OPTIMAL QUOTIENTS OF MODULAR JACOBIANS

Matthew Emerton

University of Chicago

Abstract. Let N be a positive integer, and f a normalized newform of weight two on Γ_{0}(N). Attached to f is an optimal quotient A_{f} of the Jacobian J_{0}(N). We prove two results concerning such optimal quotients. (A) Let 𝕋 denote the algebra of endomorphisms of J_{0}(N) generated by the Hecke operators, and let 𝔉_{f} denote the ideal of fusion of f in 𝕋. If Â_{f} denotes the dual abelian variety to A_{f}, then the canonical polarization of J_{0}(N) induces a polarization θ_{f} : Â_{f} → A_{f}. We show that there is an embedding ker(θ_{f}) ⊂ J_{0}(N)[𝔉_{f}] whose cokernel is supported at maximal ideals 𝔪 of 𝕋 for which J_{0}(N)[𝔪] is not two-dimensional. (B) If N is prime, let C denote the subgroup of J_{0}(N) generated by the divisor 0 − ∞ on X_{0}(N). Mazur has shown that C is equal to the full torsion subgroup of J_{0}(N)(ℚ), and that specialization modulo N induces an isomorphism of C with the group of connected components Φ_{J_{0}(N)} of the characteristic N fibre of the Néron model of J_{0}(N). We prove that analogous results hold for every optimal quotient of prime conductor, thereby confirming William Stein’s refined Eisenstein conjecture.

Let N be a positive integer, let S(N) denote the space of weight two cuspforms on Γ_{0}(N), and let 𝕋 denote the ℤ-algebra of Hecke operators acting on S(N). If f is a newform, let I_{f} denote the ideal in 𝕋 that is the kernel of the map 𝕋 → ℂ induced by the action of the Hecke operators on f. We let J_{0}(N) denote the Jacobian of the modular curve X_{0}(N), and write A_{f} = J_{0}(N)/I_{f}. Then A_{f} is referred to as the optimal quotient of J_{0}(N) attached to f. The aim of this paper is to prove two results concerning such optimal quotients, which we will describe in turn.

Our first result relates the kernel of the canonical polarization θ_{f} of A_{f} to the ideal of fusion 𝔉_{f} of f. Before stating the result, we recall the relevant notions.

We begin with the construction of the polarization θ_{f}. If A is an abelian variety, let Â denote its dual. The Jacobian J_{0}(N) is endowed with a canonical principal polarization θ : J_{0}(N)ˆ → J_{0}(N), which induces a polarization θ_{f} : Â_{f} → A_{f}. More precisely, if π_{f} denotes the surjection J_{0}(N) → A_{f}, then dualizing yields an embedding π̂_{f} : Â_{f} → J_{0}(N)ˆ . (That this is an embedding follows from the fact that ker π_{f} = I_{f}J_{0}(N) is connected.) The polarization θ_{f} is equal to the composite θ_{f} = π_{f} ∘ θ ∘ π̂_{f}.

We now define the ideal of fusion 𝔉_{f}. Let S_{f} denote the subspace of S(N) spanned by f and its Galois conjugates, and write S_{f}^{⊥} for the orthogonal complement of S_{f} under the Petersson inner product. Note that I_{f} is the annihilator of S_{f} in 𝕋; we let I_{f}^{⊥} denote the annihilator of S_{f}^{⊥} in 𝕋. We define the ideal of fusion of f to be the ideal 𝔉_{f} = I_{f} + I_{f}^{⊥} in 𝕋. Then 𝔉_{f} has finite index in 𝕋, and ‘measures’ the congruences between f and normalized Hecke eigenforms in S_{f}^{⊥}.

Finally, we introduce some terminology regarding maximal ideals of 𝕋. If 𝔪 is such a maximal ideal, then attached canonically to 𝔪 is a semi-simple representation ρ_{𝔪} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝕋/𝔪) [12, prop. 5.1]. We say that 𝔪 satisfies multiplicity one if J_{0}(N)[𝔪] is two-dimensional over the residue field 𝕋/𝔪. (The motivation for this terminology is that if ρ_{𝔪} is irreducible, then J_{0}(N)[𝔪] is isomorphic to a direct sum of copies of ρ_{𝔪} [2], and so 𝔪 satisfies multiplicity one if and only if J_{0}(N)[𝔪] is isomorphic to a single copy of ρ_{𝔪}.)

We now state our first result.

Theorem A. Let f be a normalized new form in S(N). If we embed Â_{f} into J_{0}(N) via the composite θ ∘ π̂_{f} : Â_{f} → J_{0}(N), then this restricts to an embedding ker(θ_{f}) → J_{0}(N)[𝔉_{f}] whose cokernel is supported at maximal ideals of 𝕋 that do not satisfy multiplicity one.

This theorem is related to [21, thm. 3], which (for prime N) relates the modular degree of a strong Weil curve and the modulus of congruence of the corresponding eigenform. However, even when A_{f} is an elliptic curve, the above result differs from that of [21], insofar as it relates the kernel of θ_{f} to the kernel of 𝔉_{f} on the entire Jacobian J_{0}(N). (Consider the examples discussed below, involving certain Weil curves of conductors 431 and 503.)

The strength of the result stems from the fact that “many” maximal ideals of 𝕋 are known to satisfy multiplicity one. For example, if ρ_{𝔪} is irreducible, if the residue characteristic ℓ of 𝔪 is odd, and if ℓ is prime to N, or if ℓ exactly divides N and 𝔪 is not ℓ-old, then [12, thm. 5.2 (b)] and [7, main theorem] show that 𝔪 satisfies multiplicity one. (The result of [7] is strengthened by the results of [13]. Some cases of residue characteristic two are treated in [3]. See also [15, thm. 3.5] for a related summary of results.) If N is prime, then the results of [6] show that all 𝔪 of odd residue characteristic satisfy multiplicity one.

We should point out, though, that multiplicity one does not hold for all maximal ideals (even if N is prime), and that the cokernel of the inclusion ker(θ_{f}) ⊂ J_{0}(N)[𝔉_{f}] need not vanish. We illustrate this by an example. Calculations of Kilford and Stein show that when N = 431 or 503 there is a maximal ideal 𝔪 of 𝕋 having residue characteristic two that does not satisfy multiplicity one. Furthermore, there is a strong Weil curve E arising from a newform f of level N for which the 𝔪-adic completion of 𝔉_{f} is non-trivial. Thus J[𝔪] ⊂ J[𝔉_{f}]. On the other hand, J[𝔪] cannot be contained in ker θ_{f}, since J[𝔪] ∩ E = E[2], and the fact that 𝔪 does not satisfy multiplicity one shows that the inclusion E[2] ⊂ J[𝔪] is not an equality.

The second result that we prove is Stein’s refined Eisenstein conjecture [19, conj. 1.1]. This is a conjecture that concerns optimal quotients of prime conductor. Recall that if N is prime, then the results of [6] give a very precise description of the arithmetic properties of J_{0}(N). If C denotes the subgroup of J_{0}(N) generated by image of the divisor 0 − ∞ on X_{0}(N), then C is a finite subgroup of J_{0}(N)(ℚ) that in fact equals the full torsion subgroup of J_{0}(N)(ℚ) [6, thm. III.1.2]. Also, if Φ_{J_{0}(N)} denotes the group of connected components of the characteristic N fibre of the Néron model J_{0}(N), then the specialization map C → Φ_{J_{0}(N)} is an isomorphism [6, p. 99].

Stein has conjectured that the analogous statements are true for each optimal quotient of A_{f}, and that furthermore, if {f_{i}} denotes a complete set of conjugacy class representatives of normalized eigenforms in S(N), then the product of the orders of the torsion subgroups of the A_{f_{i}}(ℚ) is equal to the order of C. We refer to [19] for a precise statement of the conjecture, or alternatively to the statement of the following theorem, which proves his conjecture in its entirety.

Theorem B. Let f be a newform of weight two and prime conductor N. Let C_{f} denote the image of C in A_{f} (so that C_{f} is the subgroup of A_{f} generated by the image of the divisor 0 − ∞ on X_{0}(N)); C_{f} is a 𝕋-submodule of the torsion subgroup A_{f}(ℚ)_{tor} of the group of ℚ-rational points on A_{f}. Let Φ_{A_{f}} (respectively ΦÂ_{f}) denote the group of connected components of the characteristic N fibre of the

Néron model of A_{f} (respectively Â_{f}).

(i) The 𝕋-module C_{f} is isomorphic to C/I_{f}.

(ii) The inclusion C_{f} → A_{f}(ℚ)_{tor} is an isomorphism of 𝕋-modules.

(iii) The 𝕋-module Â_{f}(ℚ)_{tor} is abstractly isomorphic to C_{f}.

(iv) The specialization map A_{f}(ℚ)_{tor} → Φ_{A_{f}} is an isomorphism of 𝕋-modules.

(v) The specialization map Â_{f}(ℚ)_{tor} → ΦÂ_{f} is an isomorphism.

(vi) The (étale) group schemes Φ_{A_{f}} and ΦÂ_{f} over 𝔽_{N} are constant (that is, have trivial Galois action).

(vii) If {f_{i}} is a complete set of conjugacy class representatives of the normalized eigenforms in S(N), then ∏_{f_{i}} #C_{f_{i}} = #C.

In the special case when A_{f} is an elliptic curve, parts (ii), (iii), (iv), (v) and (vi) of this result were proved by Mestre and Oesterlé [8, §5] (in this case parts (iii) and (v) follow from parts (ii) and (iv), because the elliptic curve A_{f} is self-dual). Their proof depends on calculations that are very particular to the situation of elliptic curves. The most powerful tools that they appeal to are Mazur’s classification of the possible torsion subgroups of the group of ℚ-rational points of an elliptic curve over ℚ [6, thm. III.5.1], and Ribet’s lowering-the-level theorem [12, thm. 1.1]. Our proof also depends on the results of [6], in a more direct fashion – we use crucially Mazur’s description of the commutative algebra properties of the Eisenstein ideal of 𝕋, as well as his calculation of the kernel of the Eisenstein ideal acting on J_{0}(N). We also depend on the work of both Ribet and Tate on Serre’s conjectures (in particular, [14, prop. 2.2]).

