Visibility of Shafarevich-Tate groups of modular abelian varieties and the Birch

and Swinnerton-Dyer conjecture

Amod Agashe^{∗}  William A. Stein

November 14, 2000

Abstract

We give examples of abelian subvarieties of J_{0}(N) that have nontrivial visible Shafarevich-Tate groups. These examples provide evidence for the Birch and Swinnerton-Dyer conjecture, and the methods used to compute them may lead to new connections between this conjecture and the theory of congruences between modular forms.

1 Introduction

Let N be a positive integer and let J_{0}(N) be the Jacobian of the modular curve X_{0}(N) (see, e.g., [5]). Let A be an abelian subvariety of J_{0}(N). Then the inclusion A ↪ J_{0}(N) induces a map on Galois cohomology H^{1}(Q, A) → H^{1}(Q, J_{0}(N)), which restricts to a map � (A) → � (J_{0}(N)). Following [4], we call an element c ∈ � (A) visible in J_{0}(N) if it lies in the kernel of this map. When c is visible the torsor attached to c is realized as a subvariety of J_{0}(N); it is a fiber of the quotient map J_{0}(N) → J_{0}(N)/A.

In [4], Cremona and Mazur ask the following question: If E ⊂ J_{0}(N) is a elliptic curve, how much of � (E) is visible in an abelian surface B that is itself a subvariety of J_{0}(N)? Among the 52 examples of odd nontrivial � that they consider, � is invisible in at least 8 cases. However, they warn that the elliptic curves they consider have small conductor, and the general situation is probably much different. Standard conjectures imply that when ℓ is sufficiently large, elements of order ℓ in � (E) can not be visible in an abelian surface; it is thus essential to consider visibility in higher dimensional abelian varieties.

We generalize the experiment of Cremona and Mazur by removing the constraint that only elliptic curves be considered. The numbers become large at much smaller conductor, and the proportion of � that is visible declines accordingly.

In Section 2 we give a general definition of visibility, which is motivated by a restrictionof-scalars construction of torsors. Then we prove a theorem that gives a criterion for the existence of nontrivial visible elements of Shafarevich-Tate groups. Section 3 describes the algorithms we use to enumerate modular abelian varieties, compute conjectural orders of their Shafarevich-Tate groups, and verify the visibility criterion of Section 2 when possible. Section 4 presents the results of extensive numerical investigations. It provides the impetus for the conjectures we make and questions we ask in Section 5.

∗Warning: Agashe has not yet read this draft of our preprint.

Acknowledgment. It is a pleasure to thank B. Mazur for his lectures and conversations about visibility, K. Ribet for explaining congruences to us, and R. Coleman for helping us to understand component groups. The authors would also like to thank B. Conrad, J. Ellenberg, R. Greenberg, D. Gross, L. Merel, B. Poonen, and R. Taylor for many helpful discussions.

2 Visible cohomology classes

In Section 2.1 we define the notion of visibility and observe that every cohomology class arises from a torsor that can be constructed geometrically inside of an appropriate restriction of scalars, then we define visible cohomology classes. Section 2.2, which should perhaps be omitted from a first reading, gives a criterion for the existence of visible elements of Shafarevich-Tate groups.

2.1 Geometric realization of torsors

Let A be an abelian variety over a field K. The following definition is due to Mazur.

Definition 2.1 (Visible). Let ι : A ↪ J be an embedding of A into an abelian variety J. Then the visible part of H^{1}(K, A) with respect to the embedding ι is

Vis_{J}(H^{1}(K, A)) = Ker(H^{1}(K, A) → H^{1}(K, J)).

The Galois cohomology group H^{1}(K, A) has a geometric interpretation as the group of classes of torsors X for A (see [13]). To a cohomology class c ∈ H^{1}(K, A), there is a corresponding variety X over K and a map A × X → X that satisfies axioms similar to those for a simply transitive group action. Suppose ι : A → J is an embedding and c ∈ Vis_{J}(H^{1}(K, A)). We have an exact sequence of abelian varieties 0 → A → J → B → 0. A piece of the associated long exact sequence of Galois cohomology is

⋯ → J(K) → B(K) → H^{1}(K, A) → H^{1}(K, J) → ⋯,

so, because ι_{∗}(c) = 0, there is a point x ∈ B(K) that maps to c ∈ H^{1}(K, A). Then the fiber X over x is a subvariety of J, which, when equipped with its natural action of A, lies in the class of torsors corresponding to c.

Proposition 2.2. Every element of H^{1}(K, A) is visible in some abelian variety J.

Proof. Fix c ∈ H^{1}(K, A). There is a finite extension L of K such that res_{L}(c) = 0 ∈ H^{1}(L, A). Let J = Res_{L/K}(A_{L}) be the restriction of scalars down to K of the abelian variety A, which we view as an abelian variety over L (see [2, §7.6]). Thus J is an abelian variety over K of dimension [L : K] · dim(A), and for any scheme S over K, we have a natural bijection J(S) ≈ A_{L}(S_{L}). In particular, J(A) = A_{L}(A_{L}), and there is an injection ι : A ↪ J attached to id_{A_{L}} ∈ A_{L}(A_{L}). Using the Shapiro lemma, one finds that there is a canonical isomorphism H^{1}(L, A) ≅ H^{1}(K, J) and that ι_{∗}(c) = 0 ∈ H^{1}(K, J).

Remark 2.3. In [4], J. de Jong gives a sophisticated proof of the above proposition in the special case when A is an elliptic curve. His proof involves Azumaya algebras.

2.2 Visible elements of Shafarevich-Tate groups

The Shafarevich-Tate group of an abelian variety over a number field measures the failure of the local-to-global principle for its torsors.

Definition 2.4 (Shafarevich-Tate group). Let A be an abelian variety over Q. The Shafarevich-Tate group of A is

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

where the product is over all places of Q.

If ι : A ↪ J is an embedding, then the visible part of � (A) is

Vis_{J}(� (A)) = � (A) ∩ Vis_{J}(H^{1}(Q, A)) = Ker(� (A) → � (J)).

