OPTIMAL QUOTIENTS OF MODULAR JACOBIANS

Matthew Emerton

University of Chicago

Fix a field K of characteristic zero, and an algebraic closure K̄ of K. Let G be an algebraic group over K equipped with a commutative, finite rank ℤalgebra of endomorphisms 𝕋. If I is an ideal of 𝕋, then we let G[I] denote the algebraic subgroup of G consisting of elements annihilated by I, and G[I]^{0} denote the connected component of the identity of G[I]. The K̄-valued points of the quotient G[I]/G[I]^{0} form a finite 𝕋-module, and the object of this paper is to introduce a simple technique that gives information about the support of this 𝕋module, and to apply this technique to obtain some theorems concerning optimal quotients of Jacobians of modular curves.

The technique itself is explained in the completely elementary section 1, and encapsulated in corollary 1.3. We devote the rest of this introduction to explaining its applications to optimal quotients, after first recalling the necessary notation and terminology.

Modular forms, modular Jacobians, and Hecke operators. If N is a positive integer, we let J_{0}(N) denote the Jacobian of the modular curve X_{0}(N). We let S(N) denote the space of weight two cusp forms on Γ_{0}(N), and let 𝕋 denote the finite rank, commutative ℤ-algebra of endomorphisms of S(N) generated by the Hecke operators. The algebra 𝕋 acts faithfully as endomorphisms of J_{0}(N), and there is a 𝕋-equivariant isomorphism of S(N) with the cotangent space at the origin of J_{0}(N) [5, p. 444].

Newforms and optimal quotients. If f is a normalized newform in S(N), we let S_{f} denote the subspace of S(N) spanned by f and its algebraic conjugates, and let S_{f}^{⊥} denote the orthogonal complement of S_{f} in S(N) with respect to the Petersson inner product. Each of S_{f} and S_{f}^{⊥} are closed under the action of 𝕋; let I_{f} (respectively I_{f}^{⊥}) denote the annihilator in 𝕋 of S_{f} (respectively S_{f}^{⊥}). Note that I_{f} is also the kernel of the homomorphism 𝕋 → ℂ induced by the action of the Hecke operators on the eigenform f. Since S is the direct sum of S_{f} and S_{f}^{⊥}, we find that I_{f} ∩ I_{f}^{⊥} = 0, while I_{f} + I_{f}^{⊥} has finite index in 𝕋. Let us denote the sum I_{f} + I_{f}^{⊥} by C_{f}; it is the ideal of fusion of f in 𝕋, which measures the congruences between f and Hecke eigenforms lying in the space I_{f}^{⊥}.

Write A_{f} = J_{0}(N)/I_{f} and B_{f} = I_{f}J_{0}(N); there is a short exact sequence of abelian varieties

0 → B_{f} → J_{0}(N) → A_{f} → 0.

One refers to A_{f} as the optimal quotient of J_{0}(N) corresponding to f. Dualizing this exact sequence (and taking into account the autoduality of the Jacobian J_{0}(N)) we obtain the short exact sequence

0 → Â_{f} → J_{0}(N) → B̂_{f} → 0.

Composing the injection Â_{f} → J_{0}(N) with the surjection J_{0}(N) → A_{f} yields an isogeny θ_{f} : Â_{f} → A_{f} (this is the polarization of Â_{f} induced by the canonical polarization of J_{0}(N)).

Maximal ideal in 𝕋 and Galois representations. If 𝔪 is a maximal ideal of 𝕋, then attached canonically to 𝔪 is a semi-simple representation

ρ_{𝔪} : Gal(ℚ̄/ℚ) → GL_{2}(𝕋/𝔪).

