Documenta Math. 673 Visibility of the Shafarevich–Tate Group at Higher Level

We study visibility of Shafarevich-Tate groups of modu- lar 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 certain hypothesis which can be verified explicitly. We also provide a table of examples of visible subgroups at higher level and state a conjecture inspired by our data.


Motivation
Mazur suggested that the Shafarevich-Tate group X(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 X(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 X(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 (M N ), 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 X(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.

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 5.1.3guarantees 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.
we denote the complementary isogeny by ϕ ; this is the isogeny φ : , 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 K of K, G K will be the Galois group Gal(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 ur v we always mean the maximal unramified extension of the completion

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]).

Shimura construction. Let
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 where K = Q(. . ., a n , . . . ) is the Hecke eigenvalue field.More precisely, if , 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 ).

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

Modular degree.
If A is an abelian subvariety of J 0 (N ), let be the induced polarization.The modular degree of A is See [AS02] for why m A is an integer and for an algorithm to compute it.

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.
The visible subgroup of X(K, A) relative to the embedding ι is 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 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 : 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 be a prime such that B[ ] ⊂ A and e < − 1, where e is the largest ramification index of any prime of K lying over .Suppose that Under those conditions, Agashe and Stein (see [AS02,Thm. 3.1]) construct a homomorphism B(K)/ B(K) → X(K, A)[ ] whose kernel has F -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 X(Q, A f ) at higher level (e.g., see Propositions 6.1.3and 6.2.1 below).

The main theorem
Let A /K , B /K , C /K , Q /K , N and 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 .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 are all trivial for all nonarchimedean places v of K. Then there is an injective homomorphism of R/m-vector spaces (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.

Some commutative algebra
Before proving Theorem 4.1.1we 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.
where by definition each quotient 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.
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 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.
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 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 By the long exact sequence on Galois cohomology, the quotient , 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 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 It remains to show that for any x ∈ B(K), we have ϕ(x) ∈ Vis C (X(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) = and char(v) = we arrive at the conclusion that the restriction of ϕ(x) to ). (Note that in the case char(v) = the proof uses our hypothesis that #Φ B,v (k v ).)By [Mil86, Prop I.3.8],there is an isomorphism We will use our hypothesis that 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 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 Lemma 4.2.5, our hypothesis that Φ A,v (k v )[m] = 0, and that is locally trivial, which completes the proof.
5 Strong Visibility at Higher Level

Strongly visible subgroups
Let A /Q be an abelian subvariety of J 0 (N ) /Q and let p N be a prime.Let 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 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 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.

[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, for all primes q = p with q ≤ ν(pN ).
Then there is an injective homomorphism Remark 5.1.4.In fact, we have where C ⊂ J 0 (pN ) is isogenous to A f × E.

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 2pN in an integrally closed subring O of Q that contains the ring of integers of the composite field 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 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 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 Thus we can choose an algebraic integer α such that 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 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 Let m be a maximal ideal of the Hecke algebra T(pN ) that annihilates the module , the representation ρ m occurs with multiplicity one in J 0 (pN ).Thus 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.
Proof.Let m be the degree of the isogeny ϕ : A → B. Consider the complementary isogeny ϕ : 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 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 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, ∈ λ p , so R/λ p = F .But by Lemma 5.2.3, R = T(N ), so T(N )/λ p = F , 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 where the isomorphism is an isomorphism of R[G K ]-modules.

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
Lemma 5.2.6.Let be a prime and let φ : E → E be an isogeny of elliptic curves of degree coprime to defined over a number field Proof.Consider the complementary isogeny φ : E → E. Both φ and φ induce homomorphisms ) and φ • φ and φ • φ are multiplication-by-n maps.Since (n, ) = 1 then # ker φ and # ker φ must be coprime to which implies the statement.

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 new 2 (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 .Indeed, every cyclic rational isogeny is a composition of rational isogenies of prime degree, and E admits no rational -isogeny since ρ E, is irreducible.
By Hypothesis 1 the Tamagawa numbers of E are coprime to .Since E and E f are related by an isogeny of degree coprime to , the Tamagawa numbers of E f are also not divisible by by Lemma 5.2.6.Moreover, note that Let m be the ideal of T(pN ) generated by 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 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 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/N Z) × (i.e., the Galois group of X 1 (N ) → X H is H), one should be able to give a generalization of Theorem 5.1.3which holds for newform subvarieties of J H .This requires generalizing some results from [Rib90b] to arbitrary H.