We use the following theorem to produce examples of visible elements of Shafarevich-Tate groups.

Theorem 2.5. Let A and B be abelian subvarieties of an abelian variety J such that A ∩ B is finite and A(Q) is finite. Let N be an integer divisible by the primes of bad reduction for J. Assume that B has purely toric reduction at each prime dividing N. Suppose p is a prime such that

p ∤ 2N · #(J/B)(Q)_{tor} · #B(Q)_{tor} · ∏_{ℓ|N} #Φ_{A,ℓ}(\overline{F}_{ℓ}) · #Φ_{B,ℓ}(F_{ℓ}).

Suppose furthermore that B[p] ⊂ A ∩ B. Then

B(Q)/pB(Q) ↪ Vis_{J}(� (A)).

The proof proceeds in four steps. First the torsion and p-congruence hypothesis is used to produce an injection B(Q)/pB(Q) ↪ Vis_{J}(H^{1}(Q, A)). Next we perform a local analysis at each place v of Q, which proceeds in three steps. At places v of bad reduction, we use the Mumford-Tate uniformization; at odd primes of good reduction we apply an exactness theorem about Néron models; when 2 is a place of bad reduction, we modify the situation by a 2-isogeny and apply another exactness theorem.

Proof. The quotient J/A is an abelian variety C. The long exact sequence of Galois cohomology associated to the short exact sequence

0 → A → J → C → 0

begins

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

Let ϕ be map B → C, which is obtained by composing the inclusion B ↪ J with the quotient map J → C. Since B[p] ⊂ A, we see that ϕ factors through multiplication by p. We thus obtain the following commutative diagram:

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

Using that B(Q)[p] = 0, we obtain the following diagram, all of whose rows and columns are exact:

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

where K_{0}, K_{1} and K_{2} are the indicated kernels and K_{3} is the indicated cokernel. The snake lemma gives an exact sequence

K_{0} → K_{1} → K_{2} → K_{3}.

Because B → C is an isogeny, K_{1} ⊂ B(Q)_{tor}. Since B(Q)[p] = 0 and K_{2} is a p-torsion group, the map K_{1} → K_{2} is the 0 map. The quotient J(Q)/B(Q) has no p-torsion because it is a subgroup of (J/B)(Q); also, A(Q) is a finite group, so K_{3} = J(Q)/(A(Q) + B(Q)) has no p-torsion, and the map K_{2} → K_{3} must be the 0 map. We conclude that K_{2} = 0.

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

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

Suppose that v is a place at which J has bad reduction. By hypothesis, B has purely toric reduction, so over Q_{v}^{ur} there is an isomorphism B ≅ G_{m}^{d}/Γ of Gal(\overline{Q}_{v}/Q_{v}^{ur})-modules, for some “lattice” Γ. For example, when dim B = 1, this is the Tate curve representation of B. Let n be the order of the component group of B at v; thus n equals the order of the cokernel of the valuation map Γ → Z^{d}. Choose a representative P = (x_{1}, …, x_{d}) ∈ G_{m}^{d} for the point x. Let n′ = #Φ_{B,v}(F_{v}). Then since P is rational over Q_{v}, n′P can be adjusted by elements of Γ so that each of its components x_{i} ∈ G_{m} has valuation 0; since p does not equal the residue characteristic of v, it follows that there is a point Q ∈ G_{m}^{d}(Q_{v}^{ur}) such that pQ = n′P. Thus the cohomology class res_{v}(π(n′x)) is unramified at v. By [20, Prop. 3.8],

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

where Φ_{A,v} is the component group of A at v. Since p does not divide #Φ_{A,v}(\overline{F}_{p}), and π(n′x) has order p, it follows that

0 = res_{v}(π(n′x)) = n′ res_{v}(π(x)).

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

Next suppose that J has good reduction at ℓ and that ℓ is odd. Let A, J, C, be the Néron models of A, J, C, respectively. Since ℓ is odd, 1 = e < ℓ − 1, so we may apply [2, Thm. 7.5.4] to conclude that the sequence of group schemes

0 → A → J → C → 0

is exact, in the sense that it is exact as a sequence of sheaves on the étale site (see the proof of [2, Thm. 7.5.4]). Thus it is exact on the stalks, so by [18, 2.9(d)] the sequence

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

is exact. By the Néron mapping property, the sequence

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

is also exact, so res_{v}(π(x)) is unramified. By [20, Prop. I.3.8],

H^{1}(Q_{v}^{ur}/Q_{v}, A) ≅ H^{1}(Q_{v}^{ur}/Q_{v}, Φ_{A,v}(\overline{F}_{v})) = 0,

since A has good reduction at v. Hence res_{v}(π(x)) = 0.

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

0 → Ã → J̃ → C̃ → 0

is exact, so the sequence

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

is exact. Thus the image of res_{v}(π(x)) in H^{1}(Q_{v}, Ã) is unramified. It equals 0, again by [20, Prop. 3.8], since the component group of Ã at v has order a power of 2, whereas π(x) has odd prime order p. Thus res_{v}(π(x)) = 0, since the kernel of H^{1}(Q_{v}, A) → H^{1}(Q_{v}, Ã) is a finite group of 2-power order.

2.2.1 Visibility when A also has positive rank

In Theorem 2.5, if the condition that A(Q) has rank 0 is removed, then the proof can be easily modified to show that the kernel of B(Q)/pB(Q) → Vis_{J}(� (A)) has dimension at most the rank of A(Q).

According to [3], the smallest conductor elliptic curve E of rank 3 is found in J = J_{0}(5077). The number 5077 is prime, and J decomposes up to isogeny as A×B×E, where each of A, B, and E are abelian subvarieties of J associated to newforms, which have dimensions 205, 216, and 1, respectively. The modular degree of E is 1984 = 2^{6} · 31, and the sign of the Atkin-Lehner involution on E is the same as its sign on A, so E[31] ⊂ A. The numerator of (5077 − 1)/12 is 3^{2} · 47, so 31 is coprime to the orders of any relevant component groups or torsion. Thus Vis_{J}(� (A)) contains (Z/31Z)^{2}.

