REPORT ON MOD ℓ REPRESENTATIONS OF Gal(\overline{Q}/Q)

Kenneth A. Ribet

University of California, Berkeley

In memory of Kenneth F. Ireland

1. Introduction.

Let N ≥ 1 and k ≥ 2 be integers. Let Γ_{1}(N) be the group

\{ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL(2, Z) | c ≡ 0, a ≡ d ≡ 1 mod N\}

and let S = S_{k}(Γ_{1}(N)) be the complex vector space of cusp forms of weight k on Γ_{1}(N). There is a standard action d ↦ ⟨d⟩ ∈ Aut S of the group (Z/NZ)^{∗} on S. In addition, S comes equipped with a family of Hecke operators T_{n} (n ≥ 1); the T_{n} commute with each other and with the “diamond bracket” operators ⟨d⟩. These operators are traditionally written on the right; for example, the n^{th} Hecke operator is normally written f ↦ f|T_{n}.

An eigenform in S is a non-zero cusp form f ∈ S which is an eigenvector for each of the operators T_{n} and ⟨d⟩. In particular, if f is an eigenform, then there is a Dirichlet character ϵ defined mod N such that f|⟨d⟩ = ϵ(d)f for d ∈ (Z/NZ)^{∗}. The functional equation satisfied by a weight-k cusp form implies that this “Nebentypus” character has the same parity as k: ϵ(−1) = (−1)^{k}. The eigenvalues a_{n} of the T_{n} acting on f are algebraic integers, and generate together a finite extension of Q.

Let \overline{Q} be an algebraic closure of the field of rational numbers, and consider the Galois group Gal(\overline{Q}/Q). Suppose that f ∈ S is an eigenform, and let E be a number field containing the associated eigenvalues a_{n} and ϵ(d). A construction of Deligne [10] attaches to f a family of continuous representations

ρ_{λ}: Gal(\overline{Q}/Q) → GL(2, E_{λ});

here, λ runs over the set of prime ideals of the integer ring of E, and E_{λ} denotes the completion of E at λ. Each representation ρ_{λ} is irreducible; it is characterized up to isomorphism by the identities

trace(ρ_{λ}(Frob_{p})) = a_{p}, det(ρ_{λ}(Frob_{p})) = ϵ(p)p^{k−1}

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11F11, 11R32; Secondary 11F33, 11G18.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The author was supported in part by NSF Grant #DMS 88-06815. He wishes to thank both the I.H.E.S. and the Université de Paris 7 for gracious hospitality during the period when this article was begun. Thanks are due to Fred Diamond, Bas Edixhoven and Barry Mazur for helpful comments. The referee deserves special thanks for a thorough reading of the manuscript and for many useful comments and suggestions.

for prime numbers p which are prime both to N and to the norm of λ. The symbol Frob_{p} denotes an arithmetic Frobenius element for p in Gal(\overline{Q}/Q). (For a survey of this material, the reader may wish to consult [37].)

The reduction of ρ_{λ} mod λ is a certain semisimple representation

ρ̄_{λ}: Gal(\overline{Q}/Q) → GL(2, F_{λ}),

where F_{λ} is the residue field of λ. Its construction involves a choice: one chooses a model of ρ_{λ} which has values in GL(2, O_{λ}), where O_{λ} is the ring of integers of E_{λ}, and then forms the “naive reduction” of this model, i.e., the composite of ρ_{λ} and the canonical map GL(2, O_{λ}) → GL(2, F_{λ}). The semisimplification of this naive reduction depends only on ρ_{λ}. It is the desired representation ρ̄_{λ}.

Consider the representations ρ̄_{λ} obtained for varying N, k, f, but with F_{λ} having a fixed characteristic. To assemble them, fix a prime number ℓ, and choose a place v dividing ℓ of the field of algebraic numbers in C. Let F be the residue field of v, so that F is an algebraic closure of its prime field F_{ℓ}. For each algebraic number field E contained in C, v induces a prime λ on E, together with a canonical inclusion F_{λ} ↪ F. In particular, if E is the field generated by the eigenvalues attached to an eigenform f, and λ is the prime of E induced by v, the representation ρ̄_{λ} may be viewed as taking values in GL(2, F).

