Bryden Cais Kris Klosin March 18, 2004 Mazur Seminar

1. Introduction

The goal of these notes is to show that the modular jacobian J_{0}(N)_{/Q} has a nontrivial quotient of rank 0 for any prime N such that the genus of X_{0}(N) is positive.

Recall the following theorem from James’ talk: Theorem 1.1. Let A_{/Q} be an abelian variety with good reduction outside of N and purely toric reduction at N. Suppose moreover that A[p] is admissible for some p ≠ N, where A_{/Z} is the Néron model of A. Then A_{/Q} has rank 0.

Our aim is to construct a nonzero isogeny factor J_{𝔓} of J_{0}(N) satisfying the hypotheses of this theorem. Before we construct J_{𝔓} however, we will show that the above reduction properties of A are inherited by any isogeny factor, and will discuss the relationship between isogenies A ∼ A′ × A′′ (with A, A′, A′′ abelian varieties over a field K) and idempotents in the ring of endomorphisms End^{0}(A) in the isogeny cateogory of abelian varieties over K.

2. Semi-abelian reduction

In order to apply Theorem 1.1 to the isogeny factor J_{𝔓} of J_{0}(N) that we will construct, we must show that J_{𝔓} has good reduction outside N and purely toric reduction at N, and that J_{𝔓}[p] is admissible. We first show that all isogeny factors of J_{𝔓} inherit these reduction properties.

Let K be the fraction field of a discrete valuation ring O with normalized valuation v and residue field k. Fix a separable closure K_{s} of K and a prime v_{s} of K_{s} lying over v, and denote by I the inertia group of v_{s}. Let A be an abelian variety over K, and denote by A its Neron model over O, and by A^{0} the connected component of the identity of A. Recall that we say that A has semiabelian reduction with respect to O if the reduction A_{s}^{0} is an extension of an abelian variety by a torus.

The aim of this section is to prove the following propositon: Proposition 2.1. Let the notation be as above. Suppose moreover that A is isogenous to the product A′ × A′′ of abelian varieties A′ and A′′ over K. Then if A has good (resp. semi-abelian) reduction, so do A′ and A′′.

Proof. We will only prove the case of semi-abelian reduction as the other cases are easier. Choose a prime ℓ ≠ char k. Let V_{ℓ}(A) := T_{ℓ}(A) ⊗ Q denote the rational ℓ-adic Tate module of A. First observe that

V_{ℓ}(A) = V_{ℓ}(A′ × A′′) = V_{ℓ}(A′) × V_{ℓ}(A′′).

Grothendieck’s criterion for semi-abelian reduction (cf. [1], theorem 6, page 184) asserts that A has semiabelian reduction if and only if there exists a subspace V of V_{ℓ}(A) stable under the inertia group I such that I acts trivially on both V and the quotient V_{ℓ}(A)/V. (cf. [1, Theorem 6, p. 184]) Note that V can (and will) be taken to be the maximal subspace of V_{ℓ}(A) on which the action of I is trivial.

Let V′, V′′ be the maximal subspaces of V_{ℓ}(A′) and V_{ℓ}(A′′) respectively, on which I operates trivially. Obviously V′ × V′′ = V. Moreover, as (V_{ℓ}(A′)/V′) × (V_{ℓ}(A′′)/V′′) = V_{ℓ}(A)/V, we conclude that I acts trivially on V_{ℓ}(A′)/V′ and on V_{ℓ}(A′′)/V′′. This finishes the proof. ■

Remark 2.2. Let us prove that if A has toric reduction, then so do A′ and A′′. Assume that A has toric reduction. First note that since A and A′ × A′′ are isogenous, they are both semi-abelian as their rational Tate modules are isomorphic. Denote by ϕ_{K} an isogeny A → A′ × A′′. Then there exists an isogeny ψ_{K} : A′ × A′′ → A and a nonzero integer n, such that ϕ_{K}ψ_{K} = n_{A′×A′′} and ψ_{K}ϕ_{K} = n_{A} (cf.[1], p. 169). The Neron mapping property ensures that ϕ_{K} and ψ_{K} extend to morphisms ϕ and ψ between A and the Neron model A′ × A′′ of A′ × A′′. We have Ner(A′ × A′′) = A′ × A′′ because of the Neron mapping property: if Z is a smooth O-scheme, then

Hom_{K}(Z_{K}, A′ × A′′) = Hom_{K}(Z_{K}, A′) × Hom_{K}(Z_{K}, A′′)

= Hom_{O}(Z, A′) × Hom_{O}(Z, A′′) = Hom_{O}(Z, A′ × A′′).

