EXTENSIONS OF ABELIAN VARIETIES DEFINED OVER A NUMBER FIELD

MATTHEW A. PAPANIKOLAS AND NIRANJAN RAMACHANDRAN

Abstract. We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

1. Introduction

J. Milne [15, 16] has established striking connections between arithmetic and the extensions of abelian varieties over finite fields. Our aim here is to relate extensions of abelian varieties over number fields to special values of L-functions of motives and congruences between modular forms.

For abelian varieties A and B over a field K, 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 [19, 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 [15, Thm. 3] has shown that Ext_{𝔽_{q}}^{1}(A, B) is finite and has also computed its order (see Theorem 1). Moreover, this formula for the order has now been reinterpreted as a special value of an L-function and extended to integral motives over a finite field [18]. We are naturally led to ask:

If A and B are abelian varieties over a number field K, what is the arithmetic significance of the group Ext_{K}^{1}(A, B)?

As a starting point, we see that over a number field K, not only is the group Ext_{K}^{1}(A, B) finite (Theorem 2), but its order is intimately connected with arithmetic. More precisely, Ext_{K}^{1}(A, B) is related to the following objects:

• (Theorem 3) the special value of the L-function for a (tensor) motive, described by the Bloch-Kato conjecture [1];

• (Theorem 5) congruences between modular forms, when A and B are elliptic curves over ℚ;

• (Theorems 7, 8) the congruence modulus and modular degree of an elliptic curve over ℚ.

Date: September 2, 2003.

1991 Mathematics Subject Classification. Primary 11G05; Secondary 11F33, 14K15.

Research of the first author partially supported by NSA and NSF grants. Research of the second author partially supported by MPIM.

Whereas the work of M. Flach [10] and H. Hida [11] on the special value of the symmetric square L-function of an elliptic curve over ℚ relates the first and the third items, the Ext-group relates all three, and in a manner suggestive of generalizations. For example, the order of Ext_{ℚ}^{1}(A, A) appears in the value of the symmetric square L-function of an elliptic curve A over ℚ (Theorem 3). Plus, the order of Ext_{ℚ}^{1}(A, B) conjecturally appears in the special value of the Rankin-Selberg L-function of the modular forms associated with the (non-isogenous) elliptic curves A and B over ℚ.

Another useful application is that the Ext-group enables us to interpret the denominator in the Bloch-Kato conjecture [1, Eq. (5.15.1)]. Indeed this remaining term has lacked an explicit conceptual description. Furthermore, extensions provide a way to generalize (to abelian varieties over number fields other than ℚ) the theory of congruences due to Hida [11] and K. Ribet [21], even in the absence of Fourier coefficients.

Extensions seem to be flexible enough to study congruences between automorphic forms for different groups, such as between an elliptic modular form and a Hilbert modular form. The simple behavior of the Ext-group under base-change (see end of §3) should be useful, for instance, in the study of congruences between elliptic modular forms and their lifts.

Finally, the Ext-group is finite for abelian varieties over function fields; its order should be related to special values of L-functions, as over number fields. In particular, the p-part of the Ext-group should shed light on the formulation of the p-part of the Bloch-Kato conjecture in characteristic p > 0, thus completing the non-p description [1, Prop. 5.21].

The overall goal of this paper is to investigate the role of extensions of abelian varieties in arithmetic geometry. Extensions are interesting in their own right, and even more to our delight we see that they cleanly unify the deep work of many others.

Acknowledgements. We thank J. Milne for graciously allowing the unpublished proof of Theorem 2 in §2 to be included here. We thank K. Ribet for sharing his proof of Theorem 5 in §4 with us and for permitting us to include it to supplement our own. We also thank S. Ahlgren, V. Balanzew, S. Lichtenbaum, G. Pappas, and J. Teitelbaum for their helpful remarks and suggestions.

2. Preliminaries

Extensions over finite fields. We recall the fundamental results of Milne [15], [16], which were crucial in his work on the conjecture of Birch and Swinnerton-Dyer for constant abelian varieties over finite fields.

Theorem 1 (Milne (a) [15, Thm. 3]; (b) [16]). Let A and B be abelian varieties over a finite field 𝔽_{q}.

