Visible Shafarevich-Tate Groups of Modular Abelian Varieties

Amod Agashe  William Stein

May 1999

Abstract

Building on Barry Mazur’s idea of “visibility” we perform a numerical experiment that might be used to show that for many modular abelian varieties A of rank 0 and level N ≤ 1061, that � (A) contains elements of the order predicted by the BSD conjecture. In this experiment we find other factors of J_{0}(N) of positive rank, satisfying a congruence with A, and observe that it might be possible to use them to produce elements of � (A). In carrying out these computations we were motivated to develop algorithms for computing invariants of modular abelian varieties.

Introduction

Suppose E ⊂ J_{0}(N)^{new} is a modular elliptic curve with E(Q) = 0 and � (E) = (Z/p)^{2}. John Cremona, Adam Logan, and Barry Mazur ([CM98], [Maz98]) made the surprising observation that for N ≤ 5077 there is often an elliptic curve F ⊂ J_{0}(N) with � (F) = 0, F (Q) = Z^{2}, and for which E[p] = F [p] ⊂ J_{0}(N). Frequently these hypothesis are sufficient to identify the p-Selmer groups giving a commuting diagram with exact rows:

0 → E(Q)/pE(Q) → Sel^{p}(E) → � (E)[p] → 0
||
0 → F (Q)/pF (Q) → Sel^{p}(F) → � (F)[p] → 0

In this situation, the Mordell-Weil group of F explains the Shafarevich-Tate group of E. It can be argued that exhibiting elements of the Mordell-Weil group is visibly easier than obtaining elements of � : one simply searches for points, and when a point is found it is quite visible.

But how large can p be in this picture? We know of no example of elliptic curves E, F as above with p > 13. Conjecturally, there will only be finitely many such p (see conjecture (b) of [Fre97], section 4.2 of [FM99]). Furthermore, if p does not divide the modular degree of E there can be no such F. On the other hand, if p does divide the modular degree then there must be some abelian variety B ⊂ J_{0}(N), not containing E, and such that E[p] ∩ B[p] ≠ 0.

In this paper we begin with an abelian subvariety A ⊂ J_{0}(N)^{new}. We then compute a number S which divides the order of � (A) as predicted by conjecture 1.4. For each prime p|S, we search for another abelian variety B ⊂ J_{0}(N) for which p|#(A ∩ B). In many cases we expect that one can use B to unconditionally deduce that p divides the order of � (A).

Contents

1 Modular Abelian Varieties Parametrized by Newforms 2 1.1 Optimal Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Birch and Swinnerton-Dyer Conjecture . . . . . . . . . . . . 3 1.3 Connecting Mordell-Weil and Shafarevich-Tate . . . . . . . . . . 4

2 Algorithms for Modular Abelian Varieties 4 2.1 Modular Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Method of Graphs . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Enumerating Quotients of J_{0}(N) . . . . . . . . . . . . . . . . . . 6 2.4 The Modular Polarization . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.6 Component Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.7 L(A_{f}, 1)/Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.8 The Manin Constant . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Results 9

1 Modular Abelian Varieties Parametrized by Newforms

1.1 Optimal Quotients

An abelian variety quotient of J_{0}(N) is called optimal if the kernel is connected. Let f ∈ S_{2}(Γ_{0}(N), C) be a newform, set Q_{f} = Q[a_{1}, a_{2}, …], and let f_{1}, …, f_{d} be the Gal(Q_{f}/Q)-conjugates of f. Let I_{f} = Ann(f) ⊂ T and I_{f}J_{0}(N) denote the union of the images of the endomorphism in I_{f}. We associate to f an optimal quotient A_{f} of J_{0}(N):

0 → I_{f}J_{0}(N) → J_{0}(N) → A_{f} → 0.