3 Guide to computing on J_{0}(N)

The Jacobian J_{0}(N) is equipped with an action of the Hecke algebra T. Let f ∈ S_{2}(Γ_{0}(N)) be a newform, and let I_{f} ⊂ T be the annihilator of f. The abelian variety A_{f} attached to f is the quotient J_{0}(N)/I_{f}J_{0}(N). Thus A_{f} is an abelian variety of dimension equal to the number of Galois conjugates of f and equipped with a faithful action of T/I_{f}. For the remainder of this section, A = A_{f} denotes the optimal quotient of J_{0}(N) attached to the annihilator I = I_{f} of a newform f.

3.1 The Birch and Swinnerton-Dyer conjecture

The Birch and Swinnerton-Dyer conjecture, as generalized by Tate in [28], furnishes a conjectural formula for the order of the Shafarevich-Tate group of any new optimal quotient A. In general, it is difficult given A to compute the conjectural order of � (A). However, the situation is more optimistic when A is a new modular abelian variety such that L(A, 1) ≠ 0. For these A we have devised an algorithm that we use to compute the odd part of the conjectural order of � (A) in many cases. The following is a special case of a much more general conjecture.

Conjecture 3.1 (Birch and Swinnerton-Dyer). Suppose L(A, 1) ≠ 0. Then

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

When L(A, 1) ≠ 0, work of Kolyvagin and Logachev [10, 11] implies that A(Q), A^{∨}(Q), and � (A) are all finite, so the quantities appearing in the above formula make sense. Here c_{p} = #Φ_{A,p}(F_{p}), the positive real number Ω_{A} is the measure of A(R) with respect to a basis of differentials on the Néron model of A, and A^{∨} is the abelian variety dual of A. The algorithms described below enable us in my cases to compute the conjectural order of � (A). However, for question of visibility, we instead need to compute the order of � (A^{∨}). This is no different because the Cassels-Tate pairing implies that #� (A) = #� (A^{∨}).

3.2 Modular symbols

It is not possible to compute very much about J_{0}(N) without modular symbols, which provide a finite presentation for the homology group H_{1}(X_{0}(N), Z) in terms of paths between elements of P^{1}(Q) = Q ∪ {∞}.

The modular symbol defined by a pair α, β ∈ P^{1}(Q) is denoted {α, β}. This modular symbol should be viewed as the homology class, relative to the cusps, of a geodesic path from α to β in 𝔥^{∗}. The homology group relative to the cusps is a slight enlargement of the usual homology group, in that we allow paths with endpoints in P^{1}(Q) instead of restricting to closed loops. We declare that modular symbols satisfy the following homology relations: if α, β, γ ∈ Q ∪ {∞}, then

{α, β} + {β, γ} + {γ, α} = 0.

Furthermore, the space of modular symbols is torsion free, so, e.g., {α, α} = 0 and {α, β} = −{β, α}.

Denote by M_{2} the free abelian group with basis the set of symbols {α, β} modulo the three-term homology relations above and modulo any torsion. There is a left action of GL_{2}(Q) on M_{2}, whereby a matrix g acts by

g{α, β} = {g(α), g(β)},

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

In [14], Manin proved that there is a natural injection H_{1}(X_{0}(N), Z) ↪ M_{2}(N). The image of H_{1}(X_{0}(N), Z) in M_{2}(N) can be identified as follows. Let B_{2}(N) denote the free abelian group whose basis is the finite set Γ_{0}(N)\P^{1}(Q). The boundary map δ : M_{2}(N) → B_{2}(N) sends {α, β} to [β] − [α], where [β] denotes the basis element of B_{2}(N) corresponding to β ∈ P^{1}(Q). The kernel S_{2}(N) of δ is the subspace of cuspidal modular symbols. An element of S_{2}(N) can be thought of as a linear combination of paths in 𝔥^{∗} whose endpoints are cusps, and whose images in X_{0}(N) are a linear combination of loops. We thus obtain a canonical isomorphism φ : S_{2}(N) → H_{1}(X_{0}(N), Z).

Part of the utility of modular symbols comes from the classical Abel-Jacobi theorem, which allows us to view J_{0}(N)(C) as the quotient C^{g}/H_{1}(X_{0}(N), Z), where H_{1}(X_{0}(N), Z) is embedded in C^{d} ≈ Hom(S_{2}(Γ_{0}(N)), C) using the integration pairing. Thus modular symbols give an explicit description of J_{0}(N)(C) and of its constituent parts as modules over the Hecke algebra. We can also compute Hecke operators using modular symbols.

For further introductory remarks on modular symbols, see [25], and for detailed instructions as to how to compute the space of modular symbols and the action of Hecke operators on it, see [3].

3.3 Computing with quotients and subvarieties of J_{0}(N)

First, we describe how to enumerate the newforms of level N. Then we define the modular degree, whose square annihilates the visible part of � . Finally, we describe how to intersect abelian subvarieties of J_{0}(N).

3.3.1 Enumerating quotients

Let H_{1}(X_{0}(N), Z)^{+} denote the +1-eigenspace for the action of the involution induced by complex conjugation. 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. First we compute the characteristic polynomial of T_{2}, and use it to break up the full space. We apply this process recursively with T_{3}, T_{5}, … until either we have exceeded the bound coming from [27], or we have found a Hecke operator T_{n} whose characteristic polynomial is irreducible. After computing the decomposition, we order the newforms in a way that extends the systematic ordering in [3]: 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 sign sequence, order by | Tr(a_{p})| with ties broken by taking the positive trace first.

We denote a Galois conjugacy class of newforms by a bold symbol such as 389E, which consists of the level and the isogeny class, where A denotes the first class, B the second, and so on.

As discussed in [3, pg. 5], for certain small levels the above ordering when restricted to elliptic curves does not agree with the ordering used in Cremona’s tables. For example, in the present paper our 446B is Cremona’s 446D.