We say that 𝔪 is Eisenstein if ρ_{𝔪} is reducible. 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 definition is that if 𝔪 is non-Eisenstein, so that ρ_{𝔪} is irreducible, then J_{0}(N)[𝔪] is isomorphic to a direct sum of copies of ρ_{𝔪} [1], and so 𝔪 satisfies multiplicity one if and only if J_{0}(N)[𝔪] is isomorphic to a single copy of ρ_{𝔪}.) Note that the 𝕋_{𝔪} ⊗ℚmodule T_{𝔪}J_{0}(N) ⊗ ℚ is free of rank two, and so 𝔪 satisfies multiplicity one if and only if T_{𝔪}J_{0}(N) is free of rank two over 𝕋_{𝔪}. In particular, such 𝔪 are good for J_{0}(N), in the sense of section 1.

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 [7, p. 189]. Now suppose that p exactly divides N. We say that ρ_{𝔪} is p-old if there is a maximal ideal 𝔫 in the Hecke ring of level N/p for which ρ_{𝔫} is equivalent to ρ_{𝔪}. (If 𝔪 is p-old then it is necessarily p-finite.) Continue to assume that p exactly divides N; then J_{0}(N) has semi-abelian reduction over ℤ_{p}. Let G_{0}(N)_{/ℚ_{p}} denote the semi-abelian variety over ℚ_{p} which rigid-analytically uniformizes J_{0}(N) over ℚ_{p}, and let T_{0}(N)_{/ℚ_{p}} denote the maximal torus in G_{0}(N)_{/ℚ_{p}}. By functoriality of rigid-analytic uniformization, there is an action of 𝕋 on G_{0}(N)_{/ℚ_{p}}, and so on T_{0}(N)_{/ℚ_{p}}. We say that 𝔪 satisfies toric multiplicity one at p if T_{𝔪}T_{0}(N)_{/ℚ_{p}} is free of dimension one over 𝕋_{𝔪}. Such 𝔪 are good for T_{0}(N)_{/ℚ_{p}}, in the sense of section 1.

Having filled in the necessary background, we can now discuss the applications of corollary 1.3 to optimal quotients. There are two problems that we will analyze, and we describe each problem, and the corresponding result that we obtain, in turn.

Problem I: Fix a normalized newform f in S(N). We wish to understand the relation between the kernel of the polarization θ : Â_{f} → A and the ideal of fusion C_{f} of f. More precisely, a consideration of cotangent spaces shows that B_{f} is isomorphic to the connected component of J_{0}(N)[I_{f}^{⊥}], and that Â_{f} is isomorphic to the connected component of J_{0}(N)[I_{f}]. Thus

ker(θ) = Â_{f} ∩ B_{f} ⊂ J_{0}(N)[I_{f}] ∩ J_{0}(N)[I_{f}^{⊥}] = J_{0}(N)[C_{f}],

and we wish to analyze the cokernel of this inclusion. (This problem is related to, but not the same as, that of understanding the relation between the modular degree of a strong Weil curve and the modulus of congruence of the corresponding eigenform, as discussed for example in [11].) This problem is addressed by the following result, which follows immediately from corollary 1.3, upon taking the algebraic group G to be J_{0}(N), and the ideal of endomorphisms I to be alternately I_{f} and I_{f}^{⊥}.

Theorem A. Let f be a normalized new form in S(N), and let θ : Â_{f} → A_{f} be the polarization induced by the canonical polarization of J_{0}(N). Then the cokernel of the inclusion ker(θ) ⊂ J_{0}(N)[C_{f}] is supported at maximal ideals of 𝕋 which do not satisfy multiplicity one.

We should point out that the cokernel of the inclusion ker(θ) ⊂ J_{0}(N)[C_{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 which 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 component of C_{f} is equal to 𝔪. Thus J[𝔪] ⊂ J[C_{f}]. On the other hand, J[𝔪] cannot be contained in ker θ, since J[𝔪] ∩ E = E[2], and the fact that 𝔪 does not satisfy multiplicity one shows that the inclusion E[2] ⊂ J[𝔪] cannot be an equality.