Let H = H_{1}(X_{0}(N), Z) and S_{2} = S_{2}(Γ_{0}(N), C). Integration defines a nondegenerate pairing S_{2} × H → C and hence a period map H → Hom_{C}(S_{2}, C). Composing with restriction to S_{2}[I_{f}] defines a map

Φ_{f} : H → Hom(S_{2}[I_{f}], C) ≈ C^{d}, Φ_{f}(x) = ∑_{i=1}^{d}⟨f_{i}, x⟩f_{i}^{∗}.

Let H_{f} = H/ Ker(Φ_{f}).

Theorem 1.1. A_{f} is a Q-simple abelian variety of dimension d:

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

2. Complex Tori: A_{f}(C) and A_{f}^{∨}(C) fit into exact sequences

0 → H[I_{f}] → Hom_{C}(S_{2}, C)[I_{f}] → A_{f}^{∨}(C) → 0
↓ ↓ ↓
0 → H_{f} → Hom_{C}(S_{2}[I_{f}], C) → A_{f}(C) → 0

3. Reduction of A_{f} at p:

good  if p ∤ N
purely multiplicative if p||N
purely additive  if p^{2} | N, p ≥ 5

Proof. Section 1.7 of [DDT94], [Shi73], [Shi71], [Lor].

Remark 1.2. 1) A_{f} need not be \overline{Q}-simple [Koi76].

2) The class of abelian varieties isogenous to some A_{f} is large. It is conjectured to contain all A/Q such that End(A) ⊗ Q contains a totally real number field K/Q of degree = dim A. If E is an elliptic curve whose conductor is not divisible by 27 then E is isogeneous to some A_{f} ([Wil95], [CDT99]).

1.2 The Birch and Swinnerton-Dyer Conjecture

Let A = A_{f} be the optimal quotient of J_{0}(N) corresponding to a newform f. Let A/Z be its Neron model. Let c_{p} be the number of F_{p}-rational components of the special fiber A_{F_{p}}. A basis h_{1}, …, h_{d} for the Neron differentials defines a measure μ on A(R) and we let Ω_{A} = μ(A(R)). Let � (A) = Ker(H^{1}(Q, A) → ∏_{v} H^{1}(Q_{v}, A)).)

Theorem 1.3 (Kolyvagin, Logachev). If L(A, 1) ≠ 0, then A(Q) and � (A) are both finite.

Proof. [KL89]

The following conjecture ties together the invariants of A.

Conjecture 1.4 (Birch, Swinnerton-Dyer, Tate). Assume L(A, 1) ≠ 0. Then,

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

If S is the part of L(A, 1)/Ω_{A} coprime to ∏ c_{p} · |A(Q)| · |A^{∨}(Q)| then a consequence of the BSD formula is that S divides |� (A)|.

1.3 Connecting Mordell-Weil and Shafarevich-Tate

Let f, g ∈ S_{2}(Γ_{0}(N), C) be nonconjugate newforms and A_{f}, A_{g} the corresponding optimal quotients of J_{0}(N). Let 𝔪 ⊂ T be a maximal ideal such that A_{f}^{∨}[𝔪] = A_{g}^{∨}[𝔪] ⊂ J_{0}(N). Let p be the residue characteristic of 𝔪. Assume that

p ∤ 2N · ∏_{p|N} c_{p}(A_{f})c_{p}(A_{g}).

If one wishes to deduce nontriviality of � from our computations, the following theorem is applied.

Theorem 1.5 (Mazur). We have the following commutative diagram with exact rows:

0 → A_{f}(Q)/𝔪A_{f}(Q) → H^{1}(Spec Z, A_{f}[𝔪]) → � (A_{f})[𝔪] → 0
||
0 → A_{g}(Q)/𝔪A_{g}(Q) → H^{1}(Spec Z, A_{g}[𝔪]) → � (A_{g})[𝔪] → 0