We will first show that ϕ̄ : Ā → Ā′ × Ā′′ is an isogeny. First note that ψ̄ϕ̄ = ψ̄ϕ = [n]. Since A is semi-abelian, Ā^{0} is an extension of an abelian variety by a torus. As multiplication by a nonzero integer is surjective on abelian varieties and on tori, we conclude that [n] is surjective on Ā^{0}, hence an isogeny. Since [n] = ψ̄ϕ̄, and Ā and Ā′ ×Ā′′ have the same dimension, we conclude that ϕ̄ is surjective on the identity components and has finite kernel, hence is an isogeny.

The isogeny between Ā and Ā′ × Ā′′ induces an isogeny between Ā^{0} and (Ā′ × Ā′′)^{0} = Ā′^{0} × Ā′′^{0}, so it remains to check that for connected commutative smooth algebraic groups over a field k, the property of being a torus is preserved under isogeny and under formation of direct factors. This is clear since by the structure theorem for commutative algebraic groups over k̄ such a group G is a torus if and only if G is affine and dim(V_{ℓ}(G)) = dim(G), where l ≠ char k is a prime.

3. Idempotents inside the endomorphism ring of abelian varieties

In this section we remark upon some generalities concerning the relationship between idempotents in the ring End^{0}(A) for an abelian variety A and isogenies A ∼ A′ × A′′ with A′, A′′ abelian varieties. Proposition 3.1. Let A be an abelian variety over a field K. There is a one-to-one correspondence between ordered pairs (A′, A′′) of abelian subvarieties of A such that A′ ×_{K} A′′ → A is an isogeny and idempotents in the ring End^{0}(A); the operation (A′, A′′) ↦ (A′′, A′) corresponds to e ↦ 1 − e.

Proof. Suppose we are given a pair (A′, A′′) of abelian subvarieties of A such that the natural map ϕ : A′ ×A′′ → A is an isogeny. Let ϕ^{−1} denote the inverse of ϕ inside Hom^{0}(A, A′ × A′′). Consider the composition

e : A ⟶^{ϕ^{−1}} A′ × A′′ ⟶^{pr} A′ ↪^{ι} A′ × A′′ ⟶^{ϕ} A,

where ι is inclusion and pr is projection. Note that e is an idempotent in End^{0}(A), and by definition of ϕ, it lifts the identity on A′.

Conversely, given an idempotent e ∈ End^{0}(A), choose a nonzero integer n such that ne ∈ End(A), and set A′ to be the image of ne. Note that A′ is independent of n, since multiplication by a nonzero integer is surjective on an abelian variety. The Poincare Reducibilty Theorem (c.f.[4]) guarantees the existence of a unique abelian subvariety A′′ of A, such that A′ × A′′ → A is an isogeny (the map being addition). One checks that the two processes are inverses of each other.

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4. The optimal quotient J_{𝔓} of J_{0}(N)

In this section we will construct a nonzero optimal quotient J_{𝔓} of J_{0}(N) such that the action of T on J_{0}(N) induces an action on J_{𝔓}.

Fix N and let T denote the Hecke ring. Let ℐ ⊆ T be the Eisenstein ideal let 𝔓 ⊇ ℐ be a prime of residual characteristic p ≠ N (we have seen that such a 𝔓 exists). Observe that 𝔓_{p} := 𝔓(T ⊗ Z_{p}) is a prime of T ⊗ Z_{p}.

Consider the subring T ⊗_{Z} Q ⊆ End^{0}(J_{0}(N)) (we have seen that in fact equailty holds, but we do not use this). By Proposition 3.1, the decomposition of T ⊗ Q into a product of fields gives us the isogeny decomposition of J_{0}(N) as a product of Q-simple abelian subvarieties:

(1)  J_{0}(N) ∼ ∏_{𝔮∈MinSpec(T)} J_{𝔮}.

We put

J̃_{𝔓} = ∏_{𝔮∈MinSpec(T) 𝔮⊆𝔓} J_{𝔮}.

Definition 4.1. Let A, A′, and A′′ be abelian varieties and suppose that A′ and A′′ are quotients of A. We say that A′ and A′′ are isogenous as quotients of A if there exists an isomorphism ϕ : A′ → A′′ in the isogeny category commuting (in that category) with the quotient maps A → A′ and A → A′′. Claim 4.2. Let A → A′′ be a surjective map of abelian varieties over a field K. Then there is a unique quotient abelian variety A_{opt}, isogenous to A′′ as a quotient of A, that is the quotient of A by an abelian subvariety.