One general difficulty in studying optimal quotients is that the description of the kernel of the map J_{0}(N) → A_{f} as being I_{f}J_{0}(N) is difficult to work with. A potentially more useful description is that it is equal to the connected component of J_{0}(N)[I_{f}^{⊥}]. However, a priori the latter sub-algebraic group of J_{0}(N) may not be connected, and without control of its connected component group, this characterization would again be rather unsatisfactory. We prove the following simple result, which provides the necessary control: if 𝔪 is a maximal ideal of 𝕋 that satisfies multiplicity one, then the 𝔪-adic completion of the connected component group of J_{0}(N)[I_{f}] is trivial, and so the inclusion I_{f}J_{0}(N) ⊂ J_{0}(N)[I_{f}^{⊥}] induces an isomorphism on 𝔪-divisible groups.

With this result in hand, it is easy to relate properties of optimal quotients to the commutative algebra properties of the Hecke ring (as long as one is willing to forsake information supported away from the multiplicity one maximal ideals of 𝕋). By itself it is already enough to imply theorem A, and when combined with the results of [6], to prove part (ii) of theorem B. The result itself is proved in section 2 (in a suitably abstracted situation; it is a special case of corollary 2.3). The rest of that section is devoted to elaborating on some of its consequences.

To obtain information about the connected component groups Φ_{A_{f}} we apply the results of section 2 in the context of rigid analytic uniformizations of J_{0}(N) and its optimal quotients. This is the subject of section 3. The key result is corollary 3.5.

Section 4 presents the proofs of theorems A and B. In order to prove part (vii) of theorem B, we find it necessary to generalize the notion of optimal quotient, and introduce a more flexible notion of an optimal subquotient of J_{0}(N). We then prove a more general version of theorem B for optimal subquotients (theorem 4.12), which has theorem B as an immediate corollary.

In section 1 we develop the (elementary) commutative algebra that is needed for the later sections. This section is intended to organize as usefully as possible the information on the commutative algebra of Hecke rings that can be deduced from the results of [6].

Acknowledgments. I would like to thank William Stein and Shuzo Takahashi for their comments on some earlier versions of this paper.

0. notation and conventions

All rings and algebras that we consider are commutative with unit. If I_{1} and I_{2} are two ideals of a ring R, we write (I_{1} : I_{2}) = {r ∈ R | rI_{2} ⊂ I_{1}}. If I is an ideal of R we also write I^{⊥} = (0 : I) = Ann_{R}(I).

If M is a locally compact group, we write M^{∗} for the Pontrjagin dual of M. We will apply this notation only in the cases when M is either ind-finite (that is, torsion and discrete) or pro-finite. In these cases, M^{∗} = Hom_{cont}(M, ℚ/ℤ), with ℚ/ℤ being regarded as a discrete group.

1. commutative algebra

In this section, we let R be a finite flat O-algebra, where O is a Dedekind domain with field of fractions L. If we write that an R-module is ‘torsion’, or ‘torsion-free’, we will always mean that it is O-torsion, or O-torsion-free. We say that an ideal I of R is saturated if R/I is torsion-free. Note that the association of I ⊗_{O} L to I induces a bijection between the saturated ideals of R and the ideals of R ⊗_{O} L. Note also that I^{⊥} ⊗_{O} L = (I ⊗_{O} L)^{⊥}.

Recall that R is Gorenstein if the R-module Hom_{O}(R, O) is locally free (necessarily of rank one). We say that R is monogenic if it is generated by a single element as an O-algebra. If R is monogenic, then it is Gorenstein.

Lemma 1.1. Let I_{1} and I_{2} be two ideals of R. If I_{1} is saturated then (I_{1} : I_{2}) is saturated. In particular, if I is saturated then I^{⊥} is saturated.

Proof. This is immediate from the definitions. □

Lemma 1.2. Suppose that R is either Gorenstein or reduced.

(i) If I is a saturated ideal of R, then the inclusion I ⊂ (I^{⊥})^{⊥} is an equality. (ii) If I_{1} and I_{2} are two saturated ideals of R, then (I_{2}^{⊥} : I_{1}^{⊥}) = (I_{1} : I_{2}).

Proof. Since (I^{⊥})^{⊥}/I ⊂ R/I is torsion free, by the assumption on I, it suffices to verify part (i) after tensoring over O with L. The assumption on R then implies that R ⊗_{O} L is Gorenstein, and so we may find a perfect R ⊗_{O} L-bilinear pairing (R ⊗_{O} L) × (R ⊗_{O} L) → L. This pairing puts (R ⊗_{O} L)/(I ⊗_{O} L) and I^{⊥} ⊗_{O} L = (I ⊗_{O} L)^{⊥} in duality with one another. Applying this remark also with I^{⊥} in place of I , and then counting dimensions over L, proves part (i).

To prove part (ii), note that for r ∈ R, r · I_{1}^{⊥} ⊂ I_{2}^{⊥} if and only if r · I_{1}^{⊥} · I_{2} = 0, if and only if r · I_{2} ⊂ (I_{1}^{⊥})^{⊥}. Part (i) shows that (I_{1}^{⊥})^{⊥} = I_{1}, and this establishes part (ii). □

Lemma 1.3. If R is reduced and I is a saturated ideal of R, then the natural map R → R/I ⊕ R/I^{⊥} is injective, with torsion cokernel.

Proof. It suffices to check that the map R ⊗_{O} L → (R ⊗_{O} L)/(I ⊗_{O} L) ⊕ (R ⊗_{O} L)/(I ⊗_{O} L)^{⊥} is an isomorphism. This follows from that fact that our assumption on R implies that R ⊗_{O} L is reduced, and so is a product of fields. □

If I is an ideal in R then the surjection R → R/I gives rise to an embedding Hom_{O}(R/I, O) → Hom_{O}(R, O), identifying Hom_{O}(R/I, O) with the R-module Hom_{O}(R, O)[I] of maps in Hom_{O}(R, O) that annihilate I.

Lemma 1.4. If I is a saturated ideal of R, then the inclusion Hom_{O}(R/I, O) ⊂ Hom_{O}(R, O)[I] is an equality.

Proof. Let f_{1}, …, f_{n} generated the ideal I. Since R/I is torsion-free, the diagram

R^{r} → R → R/I → 0,

in which the first arrow is defined by (r_{1}, …, r_{n}) ↦ f_{1}r_{1} + ⋯ f_{n}r_{n}, is an exact sequence of torsion-free R-modules. Thus applying Hom_{O}(–, O) yields an exact sequence

0 → Hom_{O}(R/I, O) → Hom_{O}(R, O) → Hom_{O}(R, O)^{n},

in which the third arrow is defined by ř ↦ (f_{1}ř, …, f_{n}ř). If we note that the kernel of this arrow is exactly Hom_{O}(R, O)[I], this proves the lemma. □

Corollary 1.5. If R is monogenic and I is a saturated ideal of R, then I is locally principal, and so (since it is a faithful R/I^{⊥}-module) is locally free of rank one over R/I^{⊥}.

Proof. Since R is monogenic, the same is clearly true of R/I. Since both are also finite flat O-algebras, they are both Gorenstein. Thus Hom_{O}(R, O) is isomorphic to R, and so by lemma 1.4, Hom_{O}(R/I, O) is isomorphic to I^{⊥}. Since R/I is Gorenstein, we conclude I^{⊥} is locally free of rank one over R/I. Replacing I by I^{⊥} (and noting that I = (I^{⊥})^{⊥}, since I^{⊥} is locally free over R/I, or alternatively by part (i) of lemma 1.2), we establish the corollary. □

Another proof of corollary 1.5 follows by noting that any saturated ideal in the polynomial ring O[X] is locally principal.

Corollary 1.6. If R is monogenic and I_{1} ⊂ I_{2} are saturated ideals of R, then I_{2}^{⊥} ⊂ I_{1}^{⊥}, and I_{1}^{⊥}/I_{2}^{⊥} is a free 𝕋/(I_{1} : I_{2})-module of rank one.

Proof. Since I_{1} ⊂ I_{2}, it is immediate that I_{2}^{⊥} ⊂ I_{1}^{⊥}. Corollary 1.5 shows that I_{1}^{⊥} is a locally principal ideal of R, and so I_{1}^{⊥}/I_{2}^{⊥} is a locally free 𝕋/(I_{2}^{⊥} : I_{1}^{⊥})-module of rank one. Part (ii) of corollary 1.3 now completes the proof (once we note that R is Gorenstein, since it is monogenic). □

Lemma 1.7. If I_{1} is a principal ideal of R for which R/I_{1} is torsion, and I_{2} is a saturated ideal of R, then I_{1} ∩ I_{2} = I_{1} · I_{2}.

Proof. Write I_{1} = r ·R. Since R/I_{1} is torsion, we deduce that r is a unit in R⊗_{O} L, and thus is a non-zero divisor in each of the torsion-free R-modules appearing in the exact sequence

0 → I_{2} → R → R/I_{2} → 0.

This implies that I_{2} ∩ r · R = r · I_{2}, proving the lemma. □

Corollary 1.8. Suppose that I_{1} is a principal ideal of R for which R/I_{1} is torsion, and that I_{2} is a saturated principal ideal of R. Then the surjection I_{2}^{⊥}/(I_{1} · I_{2}^{⊥}) → I_{2}^{⊥} · R/I_{1} and the inclusion I_{2}^{⊥} · R/I_{1} → R/I_{1} [I_{2}] are both isomorphisms.

Proof. Lemma 1.7 shows that I_{1} ∩ I_{2}^{⊥} = I_{1} · I_{2}^{⊥}. This implies the first statement of the lemma.

Note that there is an inclusion I_{1}+I_{2}^{⊥} ⊂ (I_{1} : I_{2}). Write I_{1} = r_{1}·R and I_{2} = r_{2}·R. Let r ∈ (I_{1} : I_{2}), so that r_{2}r = r_{1}r′ for some r′ ∈ R. Then lemma 1.7 implies that r_{1}r′ = r_{1}r_{2}r′′ for some r′′ ∈ R, and so r_{2}(r − r_{1}r′) = 0. Thus r − r_{1}r′ ∈ I_{2}^{⊥}, and so in fact I_{1} + I_{2}^{⊥} = (I_{1} : I_{2}). Thus

I_{2}^{⊥} · R/I_{1} = (I_{1} + I_{2}^{⊥})/I_{1} = (I_{1} : I_{2})/I_{1} = R/I_{1} [I_{2}].