2 Algorithms for Modular Abelian Varieties

Cremona’s [Cre97] enumeration of elliptic curves proceeds for each level N, as follows. First the q-expansion of a newforms f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N), Z) is found. The rational number L(f, 1)/Ω can then be quickly computed. The period lattice associated to E_{f} is computed and from this the invariants c_{4} and c_{6} of a Weierstrass equation are obtained. With the explicit Weierstrass equation in hand it is then possible to compute the torsion subgroup of E_{f} (Nagel-Lutz) and the Tamagawa numbers c_{p} (Tate’s algorithm). The modular degree, which controls congruences, is computed by a trick deriving from evaluating the Peterson norm in two different ways, as described in section 2.15 of [Cre97]. For elliptic curves Cremona has already computed all of the invariants we will need.

Now remove the obstruction that f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N), C) have rational Fourier coefficients. We would like still to compute L(A_{f}, 1)/Ω, the modular degree, c_{p}, and torsion. In the following sections we state algorithms for computing many of these invariants. Proofs are not always given, but will appear in forthcoming work.

2.1 Modular Symbols

Modular symbols ([Cre97], [FM99], [Mer94]) will be used in enumerating the newforms in S_{2}(Γ_{0}(N), C), computing the special values L(A_{f}, 1)/Ω, torsion and the modular degree.

Modular symbols give a presentation of the homology of the modular curve X_{0}(N), along with its structure as Hecke module. There is a pairing

⟨ , ⟩ : S_{2}(Γ_{0}(N), C) × H_{1}(X_{0}(N), Z) → C

given by integration. This pairing respects the Hecke action and is nondegenerate so knowing it on homology gives information about cusp forms. Furthermore, the pairing respect the involution ∗ induced by conjugation.

The space of modular symbols \underline{M}_{2}(Γ_{0}(N), Z) is the free abelian group generated by symbols {α, β} such that α, β ∈ P^{1}(Q) = Q ∪ {∞} subject to the relations

{α, β} + {β, γ} + {γ, α} = 0,
{α, β} = {g(α), g(β)}, all g ∈ Γ_{0}(N).

(Quotient out by any torsion.) The space of cuspidal symbols \underline{B}_{2}(Γ_{0}(N), Z) is the free abelian group generated by symbols {α}, α ∈ P^{1}(Q), modulo the relation

{α} = {g(α)}, all g ∈ Γ_{0}(N).

The cuspidal symbols \underline{S}_{2}(Γ_{0}(N), Z) are the kernel of the boundary map ∂ : \underline{M}_{2}(Γ_{0}(N), Z) → \underline{B}_{2}(Γ_{0}(N), Z) given by ∂(\{α, β\}) = \{β\} − \{α\}:

0 → \underline{S}_{2}(Γ_{0}(N), Z) → \underline{M}_{2}(Γ_{0}(N), Z) → \underline{B}_{2}(Γ_{0}(N), Z).

The Hecke operators T_{p} act on \underline{M}_{2}(Γ_{0}(N), Z) by

T_{p}(\{α, β\}) = [( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) + ∑_{r mod p} ( \begin{matrix}1 & r \\ 0 & p \end{matrix} )] \{α, β\}.

(Omit ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) if p|N.) The ∗-involution is ∗\{α, β\} = \{−α, −β\}. We denote by H_{1}(X_{0}(N), Z)^{+} the plus one eigenspace for ∗.

Theorem 2.1. Integration defines a nondegenerate pairing

S_{2}(Γ_{0}(N), C) × \underline{M}_{2}(Γ_{0}(N), Z) → C.

The image of \underline{S}_{2}(Γ_{0}(N), Z) coincides with that of H_{1}(X_{0}(N), Z) and the image of \underline{M}_{2}(Γ_{0}(N), Z) is a lattice.

Remark 2.2. In order to efficiently compute these spaces we use a trick of Manin’s which provide a finite number of generators (section 2.2 of [Cre97], section 3 of [FM99]).

2.2 The Method of Graphs

The method of graphs will be used in computing the Tamagawa numbers c_{p}.