3.3.2 The modular degree

A polarization λ of an abelian variety A over Q is an isogeny λ : A → A^{∨} such that λ\overline{Q} arises from an ample invertible sheaf on A\overline{Q} (see, e.g., [19, §13]). Since J_{0}(N) is a Jacobian, it possesses a canonical polarization arising from the θ-divisor, and this polarization induces the modular polarization θ : A^{∨} → A of A^{∨}.

\xymatrix{A^{∨} \ar@{^{(}->}[r]^{π^{∨}} \ar[dr]^{θ} & J_{0}(N)^{∨} ≅ J_{0}(N) \ar@{->>}[d]^{π} \\  & A.}

If we view A^{∨} as an abelian subvariety of J_{0}(N), then the kernel of θ is the intersection of A^{∨} with IJ_{0}(N); thus the kernel of θ measures intersections between A^{∨} and other factors of J_{0}(N).

Definition 3.2 (Modular degree). The modular degree m_{A} of A is \sqrt{deg(θ)}.

By [19, Thm. 13.3], deg(θ) is a perfect square, so m_{A} is an integer. For an algorithm to compute m_{A}, see [8].

The modular degree is of interest because its square annihilates the visible cohomology classes.

Proposition 3.3. Vis_{J_{0}(N)}(H^{1}(Q, A^{∨})) ⊂ H^{1}(Q, A^{∨})[m_{A}^{2}]

Proof. Let δ be the composite map A^{∨} → J_{0}(N) → A. There is a map δ̂: A → A^{∨} such that δ̂ ∘ δ is multiplication by deg(δ) = m_{A}^{2}. Thus Ker(H^{1}(Q, A^{∨}) → H^{1}(Q, J_{0}(N))) is contained in H^{1}(Q, A^{∨})[m_{A}^{2}].

Remark 3.4. When A has dimension one, the visible part of H^{1}(Q, A^{∨}) is contained in H^{1}(Q, A)[m_{A}]. Is this true for A of all dimensions?

3.3.3 Intersecting complex tori

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

Proposition 3.5. There is a natural isomorphism of groups

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

Proof. There is an exact sequence

0 → A ∩ B → A ⊕ B → J.

Consider the diagram

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

The snake lemma gives an exact sequence

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

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

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

Proposition 3.6. For i = 1, …, n let A_{i} = V_{i}/Λ_{i} be a subtorus of J = V/Λ, and assume that each pairwise intersection A_{i} ∩ A_{j} is finite. Then

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

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

3.4 Computing the conjectural order of � (A)

In this section, we describe how in many cases we can compute the conjectural order of � (A) when L(A, 1) ≠ 0, at least up to a power of 2.

In Section 3.4.1, we bound #A(Q) and #A^{∨}(Q). We compute each c_{p} in Section 3.4.2, for each p with p || N. When p^{2} | N, it is possible to bound c_{p}; see, e.g., [22, Cor. 15.2.1] where one finds that when dim A = 1 and p^{2} | N, we have c_{p} ≤ 4. In Section 3.4.3, we use modular symbols to compute the rational number L(A, 1)/Ω_{A}, up to a bounded Manin constant.

3.4.1 Torsion subgroup

We obtain an upper bound on #A(Q)_{tor} and #A^{∨}(Q)_{tor} as follows. The characteristic polynomial χ_{p}(X) of the Hecke operator T_{p} acting on A is a monic polynomial having integer coefficients and degree equal to the dimension of A.

Proposition 3.7. Both #A(Q)_{tor} and #A^{∨}(Q)_{tor} divide

gcd{χ_{p}(p + 1) : (p, 2N) = 1, p prime}.

Proof. Use the Eichler-Shimura relation and that for primes p for which p ∤ 2N the maps A(Q)_{tor} → Ã(F_{p}) and A^{∨}(Q)_{tor} → Ã^{∨}(F_{p}) are both injective, and that #A^{∨}(F_{p})_{tor} = #Ã^{∨}(F_{p}).

The difference of two cusps α, β ∈ X_{0}(N) defines a point (α) − (β) ∈ J_{0}(N)(C). Manin observed in [14] that (0) − (∞) is rational. The order of the image of (0) − (∞) in A(Q) can be computed as follows. Let

V = Hom(S_{2}(Γ_{0}(N)), C),

and V_{I} = Hom(S_{2}(Γ_{0}(N))[I], C). The integration pairing ⟨f, γ⟩ = 2πi ∫_{γ} f(z)dz between homology and cusp forms gives rise to a map P : H_{1}(X_{0}(N), Q) → V_{I}. By the Abel-Jacobi theory (see, e.g., [12, Thm IV.2.2]), A(C) ≅ V_{I}/P (H_{1}(X_{0}(N), Z)).

Proposition 3.8. The order of the image of (α) − (β) in A(C) equals the order of the image of the modular symbol {α, β} in P (H_{1}(X_{0}(N), Q))/P (H_{1}(X_{0}(N), Z)).

The quotient appearing in the proposition can be computed algebraically by replacing P by a map with the same kernel as P. Such a map can be computed using the Hecke operators (see [24, §3.7]).

3.4.2 Tamagawa numbers

Suppose p is a prime that exactly divides N and let Φ_{A,p} denote the component group of A at p. We have an 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. A formula for #Φ_{A,p}(\overline{F}_{p}) and, up to a power of 2, for #Φ_{A,p}(F_{p}), is given in [8] and [23].

3.4.3 Rational part of the special value

As in Section 3.4.1, let P : H_{1}(X_{0}(N), Z) → Hom(S_{2}(Γ_{0}(N))[I], C) be the map induced by integration. Let P (H_{1}(X_{0}(N), Z))^{+} denote the +1-eigenspace for the action of the involution induced by complex conjugation on the image of P.

Theorem 3.9.

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

where c_{∞} is the number of components of A(R) and c_{A} is the Manin constant of A, as defined below.