Problem II: Suppose that p is a prime which exactly divides N. Then J_{0}(N) and A_{f} both have semi-abelian reduction at p. Let Φ_{J_{0}(N)} (respectively Φ_{A_{f}}) denote the connected component group of the special fibre at p of the Néron model of J_{0}(N) (respectively of A_{f}). The surjection J_{0}(N) → A_{f} induces a morphism Φ_{J_{0}(N)} → Φ_{A_{f}}. The second problem we wish to address is that of understanding the cokernel of this map. (This problem has been raised by William Stein in the particular case that N = p [8].) It is addressed by the following result.

Theorem B. Assume as in the preceding discussion that the prime p exactly divides N. Let I be an ideal of 𝕋 such that 𝕋/I is torsion free, and let A denote the quotient J_{0}(N)/I. (For example I could be the ideal I_{f} corresponding to a normalized new form f in S(N), in which case A is equal to A_{f}.) Then any maximal ideal of 𝕋 in the support of the cokernel of the map of connected component groups Φ_{J_{0}(N)} → Φ_{A} is either p-old, or else does not satisfy toric multiplicity one.

Theorem B is proved by first applying corollary 1.3 with the algebraic group G taken to be G_{0}(N)_{/ℚ_{p}} and the ideal I taken to be I_{f}, to obtain information about the support of the 𝕋-module G_{0}(N)_{/ℚ_{p}}[I_{f}]/(G_{0}(N)_{/ℚ_{p}}[I])^{0}. An analysis of the relation between rigid analytic uniformizations of abelian varieties with semiabelian reduction and the component groups of their special fibres, carried out in section 2, then relates G_{0}(N)_{/ℚ_{p}}[I_{f}]/(G_{0}(N)_{/ℚ_{p}}[I])^{0} to the cokernel of the map Φ_{J_{0}(N)} → Φ_{A_{f}}.

In the situation of problem II, there is an approach to analyzing the support of the 𝕋-module Φ_{A} using Ribet’s lowering-the-level theorem, which we learned from [8]. The following argument is a variation of the observations of [8, §4].

Theorem C. Let A be an abelian variety subquotient of J_{0}(N) on which 𝕋 acts, and let Φ_{A} denote the connected component group of the special fibre of the Néron model of A. Then Φ_{A} is a finite 𝕋-module which is supported on the union of the Eisenstein and the p-finite locus of 𝕋.

Proof. Let us fix a maximal ideal 𝔪 of 𝕋 of residue characteristic ℓ. As explained in [8, p. 4], we have the diagram of group schemes over ℚ_{p}

[figure: 0; Hom(ΓÂ/ℓ, μ_{ℓ}); E; Φ_{A}[ℓ]; 0; 0; Hom(ΓÂ/ℓ, μ_{ℓ}); A[l]; Γ_{A}/ℓ; 0],

whose rows are exact, and in which E is the maximal finite subgroup scheme of A[l]. This diagram may be localized at 𝔪 to yield the diagram

[figure: 0; Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ}); E_{𝔪}; Φ_{A}[ℓ]_{𝔪}; 0; 0; Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ}); 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(ℚ̄/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 Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ}) and (Γ_{A})_{𝔪}/ℓ are both finite at p, the short exact sequence

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

→ 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 filtration induced by F^{•} on each of these 𝕋_{𝔪}/ℓmodules has its successive quotients being one-dimensional 𝕋/𝔪-vector spaces. (Note that these successive quotients 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 is also finite at p, and so the submodule (F^{i} ∩ E)/(F^{i+1} ∩ E) of F^{i}/F^{i+1} = ρ_{𝔪} must also be one dimensional. Thus the inclusion

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

must be an equality, and so E is also of length equal to the length of the filtration F^{•}. Hence the inclusion Hom((ΓÂ)_{𝔪}/ℓ, μ_{ℓ}) → E_{𝔪} is an isomorphism, proving that Φ_{A}[ℓ]_{𝔪} = 0. This establishes the proposition. □