Proof. We treat existence first. By the Poincaré Reducibility Theorem over K, we see that A is isogenous to a product of simple abelian subvarieties, say A ← ∏_{i=1}^{n} A_{i}, and since A′′ is a quotient of A we have (renumbering if necessary) an isogeny A′′ ∼ ∏_{i=1}^{k} A_{i}, as quotients of A, for some k ≤ n. Define A′ = ∏_{k<i≤n} A_{i}. Then we have a morphism

A′ ↪^{ι} ∏_{i=1}^{n} A_{i} ⟶^{φ} A,

where φ is an isogeny. Observe that φ ∘ ι(A′) is an abelian subvariety of A. We set

A_{opt} = A/φ ∘ ι(A′).

It is clear that A_{opt} is the quotient of A by an abelian subvariety and it follows (again from Poincaré Reducibility) that A_{opt} is isogenous to A′′ as a quotient of A.

We claim that A_{opt} has the following universal property: if q′′ : A → Ã is any quotient isogenous to the quotient q : A → A_{opt} in the sense of Definition 4.1 then q′′ factors uniquely through q in the isogeny category. Indeed, if Ã and A_{opt} are isogenous as quotients, there exists an isomorphism (In the isogeny category) ϕ : A_{opt} → Ã. By composing ϕ with multiplication by a large enough integer, we obtain honest maps n ∘ q′′ : A → Ã and ψ = n ∘ ϕ : A_{opt} → A such that ψ ∘ q = n ∘ q′′. Hence n ∘ q′′(ker q) = 0. But multiplication by n is an isogeny, and hence has finite kernel, while q′′(ker q) is connected (as ker q is an abelian subvariety of A), so that q′′(ker q) = 0.

It follows that the map A → A′′ factors uniquely through A_{opt}. From this universal property of A_{opt}, we see at once that A_{opt} is unique up to unique isomorphism. ■

Applying the above claim to A = J_{0}(N) and the isogeny factor J̃_{𝔓}, we obtain an optimal quotient J_{𝔓} of J_{0}(N). We claim that T acts on J_{𝔓}. This will follow from the following more general theorem:

Theorem 4.3. Let π : A → A′ be a surjective map of abelian varieties over a field K of characteristic 0 having an abelian variety kernel B and let T ∈ End(A). Assume there exists T′ ∈ End(A′)^{0} such that the following diagram commutes in the isogeny category:

\begin{matrix}A & ⟶_{π} & A′ \\ ^{T} ↓ & & ↓_{T′} \\ A & ⟶^{π} & A′ \end{matrix} . (2)

Then T′ ∈ End(A′).

Proof. We have T_{0} := nT′ ∈ End(A′) for some nonzero integer n, and by the universal property of the quotient map A′ → A′ having kernel A′[n], we have T′ ∈ End(A′) if and only if T_{0} kills A′[n]. Since T_{0}π = nT′π = nπT = πT n in the isogeny category, the genuine maps T_{0}π and πT n agree so that T_{0}π kills A′[n] as πT n obviously does. To conclude that T_{0} kills A′[n], it therefore suffices to show that π is faithfully flat on n-torsion, for which it is enough to show surjectivity on \overline{K}-points (as finite K-groups are étale so A[n] ⟶^{π} A′[n] is faithfully flat if and only if it is surjective on \overline{K}-points). But we have the commutative diagram

0 ⟶ B(\overline{K}) ⟶ A(\overline{K}) ⟶ A′(\overline{K}) ⟶ 0
n↓  ^{n}↓  ^{n}↓
0 ⟶ B(\overline{K}) ⟶ A(\overline{K}) ⟶ A′(\overline{K}) ⟶ 0

and since A, A′, B are abelian varieties and K is of characteristic 0, the vertical maps are all surjective, so by the Snake lemma, the map on n-torsion A[n] ⟶^{π} A′[n] is surjective as desired. This completes the proof. ■

Applying Theorem 4.3 to the optimal quotient J_{𝔓} of J_{0}(N) shows that we have an action of T on J_{𝔓}.

5. Admissibility

In Brian’s talk, it was explained why J_{𝔓}[p] is an admissible group scheme over Z. In this section we will recall the proof of this fact in more detail.