In order to define the Manin constant of A, let A denote the Néron model of A over Z.

Definition 3.10 (Manin constant). The Manin constant c_{A} of A is the index

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

In the definition, we have implicitly mapped H^{0}(A, Ω_{A/Z}) into S_{2}(Γ_{0}(N); Q) using the composition of the following maps:

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

For a discussion of why H^{0}(A, Ω_{A/Z}) is in fact contained in S_{2}(Γ_{0}(N); Z)[I], see [1].

Theorem 3.11. If ℓ | c_{A} then ℓ^{2} | 4N.

Proof. See [15, §4] when A has dimension 1, and [1] in general.

We now give the proof of Theorem 3.9.

Proof of Theorem 3.9. Let H = H_{1}(X_{0}(N), Z) and S = S_{2}(Γ_{0}(N)). There is a perfect pairing T × S → Z given by ⟨T_{n}, f⟩ = a_{n}(f), which induces a canonical isomorphism of rings T ≅ Hom_{Z}(S, Z), where Hom_{Z}(S, Z) is a ring under multiplication of functions. The subring W = Hom_{Z}(S[I], Z) of Hom_{Z}(S[I], R) is isomorphic to T/I, since S[I] is saturated in S. Thus

[W : P ({0, ∞})W] = [W : P (T{0, ∞})]
= [W : P (H)^{+}] · [P (H)^{+} : P (T{0, ∞})].

To complete the proof, observe that that Ω_{A} = [W : P (H)^{+}] · c_{∞} · c_{A} and observe that multiplication by P ({0, ∞}) has determinant ∏_{i=1}^{d} 2πi ∫_{{0,∞}} f^{(i)} = ±L(A, 1).

3.5 Emerton’s work

When N is prime, M. Emerton has proved in [6] that #A_{f}(Q) and c_{p}(A_{f}) divide the numerator of (N − 1)/12.

4 Visibility tables

The tables in this section guide and motivate the conjectures and questions of Section 5.

In Table 4.1, we list each of the 8 invisible odd Shafarevich-Tate groups found in [4], and prove^{1} that they are visible in some J_{0}(Nq).

Table 4.2 lists every quotient A_{f} of J_{0}(p) with p ≤ 2593 and L(A_{f}, 1) ≠ 0 such that the BSD conjecture predicts that #� (A_{f}^{∨}) is divisible by an odd prime. In addition, the table contains data that can frequently be used in conjuction with Theorem 2.5 to deduce that there are visible elements of � (A_{f}^{∨}). When the B column is labeled NONE then there is definitely nothing in � (A_{f}^{∨}) of the predicted order. When the B column contains an elliptic curve, its rank has been computed and is 2, so there are visible elements of � (A_{f}^{∨}). When the B column contains an abelian variety of dimension greater than 1, we have verified that L(B, 1) = 0, so the BSD conjecture predicts that B(Q) is infinite; however, we have not proved that B(Q) is infinite. If we assume that B(Q) is infinite, it follows in these cases that � (A) is visible in J_{0}(p). Note that B has rank 2 over the Hecke algebra here, so the results of [7] say nothing about B(Q).

Table 4.3 continues the computations of Table 4.2 up to level 5647. For each prime p between 2609 and 5674, we computed each factor A such that L(A, 1) ≠ 0 and the odd part of #� _{an}(A) is nontrivial. We then found all factors B such that L(B, 1) = 0 and there is a mod ℓ congruence between A and B, where ℓ | #� _{an}(A). The column labeled N gives the level, the column labeled d(A) gives the dimension of A, the column labeled d(B) gives the dimension of B, and the column labeled “cong” gives the odd part of gcd(#A ∩ B, #� _{an}(A)).

Table 4.4 lists every quotient A_{f} of J_{0}(N) with N ≤ 1642 such that L(A_{f}, 1) = 0 but the sign in the functional equation for f is +1. For each such A_{f}, we looked for an abelian variety B such that B has rank 0 and A_{f}^{∨} probably gives rise to odd visible elements of � (B). This table contains initial data towards the idea of constructing points on high-rank abelian varieties by constructing visible elements of Shafarevich-Tate groups using, e.g., Euler system methods. For example, to prove that A = 1061B really has positive rank, we consider the variety B = 1061D. To prove that A(Q) ≠ 0, it suffices to construct an appropriate element of � (B) and show that this element is visible in A + B ⊂ J_{0}(1061).

Table 4.5 suggests a first tenuous step towards a computational theory of motives attached to modular forms of weight greater than two. This table is organized like Table 4.2, except that the abelian varieties are replaced by motives attached to weight 4 modular forms. For example, at prime level 127 there is a 17-dimensional motive M such that � (M)(2) seems to contain elements of order 43. The computations used to suggest this conclusion were carried out using algorithms for higher weight modular symbols as described in [17], [24], and [26].

1This computation is currently only partially complete.

4.1 Odd invisible � in [4]

E  #� (E) mod deg(E) F  Where � (E) is visible
2849A  3^{2}  2^{5} · 5 · 61  NONE visible using an elliptic curve at level 3 · 2849
3364C  7^{2}  2^{6} · 3^{2} · 5^{2} · 7 none  visible using a 3-dimensional F at level 3364
4229A  3^{2}  2^{3} · 3 · 7 · 13 none  not visible at level 4299,

???

4343B  3^{2}  2^{4} · 1583  NONE ???
4914N  3^{2}  2^{4} · 3^{5}  none  ???
5054C  3^{2}  2^{3} · 3^{3} · 11 none  ???
5073D  3^{2}  2^{5} · 3 · 5 · 7 · 23 none  ???
5389A  3^{2}  2^{2} · 2333  NONE visible using an elliptic curve at level 7 · 5389

4.2 Visibility of � at prime level

The entries in the columns “mod deg” and “� _{an}” are only really the odd parts of “mod deg” and “� _{an}”. Theorem 2.5 does not apply to the two entries marked with a ∗.

