question in the previous remark by investigating whether or not L(f_{E}, χ, 1) = 0 which one could do using modular symbols (see [CFK06]). The authors expect that Lfunctions of twists of degree larger than three are very unlikely to vanish at s = 1 (see [CFK06]), which suggests that in general, the question might have a negative answer for cohomology classes of order larger than 3.

7.3 Visibility of Kolyvagin cohomology classes

It would also be interesting to study visibility at higher level of Kolyvagin cohomology classes. The following is a first “test question” in this direction.

Question 7.3.1. Suppose E ⊂ J_{0}(N) is an elliptic curve with conductor N, and fix a prime ℓ such that \overline{ρ}_{E,ℓ} is surjective. Fix a quadratic imaginary field K that satisfies the Heegner hypothesis for E. For any prime p satisfying the conditions of [Rub89, Prop. 5], let c_{p} ∈ H^{1}(ℚ, E)[ℓ] be the corresponding Kolyvagin cohomology class. There are two natural homomorphisms δ_{1}^{∗}, δ_{p}^{∗} : E → J_{0}(Np). When is

(δ_{1}^{∗} ± δ_{ℓ}^{∗})_{∗}(c_{ℓ}) = 0 ∈ H^{1}(ℚ, J_{0}(Np))?

When is

res_{v}((δ_{1}^{∗} ± δ_{ℓ}^{∗})_{∗}(c_{ℓ})) = 0 ∈ H^{1}(ℚ_{v}, J_{0}(Np))?

7.4 Table of Strong Visibility at Higher Level

The following is a table that gives the known examples of A_{f}_{/ℚ} with square free conductor N ≤ 1339, such that the Birch and Swinnerton-Dyer conjectural formula predicts an odd prime divisor ℓ of � (ℚ, A_{f}), but ℓ does not divide the modular degree of A_{f}. These were taken from [AS05]. If there is an entry in the fourth column, this means we have verified the hypothesis of Theorem 5.4.2, hence there really is a nonzero element in � (ℚ, A_{f}) that is not visible in J_{0}(N), but is strongly visible in J_{0}(pN). The notation in the fourth column is (p, E, q), where p is the prime used in Theorem 5.4.2, E is an elliptic curve, denoted using a Cremona label, and q ≠ p is a prime such that

⋂_{q′≤q} Ker(T′_{q}|\overline{W} − a_{q′}(E)) = 0.

A_{f}  dim ℓ | � (A_{f})_{?}  moddeg  (p, E, q)’s
551H  18  3  2^{?} · 13^{2}  (2, 1102A1, -)
767E  23  3  2^{34}  (2, 1534B1, 3)
959D  24  3  2^{32} · 583673 (2, 1918C1, 5), (7, 5369A1,2)
1337E 33  3  2^{59} · 71  (2, 2674A1, 5)
1339G 30  3  2^{48} · 5776049 (2, 2678B1, 3), (11, 14729A1,2)

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Dimitar P. Jetchev Department of Mathematics University of California Berkeley, CA 94720-3840

jetchev@math.berkeley.edu

William A. Stein

Department of Mathematics University of Washington Seattle, WA 98195-4350

                             wstein@u.washington.edu