Let M be a positive integer, p a prime not dividing M, and put N = pM. Let D be the finitely generated free abelian group on the superingular points of X_{0}(M)(\overline{F}_{p}), i.e. the enhanced elliptic curves E = (E, C) where E is a supersingular elliptic curve defined over \overline{F}_{p} and C is a cyclic subgroup of order M, and enhanced curves are identified if they are isomorphic in the evident way.

Let w_{E} = \frac{| Aut(E)|}{2} where Aut(E) is the group of \overline{F}_{p}-auotomorphisms of E. We have w_{E} ≤ 12 and if p ≥ 5 then w_{E} ≤ 3. The monodromy pairing on D is defined by

⟨[E], [E′]⟩ = \begin{cases}w_{E} & if E = E′ \\ 0 & if E ≠ E′ \end{cases}

Let

X_{J_{0}(N),p} = {∑ a_{E}[E] ∈ D : ∑ a_{E} = 0}.

For n ≥ 1 coprime to p, define T_{n} on D by

T_{n}(E) = ∑_{C_{n}}(E/C_{n}, (C + C_{n})/C_{n})

where C_{n} runs through the subgroups of order n in E such that C_{n} ∩ C = {0}. Define an involution W_{p} by W_{p} = − Frob_{p} where Frob_{p} is the endomorphism which sends (E, C) to (E^{p}, C^{p}).

Theorem 2.3. X_{J_{0}(N),p} ⊗ C is isomorphic as a Hecke module to the subspace of S_{2}(Γ_{0}(N), C) generated by newforms and oldforms of level pd for d|M.

Using an equivalence of categories it is possible to efficiently compute on X_{J_{0}(N),p} via the arithmetic of quaternion algebras. David Kohel [Koh98] has implimented in Magma algorithms which do just this and kindly provided us with his programs.

2.3 Enumerating Quotients of J_{0}(N)

We now describe how to list the optimal quotients of J_{0}(N). Fix a basis for the Q-vector space V = H_{1}(X_{0}(N), Q)^{+}. Using modular symbols (2.1) we compute matrices representing the action of the Hecke operators T_{p} on V.

Let W be a T-stable subspace of V and factor the characteristic polynomial of T_{p} as ∏_{i=1}^{n} h_{i}^{m_{i}}(x).

Proposition 2.4. Set W_{i} = Ker(h_{i}^{m_{i}}(T)). Then

W = ⨁_{i=1}^{n} Ker(h_{i}^{m_{i}}(T))

writes W as a direct sum of T-stable submodules.

Proof. Since T is commutative the kernels Ker(h_{i}^{m_{i}}) are all T-stable. To prove that W is a direct sum of the indicated subspaces proceed inductively using the following observation. If the characteristic polynomial of T is gh, with g and h coprime, then there exists polynomials a, b so that ga + hb = 1. Any w ∈ W can thus be represented as w = g(T)a(T)w + h(T)b(T)w. By Cayley-Hamilton, g(T)h(T) = 0 so the image of h is contained in the kernel of g and vice-versa. Thus g(T)a(T)w ∈ Ker(h(T)) and h(T)b(T)w ∈ Ker(g(T)).

Set p = 2. Compute the characteristic polynomial of T_{p}|W. Factor it as ∏_{i=1}^{n} h_{i}^{m_{i}}. Write V = ⊕W_{i} as in the proposition. Repeat this process with T_{3} acting on each W_{i}. Since T_{Q} is generated by finitely many Hecke operators, this process terminates.

After computing the decomposition we order the forms as follows:

1. By dimension,

2. In binary by the signs of the Atkin-Lehner involutions with + = 0 and − = 1, e.g., + + +, + + −, + − +, + − −, − + +, etc,

3. By tr(a_{p}) by absolute values, breaking ties with the positive trace being first.

For historical reasons this does not always agree with the ordering in Cremona’s tables (page 5 of [Cre97]). There is exactly one case in our tables in which the two ordering schemes disagree, our 446B is Cremona’s 446D. [Recheck this assertion when the computations are done.]