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 and that m A , where m A is the modular degree of A. Then Φ A,q (F q )[m] = 0.
Proof.The component group of Φ J 0 (N ),q (F q ) is Eisenstein by [Rib87], so By Lemma 4.2.2, the image of Φ J0(N ),q (F q ) in Φ A ∨ ,q (F q ) has no m torsion.By the main theorem of [CS01], the cokernel Φ J0(N ),q (F q ) in Φ A ∨ ,q (F q ) has order that divides m A .Since m A , it follows that the cokernel also has no m torsion.Thus Lemma 4.2.2 implies that Φ A ∨ ,q (F q )[m] = 0. Finally, the modular polarization A → A ∨ has degree m A , which is coprime to , so the induced map Φ A,q (F q ) → Φ A ∨ ,q (F q ) is an isomorphism on primary parts.In particular, that Φ A ∨ ,q (F q )[m] = 0 implies that Φ A,q (F q )[m] = 0.
If E is a semistable elliptic curve over Q with discriminant ∆, then we see using Tate curves that c p = ord p (∆).
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 be a prime.Suppose that p N is a prime, and that there is an elliptic curve E of conductor pN such that: 3. [Irreducibility] The mod representation ρ E, is irreducible.

[Noncongruence]
The representation ρ E, is not isomorphic to any representation 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 in X(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 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 ρ E, .
Proof.The divisibility assumptions of Hypothesis 2 on the c E,q imply that the Serre level of ρ E, is N and since N , the Serre weight is 2 (see [RS01, Thm.2.10]).Since 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 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 m A , so Proposition 5.4.1 implies that Φ A,q (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 c E,p = c E,p since c E,p is divisible by 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 W = W ⊗ F .We require that the intersection of the kernels of T q | W − a q (E), for q = p, has dimension 0.

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.

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 X(Q, A 23 ) and X(Q, A ∨ 23 ) are both divisible by 9.
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 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 by [ARS06] then 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,).
Finally, the Birch and Swinnerton-Dyer conjectural formula for abelian varieties of Mordell-Weil rank zero (see [AS05, Conj.2.2]) asserts that By substituting what we computed above, we obtain 3 2 | #X(Q, A).Since L(A /Q , 1) = 0, [KL89] implies that X(Q, A) is finite.By the nondegeneracy of the Cassels-Tate pairing, #X(Q, A) = #X(A ∨ /Q).Thus, if the BSD conjectural formula is true then 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 X(Q, A 23 ) which 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 X(Q, A 23 ) which are invisible at level 767, but become visible at higher level.In particular, we prove unconditionally that 3 | #X(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 X(Q, A 23 ) which is not visible in J 0 (767) but is strongly visible in J 0 (2 • 767).We apply Theorem 5.4.2 with = 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.

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 X(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 = 3 and p = 2. Consulting [Cre] we find the curve E=1918C1, with Weierstrass equation Using [Cre] we find that E has no rational 3-isogeny.The modular form attached to E is 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

The conjecture
The two examples computed in Section 6 show that for an abelian subvariety A of J 0 (N ) an invisible element of X(Q, A) at the base level N might become visible at a multiple level N M .We state a general conjecture according to which any element of X(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 ∈ X(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 , the Jacobian J h (N ) comes equipped with two morphisms α * , β * : J h (N ) → J h (N ) induced by the two degeneracy maps α, β : X h ( N ) → X h (N ) between the modular curves of levels N and N , and it is natural to consider visibility of X(Q, A) in J h (N ) 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.

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 X(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 7.1.1for h = 0 is true for all elements of order 2 and 3 in X(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 (X(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 X(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 ∈ X(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 (Res
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 ∈ X(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 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.

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.

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 of X(Q, A f ), but 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 X(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 | W − a q (E)) = 0.
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 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:

4. 3
Proof of Theorem 4.1.1Proof 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 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[ ] ⊂ 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 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, c 2 = 6, c 13 = 2, c 59 = 1.