A  dim � _{an}(A)  mod deg(A)  B  dim mod deg(B)
389E  20  5^{2}  5  389A  1  5
433D  16  7^{2}  3 · 7 · 37  433A  1  7
563E  31  13^{2}  13  563A  1  13
571D  2  3^{2}  3^{2} · 127  571B  1  3
709C  30  11^{2}  11  709A  1  11
997H  42  3^{4}  3^{2}  997B  1  3
1061D 46  151^{2}  61 · 151 · 179  1061B  2  151
1091C  62  7^{2}  1  NONE
1171D 53  11^{2}  3^{4} · 11  1171A  1  11
1283C  62  5^{2}  5 · 41 · 59  NONE
1429B  64  5^{2}  1  NONE
1481C  71  13^{2}  5^{2} · 2833  NONE
1483D 67  3^{2} · 5^{2}  3 · 5  1483A  1  3 · 5
1531D∗ 73  3^{2}  3  1531A  1  3
1559B  90  11^{2}  1  NONE
1567D 69  7^{2} · 41^{2}  7 · 41  1567B  3  7 · 41
1613D 75  5^{2}  5 · 19  1613A  1  5
1621C  70  17^{2}  17  1621A  1  17
1627C  73  3^{4}  3^{2}  1627A  1  3^{2}
1693C  72  1301^{2}  1301  1693A  3  1301
1811D 98  31^{2}  1  NONE
1847B  98  3^{6}  1  NONE
1871C  98  19^{2}  14699  NONE
1877B  86  7^{2}  1  NONE
1907D 90  7^{2}  3 · 5 · 7 · 11  1907A  1  7
1913B  1  3^{2}  3 · 103  1913A  1  3 · 5^{2}
1913E  84  5^{4} · 61^{2}  5^{2} · 61 · 103  1913A,C 1, 2 3 · 5^{2}, 5^{2} · 61
1933C∗ 83  3^{2} · 7^{2}  3 · 7  1933A  1  3 · 7
1997C  93  17^{2}  1  NONE
2027C  94  29^{2}  29  2027A  1  29
2029C  90 5^{2} · 269^{2}  5 · 269  2029A  2  5 · 269
2039F  99  3^{4} · 5^{2}  19 · 29 · 7759 · 3214201 NONE
2063C 106  13^{2}  61 · 139  NONE
2089J  91  11^{2}  3 · 5 · 11 · 19 · 73 · 139 2089B  1  11
2099B 106  3^{2}  1  NONE
2111B 112  211^{2}  1  NONE
2113B  91  7^{2}  1  NONE
2161C  98  23^{2}  1  NONE
2213C 101  3^{4}  ?  NONE
2239B 110  11^{4}  1  NONE
2251E  99  37^{2}  37  2251A  1  37
2273C 105  7^{2}  ?  NONE
2287B 109  71^{2}  1  NONE
2293C  96  479^{2}  479  2293A  2  479
2311B 110  5^{2}  1  NONE
2333C 101  83341^{2}  83341  2333A  4  83341
2339C 114  3^{8}  6791  NONE
2411B 123  11^{2}  1  NONE
2593C 109 67^{2} · 2213^{2}  67 · 2213  2593A  4  67 · 2213

4.3 More � at prime level

Only odd parts of � _{an} and congruences are given. Observe that � _{an} is only visible roughly 10 percent of the time! As the level gets large, we find that there is almost always some nontrivial � in a large-dimensional factor of J_{0}(p), and that this � is invisible. (Warning: In making this table, 53 primes below 5647 were not analyzed.)

N  d(A)  � _{an}  d(B)  cong 2609  127  192 · 612  2  19 · 61
2617  114  112 · 192  2  11 · 19
2647  117  132  NONE
2659  123  532  NONE
2663  132  432  NONE
2671  122  372  NONE
2677  115  32  1  3
2693  122  34  NONE
2699  125  192  NONE
2707  119  52  NONE
2713  118  192  NONE
2731  124  532  NONE
2749  124  72  NONE
2767  125  52  NONE
2789  136  832  NONE
2791  135  292  NONE
2797  119  112  1  11
2819  138  132  NONE
2837  128  232  1  23
2843  129  36 · 5872  NONE
2851  129  72  NONE
2861  133  114 · 612  2  11 · 61
2879  148  972  NONE
2903  150  6432  NONE
2939  150  172 · 192  NONE
2953  127  292  1  29
2963  134  52 · 312 · 612  2  31 · 61
2969  136  1032  NONE
2999  161  14592  NONE
3001  132  34  NONE
3011  146  52 · 1012  NONE
3019  130  32592  2  3259
3041  147  1032  NONE
3067  134  54  NONE
3079  148  1312  NONE
3083  141  1792  NONE
3089  135  52 · 1312  2  5 · 131
3109  136  52  NONE
3119  164  112 · 592  NONE
3181  144  432  NONE
3187  139  34  NONE
3191  167  532  NONE
3203  143  132  NONE
3221  149  72 · 412  NONE
3229  142  32  NONE
3251  166  34  NONE
3257  143  132  NONE
3271  146  74 · 432 · 712  3  7 · 43 · 71
3299  164  61312  NONE
3301  145  52  NONE
3319  158  54  NONE
3323  155  1792  NONE
3329  157  832  NONE
3331  152  9372  NONE
3343  148  72 · 532  NONE
3347  150  1392  NONE
3359  174  674  NONE
3371  159  12592  NONE
3391  159  292  NONE
3407  170  4992  NONE
3433  148  54 · 72  NONE
3449  168  1072  NONE
3461  167  832  NONE
3463  151  1992  2  199
3467  162  54  NONE
3469  151  472  NONE
3491  168  672  NONE
3511  166  372  NONE
3527  179  6592  NONE
3529  153  792  NONE
3533  164  34  NONE
3539  170  18712  NONE
3541  156  54  NONE
3557  156  2292  NONE
3559  170  11092  NONE