The lowering-the-level results of [5] (as strengthened by the main result of [4]), shows that if 𝔪 is a non-Eisenstein, p-finite maximal ideal in 𝕋 of residue characteristic ℓ, then 𝔪 is necessarily p-old, provided that ℓ is odd and either ℓ^{2} does not divide N or ℓ ≢ 1 (mod p). Combining this together with theorems B and C we see that the only possible non-p-old maximal ideals that can appear in support of the cokernel of the map Φ_{J_{0}(N)} → Φ_{A} are either Eisenstein maximal ideals not satisfying toric multiplicity one, or p-finite maximal ideals not satisfying toric multiplicity one, having residue characteristic ℓ which does not satisfy the preceding conditions.

If N = p, then certainly ℓ^{2} cannot divide N, and (as Ribet observes in [6, prop. 2.2]) the main result of [10] establishes lowering-the-level for non-Eisenstein maximal ideals also in the case that ℓ equals two. The results of [3] show that all the Eisenstein maximal ideals satisfy toric multiplicity one (as observed in the remarks following the statement of [2, thm. 0.5]). Finally, there are no p-old maximal ideals in this situation. Thus we obtain the following corollary, which confirms a conjecture of William Stein [8, conj. 4.1].

Corollary. Let N = p be prime, let I be an ideal of 𝕋 such that 𝕋/I is torsion free, and let A denote the quotient J_{0}(N)/I. Then the map of connected component groups Φ_{J_{0}(N)} → Φ_{A} is surjective.

1. connected component groups of algebraic groups and the proof of theorem a

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 geometrically 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 K̄ of K, then the K̄- 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 K̄- valued points. (This is permissible, since the finite algebraic group can be recovered up to isomorphism from its group of K̄-valued points, equipped with its natural Gal(K̄/K)-action.)

We will require the following easily proved fact:

Fact 1.1. 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 commutative ℤ-algebra of endomorphisms of 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.

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 Hom(G[ℓ^{∞}], ℚ_{ℓ}/ℤ_{ℓ}) of G[ℓ^{∞}]; this is not quite the usual ℓ-adic Tate module if G/G^{0} has a non-trivial ℓ-Sylow subgroup) 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 if T_{𝔪}G is a free 𝕋_{𝔪}-module.

Lemma 1.2. Suppose that 𝔪 is good, and that I is an ideal in 𝕋 such that the quotient 𝕋/I is torsion free (as a ℤ-module). 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] = Hom_{𝕋}(𝕋/I, Hom(T_{𝔪}G, ℚ_{ℓ}/ℤ_{ℓ})) = Hom_{𝕋}(T_{𝔪}G/I, ℚ_{ℓ}/ℤ_{ℓ}),

and so 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 1.3. Let I be an ideal in 𝕋 such that 𝕋/I is torsion free, and write H = G[I]. Then the connected component group H/H^{0} is supported at maximal ideals of 𝕋 which are not good.

Proof. This follows from fact 1.1 and lemma 1.2. □

We can now prove theorem A.

Proof of theorem A. We use the notation introduced in the discussion of problem I. As observed in that discussion, we have Â_{f} = J_{0}(N)[I_{f}]^{0} and B_{f} = J_{0}(N)[I_{f}^{⊥}]^{0}. Thus

ker(θ) = J_{0}(N)[I_{f}]^{0} ∩ J_{0}(N)[I_{f}^{⊥}].

Theorem A is follows immediately from this observation, corollary 1.3 (taking G over K to be J_{0}(N) over ℚ and I to be alternately I_{f} and I_{f}^{⊥}), and the fact that multiplicity one maximal ideals are good for J_{0}(N). □

2. exact sequences of abelian varieties with semi-abelian reduction and the proof of theorem b

In this section we assume that 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(K̄/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. The formation of Γ_{A} and G_{A} is functorial in the category of abelian varieties A having semi-abelian reduction.