(a) The group Ext_{𝔽_{q}}^{1}(A, B) is finite, and its order is given by

# Ext_{𝔽_{q}}^{1}(A, B) = ±\frac{q^{d_{A}d_{B}}}{D} ∏_{a_{i}≠b_{j}} (1 − \frac{a_{i}}{b_{j}} ) ,

where a_{i} and b_{j} are the eigenvalues of Frobenius associated with A and B over 𝔽_{q}, d_{A} and d_{B} are the dimensions of A and B, and D is the discriminant of the trace pairing Hom_{𝔽_{q}}(A, B) × Hom_{𝔽_{q}}(B, A) → End(A) → ℤ.

(b) Let J be the Jacobian of a smooth projective curve C over 𝔽_{q}. The group Ext_{𝔽_{q}}^{1}(J, A) is isomorphic to the Tate-Shafarevich group �(A/𝔽_{q}(C)) of the constant abelian variety A over the function field 𝔽_{q}(C) of C.

Finiteness of Ext_{K}^{1}(A, B) 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, and let T G = \underleftarrow{lim} G_{n} denote the total Tate module.

Theorem 2 (Milne-Ramachandran). Let A and B be abelian varieties over a number field K. Then Ext_{K}^{1}(A, B) is finite.

Proof. 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}).

Remark. This simple sequence (1), valid for any field K, is fundamental to our discussion throughout: it shows that the Ext-group is the appropriate tool to encode congruences of mod p Galois representations that do not come from global maps.

Now 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 [8] 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}^{1}(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 [14, 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 2. □

Remark. (i) In the case of elliptic curves over ℚ, the results of J.-P. Serre [23] provide upper bounds on the order of Ext_{ℚ}^{1}(A, B). (ii) The constant c = c(A, B) from [14, Cor. 1] bounds the primes dividing # Ext_{K}^{1}(A, B). (iii) The group Ext_{K}^{1}(A, B) is not invariant under isogeny.

Remark. Theorem 2 should also hold for pure integral motives [18], [24, 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 [18, Lem. 3.1], and the conjecture [24, Conj. 10.10] of Serre.

Representability issues. Fix abelian varieties A and B over a field K. The group Ext_{K}^{1}(A, B) is analogous to the group Ext_{K}^{1}(A, 𝔾_{m}), but there some differences. For example, the functor S ↦ Ext_{S}^{1}(A × S, 𝔾_{m}) on the category of schemes over K is representable (in fact, by the dual abelian variety). However, the corresponding functor for Ext_{K}^{1}(A, B) is not representable. If L is an extension of K, then the natural map Ext_{K}^{1}(A, B) → Ext_{L}^{1}(A, B) need not be injective. In fact, if L/K is a Galois extension with Galois group G, the kernel of this map can be computed via the exact sequence

0 → H^{1}(G, Hom_{L}(A, B)) → Ext_{K}^{1}(A, B) → Ext_{L}^{1}(A, B)^{G}.

Also, even over a number field K, the group Ext_{K}^{1}(A, 𝔾_{m}) can be infinite whereas Ext_{K}^{1}(A, B) is always finite. This behavior is related to the fact that the weight of the motive A is different from that of the motive 𝔾_{m}.

3. Special values of L-functions

Given that Theorem 1(a) can be reinterpreted [18] as a formula for the special value of the L-function of a motive over a finite field, it is natural to ask whether a similar phenomenon occurs over number fields. We begin with the case of selfextensions of an elliptic curve over ℚ.

Symmetric square of an elliptic curve. Let A be an elliptic curve over ℚ with End\overline{ℚ}(A) = ℤ. Let L(Sym^{2} A, s) be the symmetric square L-function of A, or rather the L-function attached to the compatible system of Galois representations Sym^{2} T_{p}A (see [10]).

The following theorem demonstrates how the Ext-group is related to special values of L-functions. At once we see that the order of Ext_{ℚ}^{1}(A, A) is incorporated in the critical value L(Sym^{2} A, 2) and that the Tate-Shafarevich group of Sym^{2} A can be reinterpreted in terms of extensions. The notations for the period Ω(2) and Tamagawa factors c_{p} are consistent with [10, Eq. (2)]. Also, for a perfect field K, let G_{K} := Gal(\overline{K}/K).