Since T is a finite Z-module, the ring T ⊗_{Z} Z_{p} is a finite Z_{p}-module and hence is semi-local. Moreover, we have seen in section 4 that T acts on J_{𝔓} in a manner that respects the quotient map J_{0}(N) → J_{𝔓} so we obtain an action of T ⊗_{Z} Z_{p} on the Tate modules T_{p}(J_{0}(N)) and T_{p}(J_{𝔓}) as G_{Q} := Gal(\overline{Q}/Q)-modules. By 8.7 and 8.15 of [3], there is a canonical isomorphism

(3)  T ⊗_{Z} Z_{p} ≃ ∏_{𝔪∈MaxSpec(T⊗_{Z}Z_{p})}(T ⊗_{Z} Z_{p})_{𝔪}.

Claim 5.1. The action of T⊗_{Z} Z_{p} on T_{p}(J_{𝔓}) factors through T_{𝔓} := (T⊗_{Z} Z_{p})_{𝔓}, so the induced action of T⊗_{Z} Z_{p} on J_{𝔓}[p] is through the quotient T_{𝔓}.

Proof. First note that the functoriality of the idempotent decomposition of T ⊗ Q implies that

V_{p}(J_{𝔓}) = ∏_{\begin{matrix}𝔮∈Spec(T⊗Q) \\ 𝔮⊂𝔓 \end{matrix}} V_{p}(J_{𝔮}) = ∏_{\begin{matrix}𝔮∈Spec(T⊗Q) \\ 𝔮⊂𝔓 \end{matrix}} V_{p}(J)_{𝔮},

where the objects V_{p}(J)_{𝔮} denote the localizations of the T ⊗ Q-module V_{p}(J) at primes 𝔮. (We will use the same letter 𝔮 to denote both a prime ideal of T ⊗ Q and its inverse image in T). From this we see that the action of T on V_{p}(J_{𝔓}) factors through ∏_{𝔮⊂𝔓}(T ⊗ Q)_{𝔮} and since every element of T − 𝔓 is mapped to a unit of ∏_{𝔮⊂𝔓}(T ⊗ Q)_{𝔮} (since 𝔮 ⊂ 𝔓), the map T → ∏_{𝔮⊂𝔓}(T ⊗ Q)_{𝔮} factors through the localization T_{𝔓}. The claim now follows after tensoring with Z_{p} and noting that T_{p}(J_{𝔓}) ⊆ V_{p}(J_{𝔓}). ■

We now prove: Lemma 5.2. Let ℐ ⊆ T be the Eisenstein ideal and 𝔓 ⊃ ℐ be any prime of T having residue characteristic p ≠ N. Then 𝔓^{r} kills J_{𝔓}[p] for some r > 0.

Proof. The action of T ⊗_{Z} Z_{p} on J_{𝔓}[p] factors through T_{𝔓}/pT_{𝔓}. As T ⊗ Z_{p} is a finite Z_{p}-module, we see that T_{𝔓}/pT_{𝔓} is a finite F_{p}-module, and consequently is a finite local ring. It follows that its maximal ideal is nilpotent. Thus, for some r > 0 we have 𝔓^{r} ⊆ pT_{𝔓}, so 𝔓^{r}J_{𝔓}[p] = 0 as claimed. ■

It follows from Lemma 5.2 that we have a filtration of T_{𝔓}[G_{Q}] modules

J_{𝔓}[p] ⊇ 𝔓J_{𝔓}[p] ⊇ 𝔓^{2}J_{𝔓}[p] ⊇ … ⊇ 𝔓^{r}J_{𝔓}[p] = 0. (4)

Observe that each quotient 𝔓^{i}J_{𝔓}[p]/𝔓^{i+1}J_{𝔓}[p] is killed by 𝔓. We will now show that this implies that the quotients are all admissible. Recall from [2] (Corollary 1.6) that admissibility of a group scheme over Z can be checked on the corresponding Galois module of \overline{Q} points. It then follows that J_{𝔓}[p] is admissible, by general results on Jordan–Hölder series. Lemma 5.3. Let G be a finite discrete G_{Q}-module of p-power order on which T acts and let 𝔓 ⊇ ℐ be a prime of T having residual characteristic p ≠ N and containing the Eisenstein ideal. Assume that for any prime ℓ |̸Np that the inertia group I_{ℓ} at ℓ acts trivially on G, and hence that we have an action of Frob_{ℓ} ∈ Gal(\overline{F}_{ℓ}/F_{ℓ}) on G. Suppose that the Eichler-Shimura relation

Frob_{ℓ}^{2} −T_{ℓ} Frob_{ℓ} +ℓ = 0

holds on G for all such ℓ. If 𝔓 kills G then G has a filtration by admissible closed subgroups with successive quotients Z/pZ or μ_{p}.

