EXTENSIONS OF ELLIPTIC CURVES AND CONGRUENCES BETWEEN MODULAR FORMS

MATTHEW A. PAPANIKOLAS AND NIRANJAN RAMACHANDRAN

Abstract. We study the group Ext_{K}^{1}(A, B) where A and B are abelian varieties over a number field K. In particular, if K = ℚ, we relate Ext_{ℚ}^{1}(A, B) to congruences between modular forms.

1. Introduction

J. Milne [11, 12] has established striking connections between arithmetic and the extensions of abelian varieties over finite fields. Our aim here is to consider extensions of abelian varieties over number fields and connect them to arithmetic. Specifically, we find that extensions of elliptic curves are an apt tool for encoding congruences between modular forms.

For abelian varieties A and B over a field K, we consider the group Ext_{K}^{1}(A, B) of isomorphism classes of Yoneda extensions of A by B in the category of commutative group schemes over K. This group is torsion by the Poincaré reducibility theorem, but it need not be finite. For instance, if A and B are non-isogenous elliptic curves over K = ℂ, then Ext_{ℂ}^{1}(A, B) ≅ (ℚ/ℤ)^{4}. Moreover, if A is an elliptic curve with complex multiplication over ℂ, then a beautiful result of S. Lichtenbaum [16, Thm. 6.1] states that Ext_{ℂ}^{1}(A, A) is naturally isomorphic to the torsion subgroup of A(ℂ).

In the case of a finite field K = 𝔽_{q}, Milne [11] has shown that Ext_{𝔽_{q}}^{1}(A, B) is finite and also computed its order. These results are key to his work [12, 13] on the conjecture of Birch and Swinnerton-Dyer and the geometric analogue of Artin-Tate. The point is, as he shows, that certain Tate-Shafarevich and Brauer groups can be identified with suitable Ext-groups. Furthermore, his formula [11, Thm. 1] for the order of Ext_{𝔽_{q}}^{1}(A, B) has now been reinterpreted as a special value of an L-function and extended to integral motives [15].

We are naturally led to ask: If A and B are abelian varieties over a number field K, how is Ext_{K}^{1}(A, B) related to the arithmetic of A and B? Our main results provide some answers.

Theorem 1 (Milne-Ramachandran). Let A and B be abelian varieties over a number field K. Then Ext_{K}^{1}(A, B) is finite.

The next three results provide a unifying way to connect congruences between modular forms to extensions of abelian varieties over ℚ.

Theorem 2. Let A and B be non-isogenous elliptic curves over ℚ of conductors M and N respectively. Let f = ∑ a_{n}q^{n} and g = ∑ b_{n}q^{n} be their associated normalized newforms [2, 26]. Then, for any integer n with gcd(n, 2MN) = 1,

Research of the first author partially supported by an NSA grant.

a_{n} ≡ b_{n} (mod e),

where e is the exponent of Ext_{ℚ}^{1}(A, B).

Let A be an elliptic curve over ℚ of conductor N, and let ϕ_{A} : X_{0}(N) → A realize A as an optimal quotient of the modular curve X_{0}(N), i.e. A is a strong Weil curve. The kernel C of the induced map J_{0}(N) ↠ A is an abelian variety over ℚ, and there is an exact sequence

η : 0 → C → J_{0}(N) → A → 0.

Let d_{A} be the degree of the modular parametrization ϕ_{A}, and let r_{A} be the restricted congruence modulus of A which measures congruences away from 2N (see §4). There is a strong relation between d_{A}, r_{A}, and the exponent e_{A} of Ext_{ℚ}^{1}(A, C).

Theorem 3. Let A/ℚ be an elliptic curve which is an optimal quotient of J_{0}(N). Then

d_{A} | e_{A} | r_{A}.

In general, e_{A} does not divide the congruence modulus m_{A} of A (see §4). The analogous result for m_{A} requires a refinement of Ext_{ℚ}^{1}(A, C), which involves the Hecke algebra 𝕋 = ℤ[T_{1}, T_{2}, …] associated with J_{0}(N). Namely, let Ext_{ℚ,𝕋}^{1}(A, C) be the Yoneda Ext-group in the category of commutative 𝕋-group schemes over ℚ, i.e. groups G together with a homomorphism 𝕋 → End_{ℚ}(G). Write e_{A,𝕋} for the exponent of Ext_{ℚ,𝕋}^{1}(A, C).

Theorem 4. Let A/ℚ be an elliptic curve which is an optimal quotient of J_{0}(N). Then

d_{A} | e_{A,𝕋} | m_{A}.