Theorem 3. If A is an elliptic curve over ℚ with End\overline{ℚ}(A) = ℤ, then up to a power of 2,

\frac{L(Sym^{2} A, 2)}{Ω(2)} = \frac{#�(Ext1 ; \overline{ℚ}(A, A))}{# Ext_{ℚ}^{1}(A, A) · # Ext1 ; \overline{ℚ}(A, A)(1)^{G_{ℚ}}} ∏_{p} c_{p}.

Proof. Since the Bloch-Kato conjecture for the symmetric square of an elliptic curve over ℚ has been proven by F. Diamond, Flach, and L. Guo [7], we need only show that the terms in [10, Eq. (2)] match up with the terms in our formula.

The total Tate module T A of A is a rank two module over ℤ̂ := \underleftarrow{lim} ℤ/nℤ. By the Weil pairing, T A is isomorphic to the Tate twist T A^{∨}(1) of the dual module T A^{∨}. From (2), we see that T Ext1 ; \overline{ℚ}(A, A) is isomorphic to the quotient of Endℤ̂(T A) by

the ℤ̂-submodule generated by the identity map. Via the self-duality Endℤ̂(T A) ≅ T A ⊗ T A^{∨}, we have

(3)  T Ext1 ; \overline{ℚ}(A, A) ≅ (Sym^{2} T A)(−1).

The terms in the denominator of [10, Eq. (2)] are #H^{0}(G_{ℚ}, (Sym^{2} T A)(−1)⊗ℚ/ℤ) and #H^{0}(G_{ℚ}, (Sym^{2} T A) ⊗ ℚ/ℤ). By (3), the former is # Ext1 ; \overline{ℚ}(A, A)^{G_{ℚ}}, and the latter is # Ext1 ; \overline{ℚ}(A, A)(1)^{G_{ℚ}}. Moreover, the numerator of [10, Eq. (2)] is

#�((Sym^{2} T A) ⊗ ℚ/ℤ) = #�((Sym^{2} T A)(−1) ⊗ ℚ/ℤ),

where the equality follows from [9, Thm. 1]. The proof of the theorem is then complete by the following lemma. □

Lemma 4. Let A be an elliptic curve over a number field K with End_{\overline{K}}(A) = ℤ. For any odd integer n, Ext_{K}^{1}(A, A)_{n} ≅ Ext1 ; \overline{K}(A, A)_{n}^{G_{K}}.

Proof. By taking the long exact sequence of G_{K}-cohomology of (1) with K = \overline{K}, we obtain an exact sequence

0 → Hom_{K}(A, A) ⊗_{ℤ} ℤ/nℤ → Hom_{K}(A_{n}, A_{n}) → Ext1 ; \overline{K}(A, A)_{n}^{G_{K}}

→ Hom(G_{K}, ℤ/nℤ) →^{ϕ} H^{1}(G_{K}, Hom_{\overline{K}}(A_{n}, A_{n})).

If f ∈ ker ϕ, then ϕ(f) is represented by a coboundary which takes values in the trace 0 space of Hom_{\overline{K}}(A_{n}, A_{n}). However, by definition ϕ(f)(σ) = f(σ) · id, for σ ∈ G_{K}. Thus for all σ ∈ G_{K}, we have 2f(σ) = 0, which implies f = 0. Therefore, ϕ is injective, and the result follows from (1). □

Remark. As noted earlier, the work of Serre [23] provides bounds on the order of Ext_{ℚ}^{1}(A, A) for any elliptic curve A over ℚ: if p is an odd prime of good reduction such that the natural map G_{ℚ} → Aut_{\overline{K}}(A_{p}) is onto, then p does not divide # Ext_{ℚ}^{1}(A, A). These considerations provide a conceptual reason for the conditions in Flach’s main result [10, Thm. 1]: they are tantamount to the triviality of H^{0}(G_{ℚ}, (Sym^{2} T A)(−1) ⊗ ℚ/ℤ), whose order appears in the denominator of [10, Eq. (2)].