N  d(A)  � _{an}  d(B)  cong 3571  163  672  NONE
3583  161  33192  2  3319
3607  159  74 · 192  NONE
3613  156  72  NONE
3617  165  32  NONE
3623  172  36  NONE
3631  172  4332  NONE
3643  160  52  NONE
3659  181  32 · 114  NONE
3671  193  5092  NONE
3691  166  3532  NONE
3701  174  34 · 2812  2  32 · 281
3709  164  312  NONE
3719  188  132 · 9772  NONE
3739  166  832  NONE
3761  176  6772  NONE
3769  168  132  NONE
3779  187  732 · 1492  1  73
3797  172  192  NONE
3803  171  25312  NONE
3821  182  3072  NONE
3823  173  72  NONE
3863  191  112 · 232 · 3112  NONE
3907  168  34  NONE
3919  182  712  NONE
3929  185  8772  NONE
3931  174  312  NONE
3943  173  24793192  4  2479319
3967  180  36 · 132  1  3 · 13
4007  195  73212  NONE
4013  176  612  NONE
4019  186  34 · 52 · 74  NONE
4021  182  54 · 712  NONE
4027  174  292 · 792  2  29 · 79
4049  186  52 · 34912  NONE
4057  173  1032  NONE
4079  212  52 · 1572 · 1792  NONE
4091  203  74  NONE
4093  174  32 · 894  2  892
4099  185  34 · 192  NONE
4111  190  2292  NONE
4139  188  292 · 672  1  67
4153  177  72  NONE
4157  193  3732  NONE
4159  188  9972  NONE
4177  183  32 · 172  NONE
4217  186  192 · 612  2  19 · 61
4219  190  712  NONE
4229  1  32  NONE
4229  194  34  NONE
4231  201  36  NONE
4253  184  36 · 28432  3  33 · 2843
4261  185  52  NONE
4271  210  1632 · 8532  NONE
4273  183  1812  NONE
4283  198  6832  NONE
4289  205  88072  NONE
4339  196  172  NONE
4349  191  1272  NONE
4357  187  72 · 132 · 172  1  7 · 13
4373  199  312 · 292  NONE
4391  222  54 · 3720372  NONE
4409  200  1572  NONE
4421  206  15232  NONE
4423  200  36 · 5872  NONE
4441  198  592 · 1012  NONE
4451  213  8092  NONE
4457  199  3372  NONE
4463  213  89512  NONE
4483  193  192 · 612  2  19 · 61
4517  201  1812  NONE
4519  202  25032  NONE
4547  205  732  1  73
4549  203  192 · 532  NONE
4591  215  63172  NONE

N  d(A)  � _{an}  d(B)  cong 4597  200  72 · 172  NONE
4603  198  8292  NONE
4621  196  132  NONE
4639  218  892  NONE
4649  215  7512  NONE
4651  210  134  NONE
4673  207  112 · 1972  NONE
4691  216  432  NONE
4729  204  6732  NONE
4733  210  172  NONE
4783  210  7972  NONE
4789  206  134  NONE
4799  230  32 · 72 · 122032  1  3 · 7
4801  213  602712  NONE
4813  207  32 · 68832  NONE
4817  214  2832  NONE
4831  217  11512  NONE
4861  216  2047492  NONE
4877  219  34 · 1032  NONE
4931  240  172 · 372 · 432  NONE
4933  211  2392  NONE
4957  212  52  NONE
4967  236  72 · 538812  NONE
4969  220  114  NONE
4973  223  52 · 112  NONE
4993  215  40132  NONE
4999  224  9851212  NONE
5003  220  972 · 18612  3  97 · 1861
5009  223  232 · 9772  NONE
5011  229  114  NONE
5021  225  16092  NONE
5023  221  514312  NONE
5039  251  1663632  NONE
5051  239  132 · 26332  NONE
5059  229  52 · 132 · 312  NONE
5077  216  2832  NONE
5081  240  192 · 1492  NONE
5099  251  74 · 112 · 4612  NONE
5113  223  192 · 612  NONE
5119  232  532 · 1032  NONE
5153  223  34 · 412  NONE
5167  231  3672  NONE
5171  249  732 · 7732  1  73
5179  226  72 · 132  NONE
5189  240  832  NONE
5197  223  372  NONE
5209  227  1812 · 14712  NONE
5227  232  32 · 77172  NONE
5231  255  45072  NONE
5233  223  1632  NONE
5237  229  72  NONE
5261  239  241032  NONE
5273  227  173892  NONE
5279  263  1204312  NONE
5281  232  672  NONE
5297  238  3972  NONE
5303  247  132 · 732 · 154672  NONE
5309  247  18226932  NONE
5323  233  34 · 1205632  3  120563
5333  237  9672  NONE
5347  231  36432  NONE
5501  250  1632  NONE
5503  241  72 · 172  NONE
5507  252  1032 · 2332  NONE
5519  278  612 · 2114692  NONE
5521  244  54  NONE
5531  253  9772  NONE
5563  246  34 · 12132  NONE
5569  239  34 · 52 · 132  NONE
5573  247  99012  NONE
5581  242  289272  NONE
5591  282  32 · 134 · 10612  NONE
5639  278  2297172  NONE
5641  244  412 · 4312  NONE
5647  245  44632  NONE

4.4 Mordell-Weil groups of positive even rank and the � they

probably induce

L(1) = 0 d L(1) ≠ 0 d cong L(1)/Ω · 2^{∗}
389A  1 389E 20 5  25/97
433A  1 433D 16 7  49/9
446B  1 446F  8 11  121/3
563A  1 563E 31 13  169/281
571B  1 571D  2 3  9
643A  1 NONE
655A  1 655D 13 9  81
664A  1 664F  8 5  25
681C  1 681B  1 3  9
707A  1 707G 15 13  169
718B  1 718F  7 7  49
794A  1 794G 12 11  121/3
817A  1 817E 15 7  49/5
916C  1 916G  9 11  121
944E  1 944O  6 7  49
997B  1 997H 42 3  81/83
997C  1 997H 42 3  81/83
1001C 1 1001F 3 3  27
1001C 1 1001L 7 7  49
1028A 1 1028E 14 11  3267
1034A 1 NONE
1041B 2 1041E 4 5  25
1041B 2 1041J 13 25  625
1058C 1 1058D 1 5  25
1061B 2 1061D 46 151 22801/265
1070A 1 1070M 7 15  75
1073A 1 NONE
1077A 1 1077J 15 9  81
1088J 1 1088R 2 3  9
1094A 1 1094F 13 11  121/3
1102A 1 1102K 4 3  9
1126A 1 1126F 11 11  121
1132A 1 1132F 12 5  225
1137A 1 1137C 14 9  81
1141A 1 1141I 22 7 1524537/41