We will show that the forgetful map Ext_{ℚ,𝕋}^{1}(A, C) → Ext_{ℚ}^{1}(A, C) is injective, and thus e_{A,𝕋} | e_{A}. Also, since d_{A} is expected to be unbounded [27, §6], Theorems 3 and 4 suggest that the order of Ext_{ℚ}^{1}(A, C) is unbounded.

These results indicate that extensions provide a convenient way to encode congruences between modular forms. Also, they suggest a way to generalize the theory of congruences due to H. Hida [8] and K. Ribet [18], even in the absence of Fourier coefficients.

We should expect that Ext_{K}^{1}(A, B) is also related to the special values of Lfunctions. The group Ext_{K}^{1}(A, B) of Theorem 1 is a variant of Ext_{K}^{1}(A, 𝔾_{m}) which, by the Weil-Barsotti formula, is the group A^{∗}(K) of rational points on the dual abelian variety A^{∗}. Moreover, the rank of A^{∗}(K) and the order of the torsion subgroup of A^{∗}(K) appear in the Birch and Swinnerton-Dyer conjecture for A.

Hida [8] has related congruence moduli to special values of L-functions, and with this in mind the order of Ext_{K}^{1}(A, B) should be related to a special value of the L-function L(A^{∗} ⊗B, s) (as in [15]). For example, for an elliptic curve A over ℚ the order of Ext_{ℚ}^{1}(A, A) appears in the denominator of the special value of L(Sym^{2} A, s) at s = 2 (cf. [7, Eq. (2)]). We plan to pursue these relationships in a future paper. Acknowledgements. We thank J. Milne for graciously allowing the unpublished proof of Theorem 1 to be included here. We also thank S. Lichtenbaum, G. Pappas, and J. Teitelbaum for their helpful remarks and suggestions.

2. Extensions over number fields

For any abelian group (scheme) G, let G_{n} denote the kernel of multiplication by the integer n. For a prime p, let T_{p}G = \underleftarrow{lim} G_{p^{m}} denote the associated Tate module.

Proof of Theorem 1. Let A and B be two abelian varieties over a number field K. For any integer n, taking Ext_{K}^{i}(−, B) of the Kummer sequence

0 → A_{n} → A →^{n} A → 0

gives a short exact sequence of abelian groups

(1)  0 → Hom_{K}(A, B) ⊗ ℤ/nℤ → Hom_{K}(A_{n}, B_{n}) → Ext_{K}^{1}(A, B)_{n} → 0,

where in the middle term we have used that Hom_{K}(A_{n}, B) = Hom_{K}(A_{n}, B_{n}).

This simple sequence (1) is fundamental to our discussion: it shows that the Extgroup is the appropriate tool to encode congruences of mod p Galois representations which do not come from global maps.

Taking the inverse limit of (1) over powers n = p^{m} of a prime p and using that Hom_{K}(A_{p^{m}}, B_{p^{m}}) = Hom_{K}(T_{p}A, B_{p^{m}}), we have an exact sequence

(2) 0 → Hom_{K}(A, B) ⊗ ℤ_{p} → Hom_{K}(T_{p}A, T_{p}B) → T_{p} Ext_{K}^{1}(A, B) → 0.

Faltings’ theorem [6] implies that the first homomorphism in (2) is in fact an isomorphism, and so T_{p} Ext_{K}^{1}(A, B) = 0 for all primes p. From (1), we see that Ext_{K}(A, B)_{n} is finite for all n. Thus the p-primary subgroup of Ext_{K}^{1}(A, B) is finite for all p.

Now by [10, Cor. 1], there is a constant c = c(A, B) so that for all primes p > c,

Hom_{K}(A, B) ⊗ ℤ/pℤ ≅ Hom_{K}(A_{p}, B_{p}).

Taking n = p in (1) gives Ext_{K}^{1}(A, B)_{p} = 0 for all p > c. This proves Theorem 1. □

Remark. In the case of elliptic curves over ℚ, the results of J.-P. Serre [20] provide upper bounds on the order of Ext_{ℚ}^{1}(A, B). Remark. Theorem 1 should also hold for pure integral motives [15], [21, p. 387] of equal weight and defined over a number field. This would follow from the properties expected of mixed motives such as semisimplicity of the category of pure isomotives, the adelic Tate conjecture [15, Lem. 3.1], and the conjecture [21, Conj. 10.10] of Serre.