This proves the second statement of the corollary. □

Corollary 1.9. Suppose that R is monogenic, that I_{1} is a principal ideal of R for which R/I_{1} is torsion, and that I_{2} ⊂ I_{3} are saturated ideals of R. Then (R/I_{1} [I_{2}])/(R/I_{1} [I_{3}]) is a locally free 𝕋/(I_{1} + (I_{2} : I_{3}))-module of rank one.

Proof. Corollary 1.5 shows that I_{2} and I_{3} are locally principal. Corollary 1.8 then shows that (R/I_{1} [I_{2}])/(R/I_{1} [I_{3}]) is (locally, and hence globally, since it is finitely generated and torsion) isomorphic to

(I_{2}^{⊥}/(I_{1} · I_{2}^{⊥}))/(I_{3}^{⊥}/(I_{1} · I_{2}^{⊥})) = (I_{2}^{⊥}/I_{3}^{⊥})/(I_{1} · (I_{2}^{⊥}/I_{3}^{⊥})).

Corollary 1.6 implies that I_{2}^{⊥}/I_{3}^{⊥} is locally free of rank one over 𝕋/(I_{2} : I_{3}). This implies the corollary. □

2. rings of operators on algebraic groups

If G is a smooth commutative algebraic group over a field K of characteristic zero, then for any integer n we let G[n] denote the the algebraic subgroup of G consisting of the n-torsion elements of G. Since we are in characteristic zero, multiplication by n is a smooth morphism from G to itself, and consequently G[n] is a finite algebraic group (since a smooth morphism from a variety to itself must have finite fibres).

If n divides n′, then G[n] ⊂ G[n′]. We will be especially interested in the case when we fix a prime ℓ, and consider the ℓ-power torsion G[ℓ^{n}] of G. We write G[ℓ^{∞}] for the ind-finite algebraic subgroup of G formed by the G[ℓ^{n}] as n ranges over all positive integers.

The formation of G[ℓ^{∞}] is evidently functorial in G. If G is connected, then multiplication by ℓ is surjective from G to itself, and so G[ℓ^{∞}] is an ℓ-divisible group.

For any G of the type we are considering, we will let G^{0} denote the connected component of the identity of G. The quotient G/G^{0} is a finite algebraic group, the group of connected components of G.

Let us remark that if we choose an algebraic closure \overline{K} of K, then the \overline{K}- valued points of any finite algebraic group over K form a finite group, and we will not distinguish between a finite algebraic group over K and its group of \overline{K}- valued points. (This is permissible, since the finite algebraic group can be recovered up to isomorphism from its group of \overline{K}-valued points, equipped with its natural Gal(\overline{K}/K)-action.)

We will require the following easily proved fact:

Fact 2.1. For any G as above and any prime ℓ, the inclusion G^{0}[ℓ^{∞}] → G[ℓ^{∞}] identifies G^{0}[ℓ^{∞}] with the maximal ℓ-divisible subgroup of G[ℓ^{∞}]. The cokernel of this inclusion is canonically isomorphic to the ℓ-Sylow subgroup of G/G^{0}.

For the remainder of this section, let us fix a group G as above. Furthermore, let 𝕋 denote a finite flat commutative ℤ-algebra that acts on G. (As indicated in the introduction, in the applications 𝕋 will be a Hecke algebra, as the notation suggests.) If I is any ideal in 𝕋, we let G[I] denote the algebraic subgroup of G consisting of elements annihilated by I. Taking R to be 𝕋 and O to be ℤ puts us in a particular case of a situation of section 1. Thus we will say that an ideal I in 𝕋 is saturated if the quotient 𝕋/I is torsion-free as a ℤ-module.

If ℓ is any prime number then the ℓ-adic completion of 𝕋 factors as the direct product of completions 𝕋_{𝔪}, where 𝔪 ranges over the maximal ideals of 𝕋 of residue characteristic ℓ. Thus any object on which this ℓ-adic completion acts similarly factors. For example, G[ℓ^{∞}] factors into such a product of ℓ-divisible subgroups. We let G[𝔪^{∞}] denote the factor corresponding to the maximal ideal 𝔪. Similarly the ℓ-adic Tate module T_{ℓ}G(N)_{/ℚ_{p}} (by which we refer to the Pontrjagin dual G[ℓ^{∞}]^{∗} = Hom(G[ℓ^{∞}], ℚ_{ℓ}/ℤ_{ℓ}) of G[ℓ^{∞}]; this is the contravariant ℓ-adic Tate module) factors into such a product, and we let T_{𝔪}J_{0}(N) denote the factor corresponding to the maximal ideal 𝔪. We say that a maximal ideal 𝔪 of 𝕋 is good (or good with respect to G, if it is necessary to specify the group G) if T_{𝔪}G is a free 𝕋_{𝔪}-module.

Lemma 2.2. Suppose that 𝔪 is good, and that I is a saturated ideal of 𝕋. Then the ind-finite group G[𝔪^{∞}, I] (which denotes of course the I-torsion subgroup of the ind-finite group G[𝔪^{∞}]) is ℓ-divisible.

Proof. This follow by Pontrjagin duality. More precisely, G[𝔪^{∞}, I] = (T_{𝔪}G)^{∗}[I] = (T_{𝔪}/I)^{∗}, showing that 𝕋_{𝔪}/I is the Pontrjagin dual of G[𝔪^{∞}, I]. Now G[𝔪^{∞}, I] is ℓ-divisible if and only if its Pontrjagin dual T_{𝔪}G/I is torsion-free. But since 𝔪 is good, this latter group is free over 𝕋_{𝔪}/I, which is torsion-free by virtue of our assumption on I. □

Corollary 2.3. Let I be a saturated ideal of 𝕋, and write H = G[I].

(i) If 𝔪 is a good maximal ideal of 𝕋 then the natural map H^{0}[𝔪^{∞}] → G[𝔪^{∞}, I] is an isomorphism. Equivalently, the natural map T_{𝔪}H → T_{𝔪}H^{0} is an isomorphism.

(ii) The connected component group H/H^{0} is supported at maximal ideals of 𝕋 that are not good.

Proof. Both claims follows immediately from fact 2.1 and lemma 2.2. □

The preceding corollary will be our basic tool in this paper. We now apply it to prove some additional results involving good maximal ideals, which will be used in what is to come.

Lemma 2.4. Let 𝔪 be a good maximal ideal in 𝕋, let I be a saturated ideal of 𝕋, let I be an ideal of finite index in 𝕋, and suppose that I𝕋_{𝔪} is a principal ideal in 𝕋_{𝔪}. If we write H = G[I], then the sequence

0 → H^{0}[I]_{𝔪} → G[I]_{𝔪} → (G/H^{0})[I]_{𝔪} → 0

is short exact. (The subscript 𝔪 denotes 𝔪-adic completions.)

Proof. Since I has finite index in 𝕋, G[I] is finite. Thus G[I]_{𝔪} = G[𝔪^{∞}, I]. A similar remark holds with H^{0} and G/H^{0} in place of G. Thus we must show that the sequence

0 → H^{0}[𝔪^{∞}, I] → G[𝔪^{∞}, I] → (G/H^{0})[𝔪^{∞}, I] → 0

is short exact.

The sequence

0 → H^{0}[𝔪^{∞}] → G[𝔪^{∞}] → (G/H^{0})[𝔪^{∞}] → 0

is short exact, as is the analogous sequence obtained by replacing H^{0} with H. Part (i) of corollary 2.3 shows that the map H^{0}[𝔪^{∞}] → H[𝔪^{∞}] is an isomorphism, and hence that (G/H^{0})[𝔪^{∞}] → (G/H)[𝔪^{∞}] is an isomorphism. Thus it suffices to prove that

0 → H[𝔪^{∞}, I] → G[𝔪^{∞}, I] → (G/H)[𝔪^{∞}, I] → 0

is short exact. This sequence is Pontrjagin dual to the sequence

0 → IT_{𝔪}G/I → T_{𝔪}G/I → T_{𝔪}G/(I + I) → 0,

which is clearly exact on the right. We must show that it is also exact on the left. Since 𝔪 is good, T_{𝔪}G is free over 𝕋_{𝔪}, and so it suffices to show that the map

I · 𝕋_{𝔪}/I → 𝕋_{𝔪}/I

is injective. This follows from lemma 1.7 □

Lemma 2.5. Suppose that G is either an abelian variety or a torus. Let Ĝ denote the dual of G, endowed with the dual 𝕋-action. Let 𝔪 be a maximal ideal of 𝕋 for which the completion 𝕋_{𝔪} is Gorenstein. Then 𝔪 is good for G if and only if it is good for Ĝ.

Proof. Let ℓ denote the residue characteristic of 𝔪. Recall that Ĝ[ℓ^{∞}] is naturally isomorphic to the Cartier dual of G[ℓ^{∞}]. Since this isomorphism is natural, it is 𝕋-equivariant, and so there is an induced isomorphism between Ĝ[𝔪^{∞}] and the Cartier dual of G[𝔪^{∞}]. Thus (if we ignore the implicit Gal(\overline{K}/K)-actions) there is an isomorphism of 𝕋_{𝔪}-modules T_{𝔪}Ĝ ⟶^{∼} Hom_{ℤ_{ℓ}}(T_{𝔪}G, ℤ_{ℓ}). Since 𝕋_{𝔪} is a Gorenstein ring, it follows that T_{𝔪}Ĝ is free over 𝕋_{𝔪} if and only if T_{𝔪}G is free over 𝕋_{𝔪}. This proves the lemma. □

Lemma 2.6. Suppose that G is either an abelian variety or a torus, and that I is an ideal of 𝕋. Let (G/I)ˆ → Ĝ be dual to the surjection G → G/I. Then this morphism is injective, and identifies (G/I)ˆ with Ĝ[I]^{0}.