Suppose that ρ: Gal(\overline{Q}/Q) → GL(2, F) is a continuous semisimple representation. When can we expect that ρ is isomorphic to one of the ρ̄_{λ}? Identify ϵ with a character defined on Gal(\overline{Q}/Q), and let χ̃ be the ℓ-adic cyclotomic character. Recall that the identity det(ρ_{λ}(Frob_{p})) = ϵ(p)p^{k−1} on Frobenius elements implies the formula det ρ_{λ} = ϵχ̃^{k−1} (1.1)

for det ρ_{λ}. On reducing this identity, one obtains det ρ̄_{λ} = ϵ̄χ^{k−1}, where χ is the mod ℓ cyclotomic character, and ϵ̄ is the reduction of ϵ mod λ (i.e., mod v). In particular, let c ∈ Gal(\overline{Q}/Q) be a complex conjugation. Then det ρ̄_{λ}(c) = ϵ̄(c)χ(c)^{k−1}. Now χ(c) = −1, and furthermore ϵ(c) is another name for ϵ(−1). Using the formula ϵ(−1) = (−1)^{k}, we obtain det ρ̄_{λ}(c) = −1. In other words, ρ̄ must be an odd representation if it is to be isomorphic to some ρ̄_{λ}.

Serre [49] has proposed that this parity condition represents a sufficient, as well as a necessary, condition for a representation ρ: Gal(\overline{Q}/Q) → GL(2, F) to arise from some eigenform. The reducible odd representations ρ arise from Eisenstein series, and will be neglected in the following discussion. For irreducible representations, we have the

(1.2) Serre Conjecture [49, (3.2.3_{?})]. Let ρ: Gal(\overline{Q}/Q) → GL(2, F) be an irreducible, odd representation. Then ρ is isomorphic to the representation ρ̄_{λ} associated to some eigenform in one of the spaces S_{k}(Γ_{1}(N)).

The associated Refined Conjecture [49, (3.2.4_{?})] asserts that ρ is a representation ρ̄_{λ} associated to an eigenform in a specific space S_{k(ρ)}(Γ_{1}(N(ρ))). The invariants k(ρ) and N(ρ) are defined by local properties of ρ. More precisely, as we recall in §2, N(ρ) is an integer prime to ℓ which depends only on the restrictions of ρ to decomposition groups in Gal(\overline{Q}/Q) for each prime number p ≠ ℓ. Similarly, k(ρ) is an integer ≥ 2 which depends only on the restriction of ρ to a decomposition group for ℓ in Gal(\overline{Q}/Q). (See [49, §2].)

MOD ℓ REPRESENTATIONS OF Gal(\overline{Q}/Q) 3

The formulation of [49, (3.2.4_{?})] specifies that ρ arise from an eigenform f ∈ S_{k(ρ)}(Γ_{1}(N(ρ))) whose character ϵ is also predicted in advance. This extra requirement means simply that ϵ can be chosen to be of order prime to ℓ. Subsequently, however, Serre found examples for ℓ ≤ 3 showing that the requirement cannot always be satisfied [50]; his examples concern the two-dimensional space S_{2}(Γ_{1}(13)). On the other hand, Carayol [7, Prop. 3] and Serre have each shown that the situation for ℓ ≥ 5 is much more favorable: the representations ρ̄_{λ} arising from a given space S_{k}(Γ_{1}(N)) all arise from eigenforms whose associated character has prime-to-ℓ order.