1143C 1 1143J 9 11  121
1147A 1 1147H 23 5  225/19
1171A 1 1171D 53 11  121/195
1246C 1 1246B 1 5  25
1309B 1 NONE
1324A 1 1324E 14 9  6561
1325E 1 1325T 11 9  2187
1363B 2 1363F 25 31  961/5
1431A 1 1431L 14 3  9
1436A 1 NONE
1443C 1 1443G 5 7  49
1446A 1 1446N 7 3  9
1466B 1 1466H 23 13 4331806939187/367
1477A 1 1477C 24 13  169
1480A 1 1480G 5 7  49
1483A 1 1483D 67 15  225/247
1525C 1 1525O 16 7  49
1531A 1 1531D 73 3  3/85
1534B 1 1534J 6 3  3
1567B 3 1567D 69 287  82369/261
1570B 1 1570J 6 11  121
1576A 1 1576E 14 11  121
1591A 1 1591F 35 31  6727/19
1594A 1 1594J 17 3 3370648239/19
1608A 1 1608J 6 13  169
1611D 1 1611O 11 9  81
1613A 1 1613D 75 5  25/403
1615A 1 1615J 13 9  102141
1621A 1 1621C 70 17  289/135
1627A 1 1627C 73 9  81/271
1633A 3 1633D 27 189  35721
1639B 1 1639G 34 17  680017/25
1641B 1 1641J 24 23  529
1642A 1 1642D 14 7  49

643A, 1034A, 1073A, 1309B all have modular degree a power of 2; 1436A has modular degree divisible by 3.

4.5 Conjecturally visible � of modular motives of weight 4

Suppose f and g are elements of S_{4}(Γ_{0}(N)) such that p^{2} | L(M_{f}, 2)/Ω and L(M_{g}, 2) = 0. If f and g satisfy a “p-congruence”, does p then divide the “visible part” of � (M_{f}(2))?

M_{f}  dim  p^{2}  M_{g}  dim
127k4C 17  43^{2}  127k4A 1
159k4E 8  23^{2}  159k4B 1
365k4E 18  29^{2}  365k4A 1
369k4I  9  13^{2}  369k4A 1
453k4E 23  17^{2}  453k4A 1
465k4H 7  11^{2}  465k4A 1
477k4L 12  73^{2}  477k4A 1
567k4G 8  13^{2}, 23^{2} 567k4A 1
581k4E 34  19^{2}  581k4A 1

5 Questions and conjectures

The following questions and conjectures were motivated by the tables above and the computations that went into creating them. The first conjecture suggests a generalization of a result of Ribet on level raising. The second conjecture asserts that � is always visible in an appropriate modular Jacobian. The third question suggests a new approach to the long-standing open problem of constructing points on abelian varieties of analytic rank greater than 1 over the Hecke algebra.

5.1 Level raising nonvanishing conjecture

Let f ∈ S_{2}(Γ_{0}(N)) be a newform such that the sign of the functional equation of L(A_{f}, s) is equal to +1, and fix a prime λ such that the associated Galois representation ρ_{f,λ} = A_{f}[λ] is irreducible. For each prime q not dividing N, let δ : J_{0}(N) → J_{0}(Nq) be the injection obtained from the sum of the two degeneracy maps. Ribet’s construction in [21] produces infinitely many primes q and newforms g ∈ S_{2}(Γ_{0}(qN)) such that

δ(A_{f}^{∨}[λ]) ⊂ δ(A_{f}^{∨}) ∩ A_{g}^{∨}

and the Tamagawa number c_{q} of A_{g}^{∨} is a power of 2.

Conjecture 5.1. Fix f and λ.

1. Then there is a g among those constructed by Ribet such that L(A_{g}, 1) ≠ 0.

2. If λ is in the support of the T-module [P (H_{1}(X_{0}(N), Z))^{+} : P (T{0, ∞})] (see Theorem 3.9), then there is a g as above such that L(A_{g}, 1) = 0.

5.2 Eventual visibility conjecture

Let S be the set of all square-free positive integers. If M, N ∈ S with M | N then there is a natural injection J_{0}(M) ↪ J_{0}(N), and hence a map � (J_{0}(M)) → � (J_{0}(N)). These maps are compatible, so the collection of groups � (J_{0}(N)), with N ∈ S, forms a directed system. Let lim_{N∈S} � (J_{0}(N)) be the direct limit of the � (J_{0}(N)).

Conjecture 5.2. lim_{N∈S} � (J_{0}(N)) = 0

If true, this would imply that if A ⊂ J_{0}(N), then each element of � (A) is visible in some J_{0}(N′), for some multiple N′ of N.

5.3 Euler systems

In [9] and [16] one finds a construction using the Heegner point Euler system of Kolyvagin of the Shafarevich-Tate groups of certain abelian varieties. Under an unverified hypothesis on Heegner points, the construction gives much of � (A_{f}/K), where K is a suitable imaginary quadratic field. Is it possible to verify the unverified hypothesis, construct Vis_{J}(� (A_{f}/K)), and thus prove that � (A_{f}^{∨}/K) contains visible elements, when the BSD conjecture suggests that it should? If so, it would follow that there is a congruent B_{f}^{∨} having positive algebraic rank, as predicted by the BSD conjecture. Thus a construction of visible elements of � (A_{f}^{∨}) also leads to a construction of points on abelian varieties of positive analytic rank.

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