Proof. Dualizing the short exact sequence 0 → IG → G → G/I → 0 yields the short exact sequence 0 → (G/I)ˆ → Ĝ → (IG)ˆ → 0. (Exactness on the left follows from the fact that IG is connected.) Let α_{1}, …, α_{r} be generators for I. Write G′ = ∏_{i=1}^{r} G. Dualizing the surjection G′ → IG defined by (g_{1}, …, g_{r}) ↦ α_{1}g_{1} + ⋯+α_{r}g_{r}, yields a morphism (IG)ˆ → Ĝ′ = ∏_{i=1}^{r} Ĝ, whose kernel is a finite subgroup of (IG)ˆ . Composing this injection with the preceding short exact sequence yields the sequence

0 → (G/I)ˆ → Ĝ → Ĝ′,

which is exact on the left, and exact in the middle up to finite index. The righthand arrow is given by ĝ ↦ (α_{1}ĝ, …, α_{r}ĝ), and so its kernel is equal to Ĝ[I]. Thus we find that (G/I)ˆ is contained in Ĝ[I] with finite index. Since it is connected, the lemma follows. □

By abuse of terminology, we say that a maximal ideal 𝔪 of 𝕋 is monogenic if the completion 𝕋_{𝔪} is monogenic as a ℤ_{ℓ}-algebra (where ℓ is the residue characteristic of 𝔪).

Lemma 2.7. Let 𝔪 be a good maximal ideal of 𝕋, and let I ⊂ 𝔪 be a saturated ideal of 𝕋. Let 𝔪′ denote the image of 𝔪 in 𝕋/I. Note that the ring 𝕋/I acts on each of G[I] and G/I.

(i) The maximal ideal 𝔪′ is good with respect to G[I]^{0}.

(ii) Suppose that G is either an abelian variety or a torus. If 𝔪 is monogenic, then 𝔪′ is also monogenic, and is good with respect to G/I.

Proof. The proof of lemma 2.2 shows that T_{𝔪′}G[I] = T_{𝔪}G/I. Thus if T_{𝔪}G is free over 𝕋_{𝔪}, we see that T_{𝔪′}G[I] is free over (𝕋/I)_{𝔪′} = 𝕋_{𝔪}/I. Also, part (i) of corollary 2.3 shows that the map T_{𝔪′}G[I] → T_{𝔪′}G[I]^{0} is an isomorphism. This proves part (i).

We now turn to proving part (ii). Since (𝕋/I)_{𝔪′} = 𝕋_{𝔪}/I, the fact that 𝔪 is monogenic immediately implies the same for 𝔪′. Both 𝕋 and 𝕋/I are torsion-free by assumption. Thus both 𝕋_{𝔪} and (𝕋/I)_{𝔪′} are Gorenstein ℤ_{ℓ}-algebras. Lemma 2.5 now shows that 𝔪 is good for Ĝ, and that to prove that 𝔪′ is good for G/I, it suffices to prove that it is good for (G/I)ˆ .

The surjection G → G/I has connected kernel, and so dualizing leads to an injection (G/I)ˆ → Ĝ. Lemma 2.6 shows that this embedding identifies (G/I)ˆ with Ĝ[I]^{0}, and it follows from part (i) that 𝔪′ is good for Ĝ[I]^{0}. This proves part (ii) of the lemma. □

We record the following result for later use.

Lemma 2.8. If G is connected, then Ann_{𝕋}(G) is a saturated ideal of 𝕋.

Proof. Let T ∈ T and n ∈ ℤ such that nT annihilates G. Then T annihilates nG. Since G is connected, nG = G. This proves the lemma. □

We finish this section by proving an abstract version of theorem A. We begin by describing the necessary set-up. To begin with, suppose that I_{1} and I_{2} are two saturated ideals of 𝕋, and that the natural map 𝕋 → 𝕋/I_{1} ⊕ 𝕋/I_{2} is injective, with finite-order cokernel.

Lemma 2.9. Suppose that G is connected. Then I_{1}G = G[I_{2}]^{0}.

Proof. The assumptions on I_{1} and I_{2} imply that I_{1} · I_{2} ⊂ I_{1} ∩ I_{2} = 0. Thus I_{1}G ⊂ G[I_{2}]. Since G is connected, I_{1}G is also connected, and so in fact I_{1}G ⊂ G[I_{2}]^{0}. To see the converse, note that the assumptions on I_{1} and I_{2} imply that I_{1} + I_{2} has finite index in 𝕋, and so we may choose n ∈ ℤ such that n𝕋 ⊂ I_{1} + I_{2}. Since G[I_{2}]^{0} is connected, we find that

G[I_{2}]^{0} = nG[I_{2}]^{0} ⊂ (I_{1} + I_{2})G[I_{2}]^{0} = I_{1}G[I_{2}]^{0} ⊂ I_{1}G.

This establishes the lemma. □

Now suppose in addition that G is either an abelian variety or a torus, and that there is a 𝕋-equivariant isomorphism ϕ : Ĝ → G. By lemma 2.6, the surjection G → G/I_{1} gives rise to an embedding (G/I_{1})ˆ → Ĝ. We let ϕ′ denote the composite

(G/I_{1})ˆ → Ĝ →^{ϕ} G → G/I_{1}.

Lemma 2.10. In the preceding situation, the restriction of ϕ to ker(ϕ′) embeds ker(ϕ′) into G[I_{1} + I_{2}]. The cokernel of this embedding is supported away from the good maximal ideals of 𝕋.

Proof. Lemma 2.9 shows that I_{1}G = G[I_{2}]^{0}. Lemma 2.6 shows that (G/I_{1})ˆ ⟶^{∼} Ĝ[I_{1}]^{0}. The isomorphism ϕ^{−1} : G → Ĝ thus identifies IG with Ĝ[I_{2}]^{0}, and so

ker(ϕ′) = Ĝ[I_{1}]^{0} ∩ Ĝ[I_{2}]^{0} = Ĝ[I_{1} + I_{2}].

It follows immediately that ϕ embeds ker(ϕ′) into G[I_{1} + I_{2}]. That the cokernel of this embedding is supported away from the good maximal ideals of 𝕋 follows from the assumptions on I_{1} and I_{2} together with part (ii) of corollary 2.3. □

3. abelian varieties with semi-abelian reduction

In this section we assume that the field K is complete with respect to a discrete valuation, and has perfect residue field. If A is an abelian variety over K having semi-abelian reduction, in the sense that the connected component of the special fibre of its Néron model over the ring of integers of K is an extension of an abelian variety by a torus, then A has a uniformization in the category of rigid analytic spaces over K in the form of a short exact sequence

0 → Γ_{A} → G_{A} → A → 0,

in which Γ_{A} is a finite rank ℤ-module equipped with a continuous Gal(\overline{K}/K)-action, G_{A} is a semi-abelian variety, and the maximal abelian variety quotient of G_{A} is isomorphic to the maximal quotient of A having good reduction (see [5, §14] and [9]). The formation of Γ_{A} and G_{A} is functorial in the category of abelian varieties A having semi-abelian reduction. (To be precise, if B → A is a morphism of abelian varieties over K having semi-abelian reduction, then there is induced functorially an algebraic morphism G_{B} → G_{A} of semi-abelian varieties over K, which restricts to a morphism Γ_{B} → Γ_{A}.)

Let 0 → C → B → A → 0 be an exact sequence of abelian varieties over K, and suppose that B (and hence also C and A) have semi-abelian reduction. Functoriality of uniformization yields a diagram with exact rows and columns

\xymatrix{0 \ar[r] & Γ_{B} \ar[d] \ar[r] & G_{B} \ar[d] \ar[r] & B \ar[d] \ar[r] & 0 \\ 0 \ar[r] & Γ_{A} \ar[d] \ar[r] & G_{A} \ar[d] \ar[r] & A \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

(the exactness of the first column follows because the kernel of the map B → A (that is, the abelian variety C) is connected.) We complete this diagram so as to obtain the following diagram in which all rows and columns are short exact:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Γ′_{C} \ar[d] \ar[r] & G′_{C} \ar[d] \ar[r] & C \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Γ_{B} \ar[r] \ar[d] & G_{B} \ar[d] \ar[r] & B \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Γ_{A} \ar[r] \ar[d] & G_{A} \ar[r] \ar[d] & A \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0 & .}

The top row of this diagram need not necessarily be the rigid-analytic uniformization of C, since the algebraic group G′_{C} need not be connected. The following lemma clarifies the situation.

Lemma 3.1. There is a canonical identification of G_{C} (the semi-abelian variety that uniformizes C) with (G′_{C})^{0} (the connected component of the identity in G′_{C}), with respect to which Γ_{C} is identified with the intersection Γ′_{C} ∩(G′_{C})^{0}. Furthermore, the induced map Γ′_{C}/Γ_{C} → G′_{C}/G_{C} is an isomorphism.

Proof. The group G′_{C} appearing in the preceding diagram is the kernel of the map of semi-abelian varieties G_{B} → G_{A}, and so is an algebraic group, of dimension equal to the difference of the dimensions of G_{B} and G_{A}, which is the difference of the dimensions of B and A, which is the dimension of C. By functoriality of uniformizations, the injection C → B induces an injection G_{C} → G_{B}. Since the image of G_{C} in G_{B} certainly maps to zero in G_{A}, the image of G_{C} lies in G′_{C}; in fact, since G_{C} is connected, it lies in (G′_{C})^{0}. Since G_{C} and (G′_{C})^{0} have the same dimension, the image of G_{C} must equal (G′_{C})^{0}. This proves the first statement of the lemma. The rest of the lemma follows from a consideration of the diagram

[figure: 0; 0; 0; 0; Γ_{C}; G_{C}; C; 0; 0; Γ′_{C}; G′_{C}; C; 0,]

whose rows and columns are exact. □

We may dualize our exact sequence of abelian varieties to obtain the exact sequence of dual abelian varieties 0 → Â → B̂ → Ĉ → 0, and a diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Γ′ ; Â \ar[d] \ar[r] & G′ ; Â \ar[d] \ar[r] & Â \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ΓB̂ \ar[r] \ar[d] & GB̂ \ar[d] \ar[r] & B̂ \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ΓĈ \ar[r] \ar[d] & GĈ \ar[r] \ar[d] & Ĉ \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0 & .}

Lemma 3.1 applies equally well in this situation, to yield isomorphisms GÂ ⟶^{∼}

(G′ ; Â)^{0} and Γ′ ; Â/Γ′ ; Â ⟶^{∼} G′ ; Â/(GÂ).