Bloch-Kato conjecture. More generally, for abelian varieties A and B over a number field K, consider the Bloch-Kato conjecture [1, Eq. (5.15.1)] which predicts the special value of the L-function associated with the motive M := A⊗B^{∨} at s = 0. Here B^{∨} is the dual motive of B; note also that M and M^{∨} = A^{∨} ⊗ B give the same L-function. Let us begin with the case Hom_{\overline{K}}(A, B) = 0. The denominator is none other than the order of Ext_{K}^{1}(A, B)! Indeed, arguments similar to those of Theorem 3 and Lemma 4 show that Ext_{K}^{1}(A, B) is canonically isomorphic to H^{0}(G_{K}, T M^{∨} ⊗ ℚ/ℤ). In fact, by [1, Conj. 5.15], the order of Ext_{K}^{1}(A, B) is the product of the Tamagawa number Tam(M) and the order of the Tate-Shafarevich group �(M) of M.

In the case that Hom_{\overline{K}}(A, B) ≠ 0, the relevant L-function is that of the quotient of M by the Artin motive corresponding to Hom_{\overline{K}}(A, B).

Remark. The previous discussion applied to the case of non-isogenous elliptic curves A and B over ℚ shows that the order of Ext_{ℚ}^{1}(A, B) appears conjecturally in the special value of L(A ⊗ B^{∨}, s), which is none other than the Rankin-Selberg convolution L-function of the corresponding modular forms.

4. Congruences between modular forms

The aim of this section is the proof of the following theorem.

Theorem 5. 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 [3, 29]. Then, for any integer n with gcd(n, 2MN) = 1,

a_{n} ≡ b_{n} (mod e),

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

We include two proofs of this theorem. The first one is indirect, its main ingredients being (i) the reduction to the case of a finite field, which depends crucially on the theory of finite flat group schemes, and (ii) Theorem 1. After reading a previous version of this paper, Ribet found a direct proof relying on the theta operator, which we also include here.

Remark. It is possible to strengthen Theorem 5. Since f and g depend only on the isogeny class of A and B and the group Ext_{ℚ}^{1}(A, B) is not invariant under isogeny (in general), one can replace e in the theorem above by (i) the exponent e′ of the group Ext_{ℚ}^{1}(A′, B′) where A′ and B′ are isogenous to A and B over ℚ; or, (ii) the least common multiple of the set of all e′.

Main reduction: 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[ \frac{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

(4) \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 [17, §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 [26, 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 [4]). The injectivity of β provides an exact sequence

(5)  0 → ker(γ) → coker(α) → coker(β) → coker(γ) → 0.

Let us also record the following observation, which follows from (4).

Proposition 6. 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 (of course 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 of ℤ_{p} is strictly less than p − 1; this fails only when p = 2.

In general for odd N, the map Ext_{ℤ[ \frac{1}{N}]}^{1}(𝒜, ℬ) → Ext_{ℚ}^{1}(A, B) may not be an isomorphism. However, it is injective, because the map Hom_{ℤ[ \frac{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 in fact 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 [26, p. 152]. Thus, the Ext-group over ℤ[\frac{1}{N}] might be smaller, and so the analogue of Theorem 5 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 [2, Ex. 4, p. 190] for a counterexample to [2, Thm. 4, p. 187]).

Proof 1 of Theorem 5. 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 (4), (5), and Theorem 2, one has the injectivity of γ : Ext_{ℚ}^{1}(A, B) ↪ Ext_{𝔽_{p}}^{1}(Ã, B̃). By Theorem 1(a),

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

(Milne remarks that this formula was also observed by J. Tate.) Moreover, since A and B are elliptic curves, Theorem 1(b) and the Cassels-Tate pairing combine to give

�(B̃/𝔽_{q}(Ã)) ≅ T × T,

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

e | (a_{p} − b_{p}), ∀p ∤ 2MN.
□

Proof 2 of Theorem 5 (Ribet). Again, because f and g are Hecke eigenforms, it suffices to show that a_{p} ≡ b_{p} (mod λ) for all primes p ∤ 2MN and prime powers λ = ℓ^{m} dividing e. Fix now such p and λ = ℓ^{m}.