3. Linking extensions over number fields and finite fields

Let A and B be abelian varieties over ℚ. Let N > 1 be the least integer such that both A and B extend to abelian schemes (denoted 𝒜 and ℬ) over ℤ[\frac{1}{N}]. Let S = 2N, and take Ext_{ℤ[1/S]}^{1}(𝒜, ℬ) for the Ext-group in the category of commutative group schemes over ℤ[\frac{1}{S}]. The change from N to S is due to the special nature of p = 2 and will be explained below.

If p ≥ 3 is a prime with p ∤ N, then both A and B have good reduction at p. Let Ã and B̃ denote the corresponding abelian varieties over 𝔽_{p}. For any positive integer n, we obtain a commutative diagram from (1) with exact rows

(3) \xymatrix{0 \ar[r] & Hom_{ℚ}(A, B) ⊗ ℤ/nℤ \ar[r] & Hom_{ℚ}(A_{n}, B_{n}) \ar[r] & Ext_{ℚ}^{1}(A, B)_{n} \ar[r] & 0 \\ 0 \ar[r] & Hom_{ℤ[ \frac{1}{S}]}(𝒜, ℬ) ⊗ ℤ/nℤ \ar[r] \ar[u]_{≀}^{a} \ar[d]_{h} & Hom_{ℤ[ \frac{1}{S}]}(𝒜_{n}, ℬ_{n}) \ar[r] \ar[u]_{≀}^{b} \ar[d]_{i} & Ext_{ℤ[ \frac{1}{S}]}^{1}(𝒜, ℬ)_{n} \ar[r] \ar[u]_{≀}^{c} \ar[d]_{j} & 0 \\ 0 \ar[r] & Hom_{𝔽_{p}}(Ã, B̃) ⊗ ℤ/nℤ \ar[r] & Hom_{𝔽_{p}}(Ã_{n}, B̃_{n}) \ar[r] & Ext_{𝔽_{p}}^{1}(Ã, B̃)_{n} \ar[r] & 0.}

The middle row is obtained by the analogue of (1) using 0 → 𝒜_{n} → 𝒜 → 𝒜 → 0 [14, §II.5]. The vertical maps are the natural restriction maps. Now, a is an isomorphism by the Néron mapping property of 𝒜 and ℬ; b is an isomorphism by [23, Cor. of Thm. 4.5.1] applied to the commutative finite flat group schemes 𝒜_{n}, ℬ_{n}; it follows that c is an isomorphism.

Put α = h ∘ a^{−1}, β = i ∘ b^{−1}, and γ = j ∘ c^{−1}. It is important to note that

β : Hom_{ℚ}(A_{n}, B_{n}) → Hom_{𝔽_{p}}(Ã_{n}, B̃_{n})

is injective: if gcd(n, p) = 1, this is clear; and if n = p^{r}, this follows from the faithful nature of the functor G ⇝ G ×_{ℤ_{p}} 𝔽_{p} which maps commutative finite flat group schemes over ℤ_{p} of p-power order to their special fiber (see the discussion following Example 4.3 on p. 409 of [3]). The injectivity of β provides an exact sequence

(4)  0 → ker(γ) → coker(α) → coker(β) → coker(γ) → 0.

Let us also record the following observation, which follows from (3).

Proposition 5. The natural map

Ext_{ℤ[ \frac{1}{S}]}^{1}(𝒜, ℬ) → Ext_{ℚ}^{1}(A, B)

is an isomorphism.

Remark. Some parts of this discussion can be generalized to other number fields.

Let us explain the reason for the switch from N to S (harmless if N is already even). Assume that N is odd, i.e. that 2 is a prime of good reduction for A and B. We have used implicitly that the absolute ramification index e of ℤ_{p} satisfies the condition e < p − 1; this fails only when p = 2.

In general for odd N, the map Ext_{ℤ[1/N]}^{1}(𝒜, ℬ) → Ext_{ℚ}^{1}(A, B) may not be an isomorphism. However, it is injective, because the map Hom_{ℤ[1/N]}(𝒜_{n}, ℬ_{n}) → Hom_{ℚ}(A_{n}, B_{n}) is injective. Given that a is an isomorphism over ℤ[\frac{1}{N}] by the Néron mapping property and that b is injective over ℤ[\frac{1}{S}], it suffices to check that the map Hom_{ℤ_{2}}(𝒜_{n}, ℬ_{n}) → Hom_{ℚ_{2}}(A_{n}, B_{n}), with n = 2^{r} a power of 2, is injective. This is clear [23, p. 152]. Thus, the Ext-group over ℤ[\frac{1}{N}] might be smaller, and so the analogue of Theorem 2 will be weaker.