More precisely, Carayol proves the following result [7, Proposition 3]:

(1.3) Theorem. Assume that ℓ ≥ 5 and that ρ: Gal(\overline{Q}/Q) → GL(2, F) is an odd irreducible representation. Suppose that ρ arises from an eigenform f ∈ S_{k}(Γ_{1}(N)) and that ϵ is the Dirichlet character which is associated to f. Let ϵ′ be a character on (Z/NZ)^{∗} which is congruent to ϵ mod v. Then ρ arises from an eigenform f′ ∈ S_{k}(Γ_{1}(N)) whose Nebentypus character is ϵ′.

In view of this result, and the counterexamples for ℓ = 2 and ℓ = 3, we will restrict attention to the invariants N(ρ) and k(ρ) in discussing Serre’s conjectures.

The Refined Conjecture has a number of striking consequences, which are catalogued in §4 of [49]. Among these are Fermat’s Last “Theorem,” the conjecture of Taniyama that all elliptic curves over Q are modular, and variants of Taniyama’s conjecture concerning other motives of rank 2.

To bridge the gap between the Serre Conjecture and its refinement, we formulate the following

(1.4) Motivating Problem. Suppose that ρ: Gal(\overline{Q}/Q) → GL(2, F) is an irreducible representation which arises from some space of cusp forms S_{k}(Γ_{1}(N)). Does ρ arise from an eigenform in S_{k(ρ)}(Γ_{1}(N(ρ))), where k(ρ) and N(ρ) are the invariants assigned to ρ by Serre?

An affirmative solution to (1.4) would imply an equivalence between Serre’s conjecture and its refinement. At the time of this writing, one is very close to a complete solution of (1.4), at least when ℓ is an odd prime. The solution involves the work of a large number of mathematicians, including N. Boston, H. Carayol, F. Diamond, B. Edixhoven, G. Faltings, B. H. Gross, B. Jordan, H. W. Lenstra, Jr., R. Livné, B. Mazur, J-P. Serre, and the author.

The goal of this article is twofold. On the one hand, we shall describe what is known about (1.4). On the other, we shall present the following new result:

(1.5) Theorem. Let ℓ ≥ 3 be a prime, and let ρ: Gal(\overline{Q}/Q) → GL(2, F) be irreducible. Suppose that ρ arises from an eigenform f of weight two and trivial character on Γ_{1}(M) ∩ Γ_{o}(p), where p is prime to ℓM. Assume that ρ is unramified at p. Then ρ arises from a weight-two eigenform with trivial character on Γ_{1}(M).

The new feature of this theorem is that the level M is not required to be prime to ℓ. In fact, M can be divisible by an arbitrarily high power of ℓ. As it stands, (1.5) applies only to forms with trivial character on Γ_{1}(M) ∩ Γ_{o}(p), i.e., to forms on Γ_{o}(Mp). However, it is very likely that the proof we give for (1.5) will generalize without difficulty to cover eigenforms on Γ_{1}(M) ∩ Γ_{o}(p) whose Nebentypus characters are arbitrary characters mod M. The discussion below shows that (1.4) will be solved for primes ℓ ≥ 5 as soon as Theorem 1.5 is generalized to this case.

One can also contemplate a variant of (1.5) which applies to forms of weight k on Γ_{1}(M) ∩ Γ_{o}(p), where k satisfies 2 ≤ k ≤ ℓ+1 and M is required to be prime to ℓ. Such a variant, which accepts arbitrary characters mod M, will again lead to a solution of (1.4). It seems very likely that the methods of Faltings, Jordan and Livné [16], [24] will lead to a variant of this type.

In light of the substantial progress made in relating Serre’s conjecture to its refinement, it is striking that there is little to report concerning the conjecture itself [49, 3.2.3_{?}], aside from the numerical evidence presented in §5 of [49].