Proof. Let Γ be the discrete finite quotient of Gal(\overline{Q}/Q) through which the Galois action on G factors. By hypothesis, for l |̸Np we have that Frob_{ℓ} acts on G and

(5) Frob_{ℓ}^{2} −T_{ℓ} Frob_{ℓ} +ℓ = 0

on G. But as 𝔓 ⊇ ℐ kills G and T_{ℓ} ≡ ℓ + 1 mod ℐ, we see that Frob_{ℓ} acts on G with eigenvalues contained in \{1, ℓ\}. Now any σ ∈ Gal(\overline{Q}/Q) also acts on G^{∨} = Hom(G, μ_{p}) via f^{σ}(g) = σf(g^{σ^{−1}}). Observe that if g is an eigenvector of σ = Frob_{ℓ} with eigenvalue ℓ then f^{σ}(g) = σf(ℓ^{−1}g) = ℓf(ℓ^{−1}g) = f(g), so that f(g) is an eigenvector with eigenvalue 1. Similarly, if g has eigenvalue 1 then f(g) has eigenvalue ℓ. It follows that the eigenvalues {1, ℓ} of Frob_{ℓ} on G × G^{∨} occur with the same multiplicity. Thus, the characteristic polynomial of Frob_{ℓ} on G × G^{∨} over F_{p} is (X − 1)^{d}(X − ℓ)^{d} for some d (which is the dimension of G as over F_{p}).

Consider now the F_{p}[G_{Q}]-module (Z/pZ)^{d} × μ_{p}^{d}. By extending Γ if necessary (and still preserving the finiteness of Γ) we can regard (Z/pZ)^{d} × μ_{p}^{d} as a finite discrete Γ-module. Since Frob_{ℓ} acts on μ_{p} with eigenvalue ℓ, we see that the characteristic polynomial of Frob_{ℓ} on (Z/pZ)^{d} × μ_{p}^{d} is also (X − ℓ)^{d}(X − 1)^{d}. Now by the Tchebotarev density theorem, every γ ∈ Γ is the image of Frob_{ℓ} for some ℓ ≠ N, p. Thus, every γ ∈ Γ has the same characteristic polynomial on the two F_{p}[Γ]-modules (Z/pZ)^{d} × μ_{p}^{d} and G. Applying the Brauer–Nesbitt theorem, we conclude that these Γ-modules have the same semisimplifications. Since (Z/pZ)^{d} × μ_{p}^{d} has a filtration as a Galois module with successive quotients isomorphic to Z/pZ or μ_{p}, so does G. ■

Corollary 5.4. Let N, p, 𝔓 be as before. Then J_{𝔓}[p] is admissible.

Proof. By [2] (Corollary 1.6), we need only check that the Galois module J_{𝔓}[p] has a Gal(\overline{Q}/Q)-stable filtration by μ_{p}’s and Z/pZ’s. Using Proposition 2.1 and Remark 2.2, we see that J_{𝔓} has toric reduction at N and good reduction outside N, so the inertia group at any ℓ |̸Np acts trivially on the Galois module J_{𝔓}[p](\overline{Q}). Moreover, the Eichler-Shimura relations hold on J_{𝔓}[p], as they do in J_{0}(N) and hence on the isogeny factor J_{𝔓}. We may therefore apply Lemma 5.3 to the situation discussed after the proof of Lemma 5.2 to conclude that J_{𝔓}[p] is admissible. ■

Corollary 5.5. The nonzero isogeny factor J_{𝔓} of J_{0}(N) has rank 0.

Proof. By Proposition 2.1 and Remark 2.2, we know that J_{𝔓} has good reduction away from N and purely toric reduction at N. As J_{𝔓}[p] is admissible by Corollary 5.4, we see that J_{𝔓} satisfies the hypotheses of Theorem 1.1 and hence has rank 0. ■

References

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[3] Matsumura, H. Commutative Ring Theory, Cambridge University Press, 1986.

[4] Milne, J. Abelian Varieties, in Arithmetic Geometry.

[5] Mumford, D. Abelian Varieties, Oxford University Press, 1970.