In addition, the definition of β over ℤ[\frac{1}{N}] is problematic, due to the non-exactness of the Néron model functor for p = 2 (see [1, Ex. 4, p. 190] for a counterexample to [1, Thm. 4, p. 187]).

Proof of Theorem 2. We shall proceed by combining the previous discussion with Milne’s work on Ext-groups. As f and g are Hecke eigenforms, it suffices to show that a_{p} ≡ b_{p} (mod e) for all primes p with p ∤ 2MN. We consider the non-trivial case that a_{p} ≠ b_{p}. This implies that

Hom_{𝔽_{p}}(Ã, B̃) = 0,

and so from (3), (4), and Theorem 1, it follows that the natural map

γ : Ext_{ℚ}^{1}(A, B) ↪ Ext_{𝔽_{p}}^{1}(Ã, B̃)

is injective. The following lemma is an application of Milne’s formula [11, Thm. 3].

Lemma 6. For non-isogenous elliptic curves E, E′ over a finite field 𝔽, we have # Ext_{𝔽}^{1}(E, E′) = (#E(𝔽) − #E′(𝔽))^{2}.

Thus, continuing with the proof of Theorem 2, it follows that

# Ext_{𝔽_{p}}^{1}(Ã, B̃) = (a_{p} − b_{p})^{2}.

Moreover, Milne [12] has shown that

Ext_{𝔽_{p}}^{1}(Ã, B̃) ≅ �(B̃/L),

where �(B̃/L) is the Tate-Shafarevich group of the constant elliptic curve B̃ over the global function field L := 𝔽_{p}(Ã). Moreover, since A and B are elliptic curves, we know via the Cassels-Tate pairing that

�(B̃/L) ≅ T × T,

for some abelian group T. Thus the exponent e of Ext_{𝔽_{p}}^{1}(Ã, B̃) divides #T = |a_{p} − b_{p}|. Combining this information with the injectivity of γ, we see that

e | (a_{p} − b_{p}), ∀p ∤ 2MN,

thus finishing the proof of Theorem 2. □

Remark. V. Vatsal [25, Cor. 1.11] has shown how congruences, like those in Theorem 2, can often be translated into congruences for the algebraic part of the values at s = 1 of L(s, f, χ) and L(s, g, χ); see [25] for notation and details. Thus, nontriviality of Ext_{ℚ}(A, B) can be linked with congruences of special values of the associated L-functions.

4. Modular parametrizations and congruence moduli

Let J_{0}(N) be the Jacobian of the modular curve X_{0}(N) over ℚ, and let S_{2}(Γ_{0}(N)) be the space of weight two cusp forms on Γ_{0}(N). For an elliptic curve A over ℚ of conductor N, let f = ∑ a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be its associated normalized newform. The congruence modulus of A [27, §5] is

m_{A} := max \{m ∈ ℤ | \begin{matrix}∃g = ∑ b_{n}q^{n} ∈ (f)^{⊥} ∩ ℤ[[q]] so that \\ b_{n} ≡ a_{n} (mod m) for all n \end{matrix} \} ,

where (f)^{⊥} ⊆ S_{2}(Γ_{0}(N)) is the orthogonal complement with respect to the Petersson inner product. The restricted congruence modulus of A is

r_{A} := max \{r ∈ ℤ | \begin{matrix}∃g = ∑ b_{n}q^{n} ∈ (f)^{⊥} ∩ ℤ[[q]] so that \\ b_{n} ≡ a_{n} (mod r) for all n with gcd(n, 2N) = 1 \end{matrix} \} .

Assume that A is an optimal quotient of J_{0}(N) and that ϕ_{A} : X_{0}(N) → A is its modular parametrization. This induces an exact sequence of abelian varieties over ℚ,

η : 0 → C → J_{0}(N) →^{ϕ_{∗}} A → 0.