The monodromy pairing [5] yields a short exact sequence of complexes (the complexes are in the vertical direction; the exact sequence is in the horizontal direction)

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Γ_{C} \ar[r] \ar[d] & Hom(ΓĈ, ℤ) \ar[r] \ar[d] & Φ_{C} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Γ_{B} \ar[r] \ar[d] & Hom(ΓB̂, ℤ) \ar[r] \ar[d] & Φ_{B} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Γ_{A} \ar[r] \ar[d] & Hom(ΓÂ, ℤ) \ar[r] \ar[d] & Φ_{A} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0 & .}

Taking the associated long exact sequence of cohomology, and using the preceding diagrams, we obtain the exact sequence

0 → ker(Φ_{C} → Φ_{B}) → Γ′_{C}/Γ_{C} → 0 → ker(Φ_{B} → Φ_{A})/ im(Φ_{C} → Φ_{B})

→ 0 → Ext^{1}(Γ′ ; Â/ΓÂ, ℤ) → coker(Φ_{B} → Φ_{A}) → 0.

Recall the isomorphisms Γ′_{C}/Γ_{C} ⟶^{∼} G′_{C}/G_{C} and Γ′ ; Â/ΓÂ ⟶^{∼} G′ ; Â/GÂ of lemma 3.1. Also recall that if M is any finite group, there is an isomorphism

Ext^{1}(M, Z) ⟶^{∼} Hom(M, ℚ/ℤ) = M^{∗}.

Using these isomorphisms, we may reorganize the information contained in the preceding exact sequence to obtain the exact sequence

(3.2)  0 → G′_{C}/G_{C} → Φ_{C} → Φ_{B} → Φ_{A} → (G′ ; Â/GÂ)^{∗} → 0.

(This exact sequence is a generalization of [16, prop. 2], which treats the case when B is the Jacobian of a Shimura curve of square-free level, and A is an elliptic curve that is an optimal quotient of B.)

The remainder of this section is devoted to applying the preceding discussion in the context of section 2. From now on we suppose that 𝕋 is a finite flat commutative ℤ-algebra that acts on B, that I_{1} and I_{2} are saturated ideals in 𝕋 for which the morphism 𝕋 → 𝕋/I_{1} ⊕ 𝕋/I_{2} is injective with finite-order cokernel, that C = I_{1}B, and that A = B/I_{1}.

The functoriality of the constructions of this section imply that 𝕋 acts naturally on G_{B}, Γ_{B}, Φ_{B}, and similarly with B replaced by any of A, C, Â, B̂, or Ĉ. The ring 𝕋 also acts naturally on G′_{C} and G′ ; Â, and all morphisms appearing in the diagrams of the preceding discussion are 𝕋-equivariant. In particular, 𝕋 acts on all the objects appearing in the exact sequence (3.2), and all the morphisms of this exact sequence are 𝕋-equivariant.

Lemma 3.3. We have the following sequences of equalities and inclusions:

G_{C} = (G′_{C})^{0} = I_{1}G_{B} = G_{B}[I_{2}]^{0} ⊂ G′_{C} ⊂ G_{B}[I_{2}],

and

GÂ = (G′ ; Â)^{0} = I_{2}GB̂ = GB̂[I_{1}]^{0} ⊂ G′ ; Â ⊂ GB̂[I_{1}],

Proof. Since I_{1} annihilates A, it also annihilates G_{A}, and so I_{1}G_{B} ⊂ G′_{C}. Since I_{1}G_{B} is connected, in fact I_{1}G_{B} ⊂ (G′_{C})^{0}. The assumptions on I_{1} and I_{2}, together with lemma 2.8, imply that I_{1}G_{B} = G_{B}[I_{2}]^{0}, and lemma 3.1 implies that G_{C} = (G′_{C})^{0}. Combining all these observations, we obtain the sequence of inclusions and equalities I_{1}G_{B} = G_{B}[I_{2}]^{0} ⊂ G_{C} = (G′_{C})^{0} ⊂ G′_{C}.

Again by lemma 2.8, we see that C = I_{1}B is also equal to B[I_{2}]^{0}. Since I_{2} annihilates C, we see that I_{2}G′_{C} ⊂ Γ′_{C}. Now I_{2}G′_{C} is an algebraic group, while Γ′_{C} is a lattice. Thus I_{2}G′_{C} = 0, and we deduce that G′_{C} ⊂ G_{B}[I_{2}]. Passing to connected components, we also find that G_{C} = (G′_{C})^{0} ⊂ G_{B}[I_{2}]^{0}. Combining these two inclusions with the result of the preceding paragraph proves the first claim of the lemma. The second claim follows from this first, once one notes that lemmas 2.6 and 2.9 imply that Â = I_{2}B̂, and hence that Ĉ = B̂/I_{2}. □

Corollary 3.4. The kernel of the map Φ_{C} → Φ_{B} is a quotient of the 𝕋-module G_{B}[I_{2}]/G_{B}[I_{2}]^{0}, while the cokernel of the map Φ_{B} → Φ_{A} is a quotient of the 𝕋module (GB̂[I_{1}]/GB̂[I_{1}]^{0})^{∗}.

Proof. Lemma 3.3 shows that there is an injection G′_{C}/G_{C} → G_{B}[I_{2}]/G_{B}[I]^{0}. The first claim of lemma now follows, since the exact sequence (3.2) identifies (G′_{C}/G_{C})^{∗} with the kernel of the map Φ_{C} → Φ_{B}. The second claim of the lemma is proved similarly. □

Corollary 3.5. Suppose that 𝔪 is a maximal ideal of 𝕋 that is good for both G_{B} and GB̂. Then the sequence

0 → (Φ_{C})_{𝔪} → (Φ_{B})_{𝔪} → (Φ_{A})_{𝔪} → 0

is short exact. (Here the subscript 𝔪 denotes 𝔪-adic completion, of course.)

Proof. This follows immediately from the exact sequence (3.2), and corollaries 3.4 and 2.3. □

We conclude this section with an analogue of lemma 2.7.

Lemma 3.6. Suppose that G_{B} is a torus, that 𝔪 is good for both G_{B} and GB̂, and that 𝔪 is monogenic. Let 𝔪_{i} denote the image of 𝔪 in 𝕋/I_{i} (for i = 1, 2). Then 𝔪_{1} (respectively 𝔪_{2}) is monogenic, and good for G_{A} (respectively G_{C}).

Proof. By lemma 3.3, G_{C} = G_{B}[I_{2}]^{0}. This equality, together with part (i) of lemma 2.7, implies that 𝔪_{2} is good for G_{C}.

To show that 𝔪_{1} is good for G_{A}, note that again by lemma 3.3, I_{1}G_{B} = G_{B}[I_{2}]^{0} ⊂ G′_{C} ⊂ G_{B}[I_{2}]. Thus there is a surjection G_{B}/I_{1} → G_{B}/G′_{C} = G_{A}, whose kernel G′_{C}/I_{1}G_{B} is a subgroup of the connected component group G_{B}[I_{2}]/G_{B}[I_{2}]^{0}. Since 𝔪 is good for G_{B}, part (ii) of corollary 2.3 shows that this group is supported away from 𝔪, and so the surjection G_{B}/I_{1} → G_{A} induces an isomorphism on 𝔪-adic Tate modules. Thus it suffices to show that 𝔪_{1} is good for G_{B}/I_{1}. This follows from part (ii) of lemma 2.7, and completes the proof of the lemma. □

4. applications to optimal quotients

In this section, we present the proofs of the two theorems stated in the introduction. We will see that theorem A is a quite straightforward consequence of the techniques of section 2. While theorem B also follows from these techniques, the process of deducing it from them is slightly more involved.

As in the introduction, we let N be a positive integer, S(N) the space of weight two cuspforms on Γ_{0}(N), and 𝕋 the Hecke algebra acting on S(N).

Lemma 4.1. Let f be a normalized Hecke eigenform in S(N), let I_{f} denote the annihilator of f in 𝕋, S_{f} denote the subspace of S(N) spanned by all the algebraic conjugates of f, and let S_{f}^{⊥} denote the orthogonal complement with respect to the Petersson inner product of S_{f} in S(N). As usual, write I_{f}^{⊥} = Ann_{𝕋}(f).

(i) I_{f}^{⊥} is the annihilator of S_{f}^{⊥} in 𝕋.

(ii) The map 𝕋 → 𝕋/I_{f} ⊕ 𝕋/I_{f}^{⊥} is injective and its cokernel has finite order.

Proof. The finite dimensional ℚ-algebra 𝕋 ⊗_{ℤ} ℚ decomposes as a product of Artin local ℚ-algebras 𝕋 ⊗_{ℤ} ℚ = ∏_{i=1}^{d} A_{i}, which are in bijection with the normalized eigenforms in S(N). In particular, the newform f corresponds to one of these factors, say A_{1}. Since f is a newform, A_{1} is a field, and so I_{f} ⊗_{ℤ} ℚ = ∏_{i=2}^{d} A_{i}, and I_{f}^{⊥} ⊗ ℚ = A_{1}. In particular, the map 𝕋 ⊗_{ℤ} ℚ → (𝕋 ⊗_{ℤ} ℚ)/(I_{f} ⊗_{ℤ} ℚ) ⊕ (𝕋 ⊗_{ℤ} ℚ)/(I_{f}^{⊥} ⊗_{ℤ} ℚ) is an isomorphism. Part (ii) follows from this.