2.4 The Modular Polarization

A polarization of an abelian variety A is an isogeny A → A^{∨} arising from a very amply invertible sheaf (see [Mil86]). Because J_{0}(N) is a Jacobian it possesses a canonical principal polarization arising from the θ-divisor and this induces the modular polarization θ_{f} : A_{f}^{∨} → A_{f}.

[figure: A_{f}^{∨}; J_{0}(N); A_{f}; π^{∨}; θ_{f}; π]

By ([Mil86], theorem 13.3) deg(θ_{f}) is a perfect square so we define the modular degree δ = \sqrt{deg(θ_{f})}. The kernel of θ_{f} is the intersection of A_{f}^{∨} with I_{f}J_{0}(N) and hence measure congruences between f and other forms.

Proposition 2.5. With notation as in theorem 1.1,

Ker(θ_{f}) = Coker(H[I_{f}] ⟶^{Φ_{f}} Φ_{f}(H))

Proof. Apply the snake lemma to the diagram in part 2 of theorem 1.1.

Thus the modular degree can be computed as follows: Let φ_{1}, …, φ_{2d} be a basis for Hom(H, Z)[I_{f}] and a_{1}, …, a_{2d} a basis for H[I_{f}]. Then δ_{f} is the square root of the absolute value of the determinant of the matrix (φ_{i}(a_{j})).

2.5 Torsion

Let χ_{p}(X) ∈ Z[X] denote the characteristic polynomial of T_{p} acting on A_{f}. It is a polynomial having integer coefficients and degree equal to dim A_{f}.

Theorem 2.6. If B is isogeneous to A then

|B(Q)_{tor}| divides gcd{χ_{p}(p + 1) : p prime, (p, 2N) = 1}.

2.6 Component Groups

Let p|N be a prime. Let Φ_{A,p} be the component group of A at p so there is an exact sequence

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

with A the Neron model of A and A_{F_{p}}^{0} connected.

Theorem 2.7 (Additive reduction). If p^{2} | N, then

Φ_{A,p}(\overline{F}_{p}) = p^{a} · |A_{f}(Q)_{tor}|

for some nonnegative a.

Proof. [Lor93]

Suppose that p | N but p^{2} ∤ N. Let X = X_{J_{0}(N),p} be the module constructed in section 2.2, equipped with its monodromy pairing ⟨ , ⟩ : X × X → Z. Let X[I_{f}] be the T-module cut out by the annihilator I_{f} of f. The monodromy pairing induces a map X → Hom(X[I_{f}], Z). Let δ_{f} be the modular degree (section 2.4), and w_{p} the sign of the Atkin-Lehner involution W_{p} on f.

Theorem 2.8 (Multiplicative reduction). If p exactly divides N, then

|Φ_{A,p}(\overline{F}_{p})| = δ_{f} · \frac{| Coker(X → Hom(X[I_{f}], Z))|^{2}}{disc(X[I_{f}] × X[I_{f}] → Z)} ,

|Φ_{A,p}(F_{p})| = \begin{cases}|Φ_{A,p}(\overline{F}_{p})| & w_{q} = −1 \\ |Φ_{A,p}(\overline{F}_{p})[2]| & w_{q} = +1 \end{cases}

2.7 L(A_{f}, 1)/Ω

Let f ∈ S_{2}(Γ_{0}(N), C) be a newform and I_{f} ⊂ T the annihilator of f. Denote by f_{1}, …, f_{d} the Gal(\overline{Q}/Q)-conjugates of f. Let Φ_{f} : \underline{M}_{2}(Γ_{0}(N), Q) → C be the period mapping Φ_{f}(x) = (⟨f_{1}, x⟩, …, ⟨f_{d}, x⟩).