This sequence is also an exact sequence of 𝕋-group schemes over ℚ. For any integer n, we have the following commutative diagram with exact rows,

\xymatrix{0 \ar[r] & Hom_{ℚ,𝕋}(A, C) ⊗ ℤ/nℤ \ar[r] \ar[d] & Hom_{ℚ,𝕋}(A_{n}, C_{n}) \ar[r] \ar[d]_{κ} & Ext_{ℚ,𝕋}^{1}(A, C)_{n} \ar[r] \ar[d]_{ι} & 0 \\ 0 \ar[r] & Hom_{ℚ}(A, C) ⊗ ℤ/nℤ \ar[r] & Hom_{ℚ}(A_{n}, C_{n}) \ar[r] & Ext_{ℚ}^{1}(A, C)_{n} \ar[r] & 0.}

The bottom row is precisely (1), and since the Kummer sequence 0 → A_{n} → A → A → 0 is a sequence of 𝕋-group schemes, the top row is obtained in the same manner as (1) but with the requirement of 𝕋-equivariance. Note also that the top row is an exact sequence of 𝕋-modules. Now since Hom_{ℚ}(A, C) = Hom_{ℚ,𝕋}(A, C) = 0 and the map κ is injective, it follows that ι is also injective. Alternately, any element of Ext_{ℚ,𝕋}^{1}(A, C) which splits in Ext_{ℚ}^{1}(A, C) is split by a 𝕋-equivariant morphism. By Theorem 1, the injectivity of ι shows that Ext_{ℚ,𝕋}^{1}(A, C) is finite.

Proposition 7. The order of η in Ext_{ℚ,𝕋}^{1}(A, C) is equal to the degree of ϕ_{A}.

Proof. Let d_{A} = deg ϕ_{A}. Because A is an optimal quotient of X_{0}(N), the dual ϕ^{∗} : A → J_{0}(N) of ϕ_{∗} is injective, and the composition ϕ_{∗} ∘ ϕ^{∗} ∈ Hom_{ℚ}(A, A) is multiplication by d_{A}. From the exact sequence 0 → A → J_{0}(N) → C^{∗} → 0 and the fact that Hom_{ℚ}(A, C^{∗}) = 0, it follows that the map

Hom_{ℚ}(A, A) →^{∼} Hom_{ℚ}(A, J_{0}(N))

induced by ϕ^{∗} is an isomorphism. Furthermore, the map

Hom_{ℚ}(A, A) → Hom_{ℚ}(A, J_{0}(N)) → Hom_{ℚ}(A, A),

induced by ϕ_{∗}∘ϕ^{∗}, is multiplication by d_{A}, the image of Hom_{ℚ}(A, J_{0}(N)) is precisely d_{A}(Hom_{ℚ}(A, A)).

The long exact sequence for Ext_{ℚ}^{i}(A, −) applied to η begins

0 → Hom_{ℚ}(A, J_{0}(N)) →^{ϕ_{∗}} Hom_{ℚ}(A, A) → Ext_{ℚ}^{1}(A, C),

and the image of an endomorphism α ∈ Hom_{ℚ}(A, A) in Ext_{ℚ}^{1}(A, C) is the pull-back α^{∗}η. Therefore, because Hom_{ℚ}(A, A) = ℤ, it follows that the image of Hom_{ℚ}(A, A) in Ext_{ℚ}^{1}(A, C) is the subgroup generated by η. Thus the order of η in Ext_{ℚ}^{1}(A, C) is d_{A}, and since η represents a class in Ext_{ℚ,𝕋}^{1}(A, C) ⊆ Ext_{ℚ}^{1}(A, C), we are done. □

There are two ways to define a 𝕋-module structure on Ext_{ℚ}^{1}(A, C), namely by pushing out along C or pulling back along A. These two 𝕋-module structures need not coincide. However, because Ext_{ℚ,𝕋}^{1}(A, C) consists of classes of extensions which are 𝕋-equivariant, it follows that the two 𝕋-module definitions restricted to Ext_{ℚ,𝕋}^{1}(A, C) are the same. In fact, Ext_{ℚ,𝕋}^{1}(A, C) is the largest subgroup of Ext_{ℚ}^{1}(A, C) on which the two 𝕋-module structures agree.

Proof of Theorem 4. By Proposition 7 it suffices now to prove that e_{A,𝕋} | m_{A}. Let I_{A} and I_{C} be the kernels of the maps 𝕋 → End_{ℚ}(A) and 𝕋 → End_{ℚ}(C). By the general considerations of [5, §2], we see that m_{A} is the exponent of 𝕋_{A,C} := 𝕋/(I_{A}+ I_{C}) as an abelian group. Now clearly both I_{A} and I_{C} annihilate Ext_{ℚ,𝕋}^{1}(A, C), and so Ext_{ℚ,𝕋}^{1}(A, C) is a 𝕋_{A,C}-module. Thus we must have e_{A,𝕋} | m_{A}. □

Proof of Theorem 3. Again by Proposition 7 it suffices to prove that e_{A} | r_{A}. Let 𝕋′ ⊆ 𝕋 be the subalgebra generated by all T_{n} with gcd(n, 2N) = 1, and let I′_{A} and I′_{C} be the kernels of the maps 𝕋′ → End_{ℚ}(A) and 𝕋′ → End_{ℚ}(C). From [5, §2], it follows that r_{A} is the exponent of 𝕋′_{A,C} := 𝕋′/(I′_{A} + I′_{C}). If we can show that Ext_{ℚ}^{1}(A, C) is a 𝕋′_{A,C}-module, then we are done. Namely, we need to show that the two operations of 𝕋′ on Ext_{ℚ}^{1}(A, C) coincide.