The decomposition of 𝕋 into a product gives rise to a decomposition S(N) = ∏_{i=1}^{d} S_{i} of the 𝕋-module S(N). Each S_{i} is a faithful A_{i} ⊗_{ℚ} ℂ-module, whose dimension over ℂ is equal to the dimension of A_{i} over ℚ. (In fact, S_{i} is canonically isomorphic as an A_{i}-module to Hom_{ℚ}(A_{i}, ℂ) [11, thm. 2.2].) Now S_{f} = S_{1}, while S_{f}^{⊥} = ∏_{i=2}^{d} S_{i}. Thus Ann_{T ⊗ℚ}(S_{f}^{⊥}) = A_{1} = I_{f} ⊗_{ℤ} ℚ. This implies part (i). □

Note that part (i) of this lemma reconciles our use of the ⊥ notation in the introduction with its use in the main body of the paper.

Proof of theorem A. We will deduce theorem A from lemma 2.10. The one complication in this is that the natural principal polarization of J_{0}(N) does not respect the action of 𝕋 on J_{0}(N). Let θ : J_{0}(N)ˆ ⟶^{∼} J_{0}(N) denote this principal polarization.

As we stated in the introduction, the action of the correspondences T ∈ 𝕋 on J_{0}(N) that we are considering is that induced via Picard functoriality. This induces a dual action of 𝕋 on J_{0}(N)ˆ . If T ∈ 𝕋 we let T^{†} denote the endomorphism of J_{0}(N) defined by θ ∘ T ∘ θ^{−1}. (Thus † denotes the Rosati involution.) As discussed in [12, pp. 443-444], the endomorphism T^{†} is the endomorphism of J_{0}(N) obtained from the correspondence T via Albanese functoriality, and in general T and T^{†} are not the same endomorphism. However, it is pointed out in [12, p. 444, eqns. (3), (4)] that one has the equality T^{†} = w_{N}T w_{N}, where w_{N} is the Atkin-Lehner involution of J_{0}(N). Thus if we write ϕ = w_{N} ∘ θ : J_{0}(N)ˆ → J_{0}(N), then ϕ is 𝕋-equivariant. Note also that since w_{N} is an involution, it is fixed by the Rosati involution, and so we may equally well write ϕ = θ ∘ w_{N}.

Now let f be a newform in S(N), corresponding to the ideal I_{f} in 𝕋. Note that by part (ii) of lemma 4.1, I_{f} and I_{f}^{⊥} satisfy the hypotheses of lemma 2.10. Thus, if we let ϕ′ : Â_{f} → A_{f} be the polarization of Â_{f} induced by the principal polarization ϕ of J_{0}(N), we deduce from lemma 2.10 that ϕ embeds ker ϕ′ into J_{0}(N)[I_{f} +I_{f}^{⊥}] = J_{0}(N)[𝔉_{f}], with cokernel supported away from the good maximal ideals of 𝕋.

To complete the proof of theorem A, it suffices to prove the following two claims. First, that each of I_{f}J_{0}(N) ⊂ J_{0}(N) and Â_{f} ⊂ J_{0}(N)ˆ is invariant under w_{N}, so that ϕ(ker(ϕ′)) = θ(ker(θ_{A})). Second, that if 𝔪 satisfies multiplicity one, then it is good for J_{0}(N).

One may regard the exact sequence of abelian varieties

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

as arising by applying the exponential map to the exact sequence of tangent spaces

0 → S_{f}^{⊥} → S(N) → S_{f} → 0.

It follows from the results of [1] and the fact that f is a newform that each of S_{f} and S_{f}^{⊥} is invariant under the action of w_{N}. As a consequence, we deduce that w_{N} restricts to an automorphism of I_{f}J_{0}(N). Dualizing, we observe that w_{N} restricts to an involution of Â_{f}. This proves the first of the above claims.

To prove the second of the above claims, note that the 𝕋_{𝔪}⊗ℚ-module T_{𝔪}J_{0}(N)⊗ ℚ is free of rank two, and so if 𝔪 satisfies multiplicity one then T_{𝔪}J_{0}(N) is free of rank two over 𝕋_{𝔪}. In particular, such 𝔪 are good for J_{0}(N). □

As already mentioned, the proof of the theorem B is more involved. We begin by recalling some additional terminology concerning maximal ideals of the Hecke ring. For any prime p, we say that a maximal ideal 𝔪 of 𝕋 is p-finite if the representation ρ_{𝔪} is finite at p [17, p. 189]. We say that a maximal ideal 𝔪 of 𝕋 is Eisenstein if ρ_{𝔪} is reducible.

We now prove a result that controls the support of the connected component groups of special fibres of Néron models of certain 𝕋-equivariant subquotients of J_{0}(N). It is a minor variation on the argument of [19, § 4].

Proposition 4.2. Let N be a positive integer, and let A be a subquotient of J_{0}(N) in the category of abelian varieties with 𝕋-action. Suppose that A has semi-abelian reduction at a prime p, and let Φ_{A} denote the 𝕋-module of connected components of the characteristic p fibre of the Néron model of A. Then Φ_{A} is supported on the p-finite and Eisenstein locus of 𝕋.

Proof. We begin with a slight generalization of the discussion of [19, p. 4]. Consider the p-adic rigid analytic uniformization 0 → Γ_{A} → G_{A} → A → 0 of A. If ℓ is a prime, then applying Hom(ℤ/ℓ, –) to this short exact sequence yields the short exact sequence of Gal(\overline{ℚ}_{p}/ℚ_{p})-modules

0 → G_{A}[ℓ] → A[ℓ] → (Γ_{A}/ℓ)^{∗} → 0.

Both G_{A}[ℓ] and (Γ_{A}/ℓ)^{∗} are finite Gal(\overline{ℚ}_{p}/ℚ_{p})-modules. Before proving this, we introduce some notation. We let A denote the Néron model of A over ℤ_{p}, and let A^{0} denote the connected component of the identity in A. Now to see the asserted finiteness, note that the Galois action on Γ_{A}, and hence on (Γ_{A}/ℓ)^{∗}, is even unramified, while G_{A}[ℓ] extends over ℤ_{p} to the finite part of the group scheme A^{0}[ℓ].

Applying Hom(ℤ/ℓ, –) to the exact sequence 0 → Γ_{A} → Hom(ΓÂ, ℤ) → Φ_{A} → 0 yields an injection 0 → Φ_{A}[ℓ] → (Γ_{A}/ℓ)^{∗}. Pulling back the above short exact sequence by this map yields a morphism of short exact sequences of Gal(\overline{ℚ}_{p}/ℚ_{p})- modules

[figure: 0; G_{A}[ℓ]; E; Φ_{A}[ℓ]; 0; 0; G_{A}[ℓ]; A[l]; Γ_{A}/ℓ; 0.]

Note that E is also finite at p, since it extends over ℤ_{p} to the finite part of the group scheme A[ℓ].

The above diagram may be localized at any maximal ideal 𝔪 of 𝕋 having residue characteristic ℓ to yield the diagram

[figure: 0; G_{A}[ℓ]_{𝔪}; E_{𝔪}; Φ_{A}[ℓ]_{𝔪}; 0; 0; G_{A}[ℓ]_{𝔪}; A[l]_{𝔪}; (Γ_{A})_{𝔪}/ℓ; 0.]

Now suppose that 𝔪 is not Eisenstein. Then A[ℓ]_{𝔪} has a filtration F^{•} as a 𝕋_{𝔪}/ℓ [Gal(\overline{ℚ}/ℚ)]-module whose successive quotients F^{i}/F^{i+1} are isomorphic to ρ_{𝔪}.

Suppose in addition that 𝔪 is not p-finite. Since G_{A}[ℓ]_{𝔪} and (Γ_{A})_{𝔪}/ℓ are both finite at p, the short exact sequence

0 → (F^{i} ∩ G_{A}[ℓ]_{𝔪})(F^{i+1} ∩ G_{A}[ℓ]_{𝔪})

→ F^{i}/F^{i+1} → F^{i}/(F^{i+1} + F^{i} ∩ Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ})) → 0,

which exists for each i, shows that the graded pieces of the filtration induced by F^{•} on each of these 𝕋_{𝔪}/ℓ-modules are one-dimensional 𝕋/𝔪-vector spaces. (Note that these graded pieces are precisely the kernel and cokernel in the preceding short exact sequence.) Thus each of these 𝕋_{𝔪}/ℓ-modules is of the same length; that is, they are both of length equal to the length of the filtration F^{•}. Now E_{𝔪} is also finite at p, and so the submodule (F^{i} ∩ E_{𝔪})/(F^{i+1} ∩ E_{𝔪}) of F^{i}/F^{i+1} = ρ_{𝔪} must again be one dimensional. Thus E_{𝔪} is also of length equal to the length of the filtration F^{•}, the inclusion Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ}) → E_{𝔪} must be an isomorphism, and so Φ_{A}[ℓ]_{𝔪} = 0. This establishes the proposition. □

We now assume that N is prime. Our first object will be to obtain a correspondence between saturated prime ideals of 𝕋 and abelian subvarieties of J_{0}(N). In order to apply some results of section 1, it will be useful to note that since N is prime, the Hecke algebra 𝕋 is reduced.

Proposition 4.3. If A is an abelian subvariety of J_{0}(N) then A is defined over ℚ, and is closed under the action of 𝕋 on J_{0}(N).

Proof. Let us remark that if A is an abelian subvariety of any abelian variety J, then there is an endomorphism of J whose image is equal to A. (Since the category of abelian varieties up to isogeny is semi-simple, we may find a map σ : J/A → J whose composition with the projection π : J → J/A is multiplication by an integer m on J/A; then m − σ ∘ π is the desired endomorphism.) In our case J = J_{0}(N), and so [10, prop. 3.1] shows that any endomorphism of J_{0}(N) is defined over ℚ, and hence so is its image. This proves that any abelian subvariety A of J_{0}(N) is defined over ℚ.

Now define 𝕋′ to be the subring of 𝕋 generated by the Hecke operators T_{n} for those n prime to N. Since there are no oldforms in S(N), 𝕋′ has finite index in 𝕋; let m be a non-zero integer for which m𝕋 ⊂ 𝕋′. From [18, prop. 7.19] we see that A is closed under the action of 𝕋′. Since A is connected, mA = A, and so 𝕋A = 𝕋mA ⊂ 𝕋′A ⊂ A. This completes the proof of the proposition. □

If A is an abelian subvariety of 𝕋, we may associate to A the ideal Ann_{𝕋}(A), the annihilator in 𝕋 of A. Lemma 2.8 shows that this is a saturated ideal of 𝕋. Conversely, if I is a saturated ideal of 𝕋, then J_{0}(N)[I]^{0} is an abelian subvariety of J_{0}(N).