Suppose first that p ≠ ℓ. Letting σ ∈ G_{ℚ} be a Frobenius element at p, the Eichler-Shimura relations [12, Thm. 4.2.2] dictate that

σ^{2} − a_{p}σ + p = 0 on A_{λ},
σ^{2} − b_{p}σ + p = 0 on B_{λ}.

Now from (1) with n = λ, it follows that there is a Galois equivariant morphism α : A_{λ} → B_{λ} with exponent λ. Because α is G_{ℚ}-equivariant, it follows that the operators a_{p}σ and b_{p}σ agree on the image of α in B_{λ}, which has exponent λ. Hence a_{p} ≡ b_{p} (mod λ).

Now suppose that p = ℓ. In particular, ℓ ∤ 2MN. Now by the preceding paragraph, we see that a_{n} ≡ b_{n} (mod λ) whenever n is coprime to 2MNℓ. Let f̂ and ĝ be modular forms with Fourier expansions

f̂ = ∑_{\begin{matrix}n=1 \\ gcd(n,2MN)=1 \end{matrix}}^{∞} a_{n}q^{n},  ĝ = ∑_{\begin{matrix}n=1 \\ gcd(n,2MN)=1 \end{matrix}}^{∞} b_{n}q^{n}.

(These modular forms can be obtained by twisting twice by appropriate quadratic characters, and as such have level dividing say 256M^{5}N^{5}.) Thus, if we let Θ = q \frac{d}{dq} be the usual Θ operator on modular forms, we see that Θ^{m} annihilates f̂ − ĝ modulo λ. In particular, as power series in q,

Θ(f̂− ĝ) ≡ 0 (mod ℓ).

Since ℓ is odd, the Θ operator is injective modulo ℓ on modular forms of weight 2 [13, §II]. Therefore f̂ ≡ ĝ (mod ℓ). Moreover, if m > 1, we see that Θ annihilates (f̂− ĝ)/ℓ modulo ℓ, and thus f̂ ≡ ĝ (mod ℓ^{2}). By induction, f̂ ≡ ĝ (mod λ). □

Remark. In comparing the two proofs, Ribet notes that, when e is odd, the arguments in both proofs of Theorem 5 imply that a_{n} ≡ b_{n} (mod e) for n coprime to MN, instead of to 2MN. In some sense, having the Θ operator injective modulo odd primes is analogous to having the absolute ramification index of ℤ_{p} strictly less than p − 1 in Proposition 6.

Remark. V. Vatsal [28, Cor. 1.11] has shown how congruences, like those in Theorem 5, 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 [28] for notation and details. Thus, the nontriviality of Ext_{ℚ}^{1}(A, B) can be linked with congruences of special values of the associated L-functions.

5. Modular parametrizations and congruence moduli

We now turn to describing how the theory of congruence modulus of Hida [11], Ribet [21] and D. Zagier [30] fits in with the Ext-group.

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 [30, §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) (i.e. a strong Weil curve) and that ϕ_{A} : X_{0}(N) → A is its modular parametrization. This induces an exact sequence of abelian varieties over ℚ,

(6)  η : 0 → C → J_{0}(N) →^{ϕ_{∗}} A → 0.

Recall the well-known result observed by Ribet and Zagier [30]: letting d_{A} be the degree of ϕ_{A}, we have

d_{A} | m_{A}.

Of course m_{A} divides r_{A}. Our next result, that the exponent e_{A} of Ext_{ℚ}^{1}(A, C) sits nicely between d_{A} and r_{A}, ultimately relies on the Eichler-Shimura relations.

Theorem 7. With notations as above, one has

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

In general, e_{A} does not divide the congruence modulus m_{A}. 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). We show below that the natural forgetful map Ext_{ℚ,𝕋}^{1}(A, C) → Ext_{ℚ}^{1}(A, C) is injective, and thus e_{A,𝕋} | e_{A}.

Theorem 8. With notations as above, one has

d_{A} | e_{A,𝕋} | m_{A}.

Since d_{A} is unbounded, Theorems 7 and 8 show that the order of Ext_{ℚ}^{1}(A, C) is unbounded.

The sequence (6) 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) = 0 = Hom_{ℚ,𝕋}(A, C) 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 2, the injectivity of ι shows that Ext_{ℚ,𝕋}^{1}(A, C) is finite.