Consider T_{p} with p ∤ 2N. Since Hom_{ℚ}(A, C) = 0, (1) implies that Ext_{ℚ}^{1}(A, C)_{n} ≅ Hom_{ℚ}(A_{n}, C_{n}). From the Eichler-Shimura relations [9, Thm. 4.2.1], we have that

(5)  T_{p} = F + V ∈ End_{𝔽_{p}}(J̃),

where J̃ is the reduction of J_{0}(N) modulo p, and F and V are the p-th power Frobenius and Verschiebung. As in (3), the natural map

(6)  Hom_{ℚ}(A_{n}, C_{n}) ↪ Hom_{𝔽_{p}}(Ã_{n}, C̃_{n})

is injective. Since both A and C are subabelian varieties of J_{0}(N), the action of T_{p} on Ã_{n} and C̃_{n} is determined by the restriction of (5). In particular, for any α ∈ Hom_{ℚ}(A_{n}, C_{n}), its image α̃ ∈ Hom_{𝔽_{p}}(Ã_{n}, C̃_{n}) satisfies

α̃ ∘ T_{p}|Ã_{n} = T_{p}|C̃_{n} ∘ α̃.

The injectivity of (6) shows that in fact α∘T_{p}|_{A_{n}} = T_{p}|_{C_{n}} ∘α, and we are done. □

Corollary 8. If the conductor N of A is square-free, then d_{A} = e_{A,𝕋} = m_{A}.

Proof. For prime N, this follows from Theorem 4 and [27, Thm. 3], and for general N, by the recent work of Ribet [19] which resolves a conjecture of A. Agashe and W. Stein. □

Remark. If N is prime and if the form g = ∑ b_{n}q^{n} ∈ (f)^{⊥} ∩ ℤ[[q]], which gives b_{n} ≡ a_{n} (mod r_{A}) for all n with gcd(n, 2N) = 1, also satisfies b_{2} ≡ a_{2} (mod r_{A}), then in fact d_{A} = e_{A,𝕋} = m_{A} = e_{A} = r_{A} by the Sturm bound [22]. Example. Theorems 3 and 4 are best possible in the following sense. The mod 3 representation of the elliptic curve 90A1 in [4] of conductor 90, A : y^{2} + xy = x^{3} − x^{2} + 6x, has image ( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix}) [17, Thm. 3.2]. Likewise, the mod 3 representation of the elliptic curve 90C1 in [4], also of conductor 90, B : y^{2}+xy+y = x^{3}−x^{2}+13x−61, has image ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ). Thus there is a non-trivial element α ∈ Hom_{ℚ}(A_{3}, B_{3}), which induces a homomorphism α : A_{3} → C_{3}, where C is the kernel of J_{0}(90) → A. Thus Ext_{ℚ}^{1}(A, C)_{3} ≅ Hom_{ℚ}(A_{3}, C_{3}) has a non-trivial element of order 3. However, direct computation provides d_{A} = 8 and m_{A} = 8. In particular, e_{A} ∤ m_{A}. Remark. If A is an elliptic curve over ℚ and ϕ : J → A realizes A as an optimal quotient of the Jacobian J of a Shimura curve, then we have an exact sequence

η : 0 → C → J →^{ϕ} A → 0.

Theorem 3 suggests that the order of Ext_{ℚ}^{1}(A, C) provides a “congruence modulus” even in this setting where the lack of Fourier coefficients makes the theory of Hida-Ribet inapplicable. In fact, the Hida constant c_{A} defined by E. Ullmo [24, p. 326] is the order of η ∈ Ext_{ℚ}^{1}(A, C).

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Department of Mathematics, Brown University, Providence, RI 02912 USA

E-mail address: map@math.brown.edu

Department of Mathematics, University of Maryland, College Park, MD 20742 USA

EXTENSIONS OF ELLIPTIC CURVES 9

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

E-mail address: atma@math.umd.edu