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

(the exactness of the top row follows because the kernel of the map B → A is connected; see [9, prop. 4.12]). We can fill this diagram in 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] & Γ_{B} \ar[d] \ar[r] & Γ_{A} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & G′_{C} \ar[r] \ar[d] & G_{B} \ar[d] \ar[r] & G_{A} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & C \ar[r] \ar[d] & B \ar[r] \ar[d] & A \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0 & .}

The algebraic group G′_{C} need not be connected; functoriality of uniformization yields a map G_{C} → G′_{C}, which induces an isomorphism G_{C} ⟶^{∼} (G′_{C})^{0}, as well as a map Γ_{C} → Γ′_{C}, whose cokernel is finite and is canonically isomorphic to the connected component group G′_{C}/(G′_{C})^{0}.

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

Again we obtain an isomorphism GÂ ⟶^{∼} (G′ ; Â)^{0}, and a short exact sequence

0 → ΓÂ → Γ′ ; Â → GÂ/(G′ ; Â) → 0.

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

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

(2.1)  0 → G′_{C}/G_{C} → Φ_{C} → Φ_{B} → Φ_{A} → (G′ ; Â/GÂ)^{∗} → 0.

There is a particular deduction that we wish to make from the exact sequence (2.1). We will state it as a lemma, after introducing the relevant assumptions and notation.

Suppose that 𝕋 is an algebra of endomorphisms of B, that I is an ideal in 𝕋, and that C = IB and A = B/I. Note that by functoriality, 𝕋 acts on GB̂.

Lemma 2.2. In the preceding situation, the cokernel of the map Φ_{B} → Φ_{A} is a quotient of the group (GB̂[I]/(GB̂[I])^{0})^{∗}.

Proof. Since I annihilates A, we see that IG′ ; Â ⊂ Γ′ ; Â. Now IG′ ; Â is an algebraic group, while Γ′ ; Â is a lattice. Thus IG′ ; Â = 0, and we deduce that G′ ; Â ⊂ GB̂[I]. A consideration of cotangent spaces shows that

(GB̂[I])^{0} = (G′ ; Â)^{0} = GÂ.

Thus we have an injection

G′ ; Â/GÂ → GB̂[I]/(GB̂[I])^{0},

and hence a surjection

(GB̂[I]/(GB̂[I])^{0})^{∗} → (G′ ; Â/GÂ)^{∗}.

The lemma now follows, since the exact sequence (2.1) identifies (G′ ; Â/GÂ)^{∗} with the cokernel of the map Φ_{B} → Φ_{A}. □

We can now prove theorem B.

Proof of theorem B. As in the statement of the theorem, we let I be an ideal in the Hecke algebra 𝕋 such that 𝕋/I is torsion-free, and let A = J_{0}(N)/I. Lemma 2.2 (and the self-duality of the Jacobian J_{0}(N)) implies that any maximal ideal in the support of the cokernel of the map

Φ_{J_{0}(N)} → Φ_{A}

is in the support of Ext^{1}(G_{0}(N)_{/ℚ_{p}}[I]/(G_{0}(N)_{/ℚ_{p}}[I])^{0}, ℤ), and so in the support of G_{0}(N)_{/ℚ_{p}}[I]/(G_{0}(N)_{/ℚ_{p}}[I])^{0}. Now corollary 1.3 (taking G_{/K} to be G_{0}(N)_{/ℚ_{p}}) implies that any such 𝔪 is not good for G_{0}(N)_{/ℚ_{p}}. If 𝔪 is not p-old, then the map T_{𝔪}T_{0}(N)_{/ℚ_{p}} → T_{𝔪}G_{0}(M)_{ℚ_{p}} is an isomorphism, and so such an 𝔪 must not be good for T_{0}(N)_{/ℚ_{p}} either. In particular, it must not satisfy toric multiplicity one. □

References

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