Proposition 9. 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}, and so 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. □

Now 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 8. By Proposition 9 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 [6, §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 7. Again by Proposition 9 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 [6, §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 [12, Thm. 4.2.1], we have that

(7)  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 (4), the natural map

(8)  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 (7). 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 (8) shows that in fact α∘T_{p}|_{A_{n}} = T_{p}|_{C_{n}} ∘α, and we are done. □

Corollary 10. 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 8 and [30, Thm. 3], and for general N, by the recent work of Ribet [22] 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 [25].

Example. Theorems 7 and 8 are best possible in the following sense. The mod 3 representation of the elliptic curve 90C1 in [5] of conductor 90,

A : y^{2} + xy + y = x^{3} − x^{2} + 13x − 61,

has image ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ) [20, Thm. 3.2]. Likewise, the mod 3 representation of the elliptic curve 90A1 in [5], also of conductor 90,

B : y^{2} + xy = x^{3} − x^{2} + 6x,

has image ( \begin{matrix}1 & ∗ \\ 0 & ∗ \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} = 16 and m_{A} = 16. In particular, d_{A} ≠ m_{A} and 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 7 suggests that the exponent 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 [27, p. 326] is the order of η ∈ Ext_{ℚ}^{1}(A, C).

References

[1] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, Progress in Mathematics, 90, Birkhauser, Boston, 1990, pp. 33–400.

[2] S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer-Verlag, Berlin, 1990.

[3] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.

[4] B. Conrad, The flat deformation functor, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 373–420.

[5] J. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997.

[6] F. Diamond, On congruence modules associated to Λ-adic forms, Compositio Math. 71 (1989), 49–83.

[7] F. Diamond, M. Flach, and L. Guo, Adjoint motives of modular forms and the Tamagawa number conjecture, preprint, 2001.

[8] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.

[9] M. Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990), 113–127.

[10] M. Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), 307–327.

[11] H. Hida, Congruence of cusp forms and special values of their zeta functions, Invent. Math. 63 (1981), 225–261.

[12] H. Hida, Geometric modular forms and elliptic curves, World Scientific Publishing, River Edge, NJ, 2000.

[13] N. M. Katz, A result on modular forms in characteristic p, Modular functions of one variable, V, Lecture Notes in Math., vol. 601, Springer, Berlin, 1977, pp. 53–61.

[14] D. W. Masser and G. Wüstholz, Refinements of the Tate conjecture for abelian varieties, Abelian varieties (Egloffstein, 1993), de Gruyter, Berlin, 1995, pp. 211–223.

[15] J. S. Milne, Extensions of abelian varieties defined over a finite field, Invent. Math. 5 (1968), 63–84.

[16] J. S. Milne, The Tate-Šafarevič group of a constant abelian variety, Invent. Math. 6 (1968), 91–105.

[17] J. S. Milne, Arithmetic duality theorems, Academic Press Inc., Boston, MA, 1986.

[18] J. S. Milne and N. Ramachandran, Integral motives and special values of zeta functions, preprint, math.NT/0204065, 2002.

[19] M. A. Papanikolas and N. Ramachandran, A Weil-Barsotti formula for Drinfeld modules, J. Number Theory 98 (2003), 407-431.

[20] A. Reverter and N. Vila, Images of mod p Galois representations associated to elliptic curves, Canad. Math. Bull. 44 (2001), 313–322.

[21] K. A. Ribet, Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 503–514.

[22] K. A. Ribet, Congruences and the modular degree, in preparation.

[23] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.

[24] J.-P. Serre, Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 377–400.

[25] J. Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985), Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280.

[26] J. Tate, Finite flat group schemes, Modular forms and Fermat’s last theorem (Boston, MA, 1995), Springer, New York, 1997, pp. 121–154.

[27] E. Ullmo, Sur la constante de Hida des courbes modulaires et des courbes de Shimura, J. Théor. Nombres Bordeaux 13 (2001), 325–337.

[28] V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–419.

[29] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.

[30] D. Zagier, Modular parametrizations of elliptic curves, Canad. Math. Bull. 28 (1985), 372– 384.

Department of Mathematics, Texas A&M University, College Station, TX 77843

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

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

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

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