Proposition 4.4. The preceding correspondences between abelian subvarieties of J_{0}(N) and saturated ideals of 𝕋 put these two sets of objects in order-reversing bijection with one other. More precisely:

(i) If A is an abelian subvariety of J_{0}(N) and I = Ann_{𝕋}(A), then A = J_{0}(N)[I]^{0}. (ii) If I is a saturated ideal of 𝕋, then Ann_{𝕋}(J_{0}(N)[I]^{0}) = I.

(iii) If I_{1} and I_{2} are two saturated ideals of 𝕋, then I_{1} is contained in I_{2} if and only if J_{0}(N)[I_{1}]^{0} contains J_{0}(N)[I_{2}]^{0}.

Proof. This is an easy consequence of the results of [10]; see the discussion of [6, II.10]. □

It is useful to note the following alternative description of J_{0}(N)[I]^{0}.

Lemma 4.5. If I is a saturated ideal of 𝕋, then the inclusion I^{⊥}J_{0}(N) ⊂ J_{0}(N)[I]^{0} is an equality.

Proof. Since 𝕋 is reduced, this follows from lemmas 1.3 and 2.9. □

We now generalize the notion of optimal quotient of J_{0}(N). If A_{1} and A_{2} are two abelian subvarieties of J_{0}(N), such that A_{2} ⊂ A_{1}, we refer to the quotient A = A_{1}/A_{2} as an optimal subquotient of J_{0}(N). An important invariant of the optimal subquotient A of J_{0}(N) is the saturated (by lemma 2.8) ideal I = Ann_{𝕋}(A). The following result describes this ideal in terms of the annihilators of A_{1} and A_{2}.

Lemma 4.6. Let A = A_{1}/A_{2} be an optimal subquotient of J_{0}(N). If we write I = Ann_{𝕋}(A), and I_{i} = Ann_{𝕋}(A_{i}) (for i = 1, 2), then I = (I_{1} : I_{2}).

Proof. An alternative description of I is that it is the ideal of elements of 𝕋 that multiply A_{1} into A_{2}. Thus I · I_{2} annihilates A_{1}, and so I · I_{2} ⊂ I_{1}. This proves that I ⊂ (I_{1} : I_{2}).

Proposition 4.4 and lemma 4.5 together show that A_{1} = I_{1}^{⊥}J_{0}(N) and that A_{2} = I_{2}^{⊥}J_{0}(N). Part (ii) of lemma 1.2 shows that (I_{1} : I_{2}) = (I_{2}^{⊥} : I_{1}^{⊥}). Thus

(I_{1} : I_{2})A_{1} = (I_{2}^{⊥} : I_{1}^{⊥}) · I_{1}^{⊥}J_{0}(N) ⊂ I_{2}^{⊥}J_{0}(N) = A_{2}.

Thus (I_{1} : I_{2}) ⊂ I. Combining this with the result of the preceding paragraph proves the lemma. □

The following result shows that the collection of optimal subquotients of J_{0}(N) is closed under passing to duals.

Lemma 4.7. If A is an optimal subquotient of J_{0}(N), then Â is naturally isomorphic (as an abelian variety with 𝕋-action) to an optimal subquotient of J_{0}(N).

Proof. Write A = A_{1}/A_{2}, B_{1} = J_{0}(N)/A_{1}, and B_{2} = J_{0}(N)/A_{2}. The surjections J_{0}(N) → B_{2} → B_{1} each have connected kernels, and so dualize to yield injections B̂_{1} → B̂_{2} → J_{0}(N)ˆ . Dualizing the short exact sequence

0 → A → B_{2} → B_{1} → 0

(and using the fact that A is connected) yields the short exact sequence

0 → Â → B̂_{2} → B̂_{1} → 0,

and so we have an isomorphism B̂_{2}/B̂_{1} ⟶^{∼} Â. The lemma follows once we note that J_{0}(N)ˆ is 𝕋-equivariantly isomorphic to J_{0}(N). (See the proof of theorem A. In fact, since w_{N} commutes with 𝕋 when N is prime, the canonical polarization of J_{0}(N) yields such an isomorphism.) □

Let I denote the Eisenstein ideal of 𝕋 [6, p. 95]. We recall the following consequences of the results of [6], concerning the Eisenstein ideal I and the Eisenstein maximal ideals of 𝕋.

Proposition 4.8. (i) If 𝔪 is an Eisenstein maximal ideal of 𝕋 of residue characteristic ℓ, then T_{𝔪} is a monogenic ℤ_{ℓ} algebra, and I𝕋_{𝔪} is a principal ideal of 𝕋_{𝔪}.

(ii) If 𝔪 is a non-Eisenstein maximal ideal of 𝕋, then I𝕋_{𝔪} is the unit ideal of 𝕋_{𝔪}.

(iii) Let A be an optimal subquotient of J_{0}(N), and let I = Ann_{𝕋}(A). If 𝔪 is a maximal ideal of 𝕋, let 𝔪′ denote the image of 𝔪 in 𝕋/I. As in section 3, let G_{A} denote the semi-abelian variety over ℚ_{N} that rigid analytically uniformizes A over ℚ_{p}. Note that 𝕋/I acts on both A and (by functoriality) on G_{A}. If 𝔪 is an Eisenstein maximal ideal of 𝕋, then 𝔪′ is good for both A and G_{A}.

Proof. Part (i) is given by [6, cor. II.16.2, thm. II.18.10]. Part (ii) follows from [6, prop. II.14.1], which reconciles our definition of Eisenstein maximal ideals as being those whose associated Galois representation is reducible with that of Mazur as being those in the support of 𝕋/I.

For part (iii), note that the results of [6] imply that every Eisenstein maximal ideal is good for both J_{0}(N) and G_{J_{0}(N)} (see [4, thm. 0.5] and the discussion following its statement). Now repeated applications of lemmas 2.7 and 3.6 prove part (iii). □

Recall that C denotes the subgroup of J_{0}(N) generated by the image of the divisor 0 − ∞ on X_{0}(N). We know that C is a free 𝕋/I-module [6, props. II.9.7, II.11.1].

If A = A_{1}/A_{2} is an optimal subquotient of J_{0}(N), we define C_{A} to be the quotient C ∩ A_{1}/C ∩ A_{2}. Note that C_{A} is a subgroup of A(ℚ)_{tor}. As in section 3, we also let Φ_{A} denote the group of connected components of the characteristic p fibre of the Néron model of A.

Lemma 4.9. If A is an abelian subvariety of J_{0}(N), with annihilator I = Ann_{𝕋}(A), and H is a finite order 𝕋-submodule of J_{0}(N) that is supported on the Eisenstein locus of 𝕋, then H ∩ A = H[I].

Proof. Proposition 4.4 shows that A = J_{0}(N)[I]^{0}. Part (iii) of proposition 4.8, together with corollary 2.3, shows that J_{0}(N)[I]/J_{0}(N)[I]^{0} is supported away from the Eisenstein maximal ideals of 𝕋. By assumption H is supported at the Eisenstein maximal ideals of 𝕋. Thus we conclude that

H ∩ A = H ∩ J_{0}(N)[I]^{0} = H ∩ J_{0}(N)[I] = H[I],

proving the lemma. □

Corollary 4.10. If A is an abelian subvariety of J_{0}(N), with annihilator I = Ann_{𝕋}(A), then C ∩ A = C[I].

Proof. Part (ii) of proposition 4.8 shows that C is supported at the Eisenstein maximal ideals of 𝕋. The corollary is thus a special instance of lemma 4.9. □

Lemma 4.11. If A is an optimal subquotient of J_{0}(N), then A(ℚ)_{tor} is supported on the Eisenstein locus of 𝕋.

Proof. This follows from [6, lem. III.1.1]. □

We can now state and prove our main theorem concerning optimal subquotients of J_{0}(N), which will have theorem B as a corollary.

Theorem 4.12. Let A be an optimal subquotient of J_{0}(N), and let I = Ann_{𝕋}(A). (i) C_{A} is a free 𝕋/(I + I)-module of rank one. (ii) The specialization map C_{A} → Φ_{A} is an isomorphism (iii) The (étale) group scheme Φ_{A} over 𝔽_{N} is constant (that is, has trivial Galois

action).

(iv) The inclusion C_{A} ⊂ A(ℚ)_{tor} is an isomorphism.

Proof. For the duration of the proof, write A = A_{1}/A_{2}, and write I_{i} = Ann_{𝕋}(A_{i}) (for i = 1, 2). We will prove each part of the theorem in turn.

For part (i), note that lemma 4.9 shows that C_{A} = C[I_{1}]/C[I_{2}]. Thus C_{A} is a 𝕋/(I_{1} : I_{2})-module. Lemma 4.6 shows that (I_{1} : I_{2}) = I, and so C_{A} is a 𝕋/I-module. Since it is certainly annihilated by I, it is in fact a 𝕋/(I + I)-module.

To show that it is free of rank one, it suffices to check this after completing at each of the Eisenstein maximal ideal 𝔪, since 𝕋/I is supported at these maximal ideals. If we complete at such a maximal ideal, part (i) of proposition 4.8 shows that we are in the situation of corollary 1.9, and the conclusion of that result gives us what we want. This proves part (i).

We begin our proof of part (ii) by noting that by [14, prop. 2.2], no non-Eisenstein maximal ideal of 𝕋 is N-finite. Thus proposition 4.2 implies that Φ_{A}, Φ_{A_{1}} and Φ_{A_{2}} are all supported on the Eisenstein locus of 𝕋. Let 𝔪 be an Eisenstein maximal ideal of 𝕋, and let 𝔪_{1} denote the image of 𝔪 in 𝕋/I_{1}. Part (iii) of proposition 4.8, together with lemma 4.7, shows that 𝔪_{1} is good for both A_{1} and Â_{1}. Thus corollary 3.5 implies that the sequence

0 → (Φ_{A_{2}})_{𝔪} → (Φ_{A_{1}})_{𝔪} → (Φ_{A})_{𝔪} → 0

is short exact. Since this is true for every Eisenstein maximal ideal, we deduce that

0 → Φ_{A_{2}} → Φ_{A_{1}} → Φ_{A} → 0

is short exact.