Theorem 2.9. Let φ_{1}, …, φ_{n} be a Q-basis for Hom(\underline{M}_{2}(Γ_{0}(N), Q), Q)[I_{f}] and Ψ = φ_{1} × ⋯ × φ_{n} : \underline{M}_{2}(Γ_{0}(N), Q) → Q^{n}. Then n = d and Ker(Φ_{f}) = Ker(Ψ).

Define a set theoretic map φ_{f} : \underline{M}_{2}(Γ_{0}(N), Q)^{+} → Q_{≥0} by

φ_{f}(x) = [Ψ(\underline{M}_{2}(Γ_{0}(N), Z)^{+}) : Ψ(Tx)]

where the lattice index is taken inside of the Q-vector space Ψ(\underline{M}_{2}(Γ_{0}(N), Q))^{+}. The lattice index doesn’t depend on the choice of Ψ.

Let S_{2}(Γ_{0}(N), Z) denote the space of cusp forms whose q-expansions at infinity have integer coefficients. Let μ_{∞} be the measure on A_{f} given by choosing an integral basis for S_{2}(Γ_{0}(N), Z)[I_{f}]. Let Ω_{f} = μ_{∞}(A_{f}(R)^{0}) (real volume) where A_{f}(R)^{0} denotes the identity component of A_{f}(R).

Theorem 2.10.

\frac{L(A_{f}, 1)}{Ω_{f}} = ±φ_{f}(\{0, ∞\}) ∈ Q

2.8 The Manin Constant

Let S_{f}(C) ⊂ S_{2}(Γ_{0}(N), C) denote the subspace of cusp forms spanned by the conjugates of f. There are two lattices in S_{f}(C). One is the lattice S_{f}(Z) of cusp forms with integer Fourier expansion at infinity. The other S_{f}(A/Z) is the pullback of the Neron differentials defined above. The Manin constant is

c_{f} = [S_{f}(Z) : S_{f}(A/Z)].

We are aware of no examples of newforms f for which c_{f} ≠ 1. It is very likely that one can extend methods known for elliptic curves (e.g., appendix of [Maz78]) to show that c_{f} is at least coprime to 2N. To conclude that the computations of this paper verify the BSD conjecture in certain cases, one must show something of this form.

3 Results

Step 1: Possible � All A_{f} of level N ≤ 1061 for which L(A_{f}, 1)/Ω_{f} is divisible by an odd square n_{f}.

Step 2: Possibly Explainable � All A_{f} from the previous list such that for each p|n_{f}, there exists a new factor B_{g} of positive analytic rank such that (A_{f} ∩ B_{g})[p] ≠ 0.

Step 3: Nontrivial Probably Explainable � For each f in the previous step we compute the geometric component groups and torsion of the A_{f} and B_{g}, and the part s_{f} of n_{f} coprime to all of these numbers, and to 2N. This table lists those f for which s_{f} > 1.

A_{f}  s_{f} dim w_{q} g_{p}  T T L/Ω  δ_{A}  B_{g}
389E  5^{2} 20 − 97  97  5^{2}  5  389A
433D  7^{2} 16 −  3^{2}  3^{2}  7^{2}  3 · 7 · 37  433A
446F 11^{2} 8 +− 1, 3  3  11^{2} 11 · 359353 446B
664F  5^{2}  8 −+ a, 1  1  5^{2}  5  664A
1061D 151^{2} 46 − 5 · 61 5 · 53 151^{2} 61 · 151 · 179 1061A

Step 4: Explanatory Factors More information about the factors B_{g}.

B_{g}  rank dim w_{q} g_{p} T δ_{B}  Comments
389A  2  1 − 1 1 5  first curve of rank 2
433A  2  1 − 1 1 7
446B  2  1 +− 1, 1 1 11
664A  2  1 −+ 1, 1 1 5
1061B 4  2 − 1 1 151 first surface of rank 4 [Bru95]

Factorizations

446 = 2 · 223, 664 = 2^{3} · 83

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