Now consider the morphism of short exact sequences

\xymatrix{0 \ar[r] & C_{A_{2}} \ar[r] \ar[d] & C_{A_{1}} \ar[r] \ar[d] & C_{A} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Φ_{A_{2}} \ar[r] & Φ_{A_{1}} \ar[r] & Φ_{A} \ar[r] & 0,}

in which the vertical arrows are provided by specialization to characteristic p. Corollary 4.10 implies that C_{A_{2}} = C_{A_{1}}[I_{2}], while Φ_{A_{2}} ⊂ Φ_{A_{1}}[I_{2}], since I_{2} annihilates A, and hence Φ_{A}. Thus if we knew that the central vertical arrow were an isomorphism, we would conclude that the same is true of the left-hand vertical arrow, and hence also of the right-hand vertical arrow. Applying the same argument with A_{1}, J_{0}(N) and J_{0}(N)/A_{1} in place of A_{2}, A_{1} and A, and using the fact that we know that part (iii) holds for J_{0}(N) [6, p. 99], we deduce that the central arrow is an isomorphism, and thus that part (iii) holds for A. This completes the proof of part (ii).

Part (iii) is an immediate consequence of part (ii), which shows that Φ_{A} is obtained by specializing a group, all of whose points are defined over ℚ.

To prove part (iv), we begin by noting that C_{A} is a direct factor of A(ℚ)_{tor}. The argument is the same as that of [6, p. 99], namely that the composite of the specialization map A(ℚ)_{tor} → Φ_{A} with the inverse of the isomorphism C_{A} ⟶^{∼} Φ_{A} of part (ii) yields a retraction to the inclusion C_{A} ⊂ A(ℚ)_{tor}. Write A(ℚ)_{tor} = C_{A} ⊕ M_{A}.

Observe that C_{A} ⊂ A(ℚ)[I]. Suppose that we could prove that this inclusion were an equality. Then we would conclude that M_{A}[I] = 0, and so M_{A}[𝔪] would be trivial for every Eisenstein maximal ideal 𝔪. Since M_{A} is supported at the Eisenstein maximal ideals of 𝕋, by lemma 4.11, we would conclude that M_{A} = 0, and part (iv) would be proved. Thus we turn to proving that the inclusion C_{A} ⊂ A(ℚ)[I] is an equality.

It suffices to prove that for each Eisenstein maximal ideal 𝔪 of 𝕋, the corresponding inclusion of 𝔪-adic completions (C_{A})_{𝔪} → A(ℚ)[I]_{𝔪} is an equality. Fix one such 𝔪. We claim that the sequence

(4.13)  0 → A_{2}[I]_{𝔪} → A_{1}[I]_{𝔪} → A[I]_{𝔪} → 0

is exact. Indeed, this follows from lemma 2.4, once we note that the image 𝔪_{1} of 𝔪 in 𝕋/I_{1} is good for A_{1} by part (iii) of proposition 4.8, that I_{2}/I_{1} is a saturated ideal of 𝕋/I_{1}, that I(𝕋/I)_{𝔪_{1}} is principal by part (i) of proposition 4.8, and that A_{2} = A_{1}[I_{2}/I_{1}]. Now A_{i}[I]_{𝔪} = J_{0}(N)[I + I_{i}]_{𝔪}, by lemma 4.9, for i = 1, 2. Since (C_{A_{i}})_{𝔪} is equal to the Gal(\overline{ℚ}/ℚ)-invariants of J_{0}(N)[I + I_{i}]_{𝔪}, we are reduced to proving that the morphism J_{0}(N)[I+I_{1}]_{𝔪} → J_{0}(N)[I+I_{1}]_{𝔪}/J_{0}(N)[I+I_{2}]_{𝔪} induces a surjection after passing to Gal(\overline{ℚ}/ℚ)-invariants.

Corollary 1.5 implies that I_{2}𝕋_{𝔪} is principal; write I_{2}𝕋_{𝔪} = α𝕋_{𝔪}. Multiplication by α induces an isomorphism J_{0}(N)[I +I_{1}]_{𝔪}/J_{0}(N)[I +I_{2}]_{𝔪} ⟶^{∼} I_{2}J_{0}(N)[I +I_{1}]_{𝔪}. Thus we must show that multiplication by α induces a surjection from the Galois invariants of J_{0}(N)[I + I_{1}]_{𝔪} onto those of I_{2}J_{0}(N)[I + I_{1}]_{𝔪}.

Since T_{𝔪}J_{0}(N) is free of rank two over 𝕋_{𝔪} [6, cor. II.16.3], we see that J_{0}(N)[I + I_{1}]_{𝔪} is free of rank two over 𝕋_{𝔪}/(I +I_{1}). Thus corollary 1.8 shows tha I_{2}J_{0}(N)[I + I_{1}]_{𝔪} = J_{0}(N)[I + I_{1} + I_{2}^{⊥}]_{𝔪}. The Galois invariants of J_{0}(N)[I + I_{1}]_{𝔪} are equal to C[I_{1}]_{𝔪}, and so the Galois invariants of J_{0}(N)[I +I_{1}+I_{2}^{⊥}]_{𝔪} are equal to C[I_{1}+I_{2}^{⊥}]_{𝔪}. Hence we must show that multiplication by α induces a surjection from C[I_{1}]_{𝔪} onto C[I_{1}+I_{2}^{⊥}]_{𝔪}. Since C[I_{1}]_{𝔪} is free of rank one over 𝕋_{𝔪}/(I +I_{1}), by part (i), this again follows from lemma 1.8. Thus part (iv) is proved. (We remark that rather than appealing to lemma 1.8, we could also have used the fact that 𝕋/I is isomorphic to ℤ/n for an integer n [6, prop. II.9.7]; this allows one to make an analogous argument, using the elementary and explicit description of ideals in ℤ/n.) □

Proof of theorem B. Let f be a normalized eigenform in S(N). Then A_{f} and Â_{f} are optimal subquotients of J_{0}(N), both of whose annihilators in 𝕋 are equal to I_{f}. The first six parts of theorem B thus follow immediately from theorem 4.12.

It remains to prove part (vii). Let {f_{1}, …, f_{d}} be a complete set of conjugacy class representatives of the normalized eigenforms in S(N). Observe that we may find a descending sequence of saturated ideals 0 = I_{0} ⊂ I_{1} ⊂ ⋯ ⊂ I_{d} = 𝕋 of 𝕋 such that (I_{i−1} : I_{i}) = I_{f_{i}}. Let A_{i} = J_{0}(N)[I_{i−1}]^{0}/J_{0}(N)[I_{i}]^{0} (for i = 1, …, d). Lemma 4.6 shows that Ann_{𝕋}(A_{i}) = I_{f_{i}}. Thus part (i) of theorem 4.12 shows that C_{A_{i}} and C_{f_{i}} = C_{A_{f_{i}}} are isomorphic 𝕋-modules, and so in particular have the same order. Also, corollary 4.10 shows that C_{A_{i}} = C[I_{i}]/C[I_{i−1}]. Thus we conclude that

∏_{f_{i}} #C_{f_{i}} = ∏_{i=1}^{d} \frac{#C[I_{i}]}{#C[I_{i−1}]} = #C.

This proves part (vii). □

References

1. A. O. L. Atkin, J. Lehner, Hecke operators on Γ_{0}(m), Math. Ann. 185 (1970), 134–160.

2. N. Boston, H. W. Lenstra, Jr., K. A. Ribet, Quotients of group rings arising from twodimensional representations, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), 323–328.

3. K. Buzzard, On level-lowering for mod 2 representations, Math. Res. Lett. 7 (2000), 95–110.

4. M. Emerton, Supersingular elliptic curves, theta series, and weight two modular forms, preprint (2000), submitted for publication.

5. A. Grothendieck, SGA 7, Exposé IX, SLN 288 (1972), 313–523.

6. B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. Inst. Hautes Étud. Sci. 47 (1977), 33–186.

7. B. Mazur K. A. Ribet, Two-dimensional representations in the arithmetic of modular curves, Astérisque 196–197 (1991), 215–255.

8. J. -F. Mestre, J. Oesterlé, Courbes de Weil semi-stables de discrimnant une puissance m-ième, J. Reine Angew. Math., 400 (1989), 173–184.

9. M. Raynaud, Variétés abéliennes et géométrie rigide, Actes, Congrès Intern. Math., 1970, Tome 1, 473–477.

10. K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math. 101 (1975), 555–562.

11. , Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), 193–205.

12. , On modular representations of Gal(\overline{ℚ}/ℚ) arising from modular forms, Invent. Math. 100 (1990), 431–476.

13. , Multiplicites of p-finite mod p Galois representations in J_{0}(Np), Bol. Soc. Brasil. Mat. (N. S.) 21 (1991), 177–188.

14. , Torsion points on J_{0}(N) and Galois representations, Arithmetic Theory of Elliptic Curves (J. Coates, ed.), SLN, vol. 1716, 1999.

15. K. A. Ribet, W. A. Stein, Lectures on Serre’s conjectures, Proceedings of the 1999 IAS/Park City Mathematics Institute, 2000, to appear.

16. K. A. Ribet, S. Takahashi, Parametrizations of elliptic curves by Shimura curves and by classical modular curves, Proc. Natl. Acad. Sci. 94 (1997), 11110–11114.

17. J. -P. Serre, Sur les représentations modularies de degré 2 de Gal(\overline{ℚ}/ℚ), Duke Math. J. 54 (1987), 179–230.

18. G. Shimura, Publ. Math. Soc. Japan 11, Tokyo-Princeton, 1971.

19. W. A. Stein, The refined Eisenstein conjecture , preprint (1999), available at

http://shimura.math.berkeley.edu/~ was/Tables/Notes/refinedeisen.html.

20. W. A. Stein, Explicit approaches to modular abelian varieties, Berkeley Ph. D. thesis (2000).

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

University of Chicago Department of Mathematics 5734 S. University Ave. Chicago, IL 60637

E-mail address: emerton@math.uchicago.edu