2. Stripping powers of ℓ from the level.

Let ρ: Gal(\overline{Q}/Q) → GL(2, F) be an irreducible odd representation, as above, and let N(ρ) and k(ρ) be the invariants assigned to ρ by Serre. Recall [49, §1] that N(ρ) is a product ∏ p^{n(p,ρ)} extended over the set of prime numbers p ≠ ℓ. The integer n(p, ρ) is defined by restricting ρ to a decomposition group D_{p} for p in Gal(\overline{Q}/Q). Consider the ramification subgroup G_{0} of D_{p}, along with the higher ramification groups G_{i} (i > 0). Let V be a two-dimensional F-vector space affording the representation ρ. For each i ≥ 0, let V_{i} be the subspace of V consisting of those v ∈ V which are fixed by all elements of G_{i}. Then

n(p, ρ) = ∑_{i≥0} \frac{1}{(G_{0} : G_{i})} dim V/V_{i}.

In particular, N(ρ) is prime to ℓ, so that Serre’s refined conjecture implies the weaker statement that ρ arises from some space of cusp forms S_{k}(Γ_{1}(N)), with N prime to ℓ. Let us say simply that ρ arises from a subgroup Γ of SL(2, Z) if it arises from the space of weight-k cusp forms on Γ, for some k ≥ 2. Then the weak statement implied by Serre’s conjecture is that ρ arises from the group Γ_{1}(N) for some N which is prime to ℓ. We shall now show (at least when ℓ ≥ 3) that any ρ which is modular in the sense that it arises from some Γ_{1}(N) with N not necessarily prime to ℓ is automatically modular in the apparently stronger sense that it arises from Γ_{1}(N) for some N which is prime to ℓ. This is certainly a “well known” fact, which is close to results which are already in the literature, cf. [49, Remarque, p. 195].

(2.1) Theorem. Assume ℓ ≥ 3. Suppose that ρ arises from Γ_{1}(M), where M is the product Nℓ^{α} with (N, ℓ) = 1. Then ρ arises from Γ_{1}(N).

We prove (2.1) using the concrete techniques of Serre [46, §3], [47, Theorem 5.4] and Queen [35, §3]. An alternative method would be based on results of N. Katz: see the appendices of [25] and the discussions in [20, §1] and [21, §1]. (Thanks are due to Fred Diamond for pointing out these references.) Note that the assumption ℓ ≥ 3 is made principally for convenience; it should be possible to analyze the case ℓ = 2 without great difficulty.

Our proof consists of several independent steps. Before beginning it, we shall normalize the eigenforms which occur in the proof. Recall that if f satisfies f|T_{n} = a_{n}f for all n ≥ 1, then we may scale f by a constant so that its Fourier coefficients are the eigenvalues a_{n}. We will assume that all eigenforms are scaled in this way, i.e., that they are “normalized eigenforms.” We represent these eigenforms as series ∑_{n≥1} a_{n}q^{n}; the variable q initially represents e^{2πiτ}, where τ is an element of the complex upper half plane, but rapidly becomes a formal variable. In particular, we say that two forms are congruent mod ℓ or mod v if their Fourier series are formally congruent. Similarly, the mod v reduction of an eigenform f is meant to be the formal power series ∑ ā_{n}q^{n}, where “¯ ” denotes the reduction map mod v.

Step 1. The representation ρ arises from Γ_{o}(ℓ^{r}) ∩ Γ_{1}(ℓN) for some r ≥ 0.

When ℓ ≥ 5, the assertion follows from (1.3). In that case, we can take r = α. It can be proved in an elementary manner, for all ℓ ≥ 3, by the following argument.

Suppose that ρ arises from an eigenform f in S_{k}(Γ_{1}(M)) and that κ is the associated Dirichlet character mod M. We assume that k ≥ 2 and that α is positive (i.e., that M is divisible by ℓ). Decompose κ as a product ϵηω^{i}, where: ϵ has conductor dividing N, η has ℓ-power order and ℓ-power conductor, and ω is the “Teichmüller” character, i.e., that character of conductor ℓ and order ℓ−1 which is congruent to the identity function mod v. The exponent i is naturally an integer mod ℓ−1. Because η has, in particular, odd order, we may write it in the form ξ^{−2}, where ξ is a character of ℓ-power order. The cusp form f ⊗ ξ is a form of type (k, ϵω^{i}) (i.e., weight k and character ϵω^{i}) which gives rise to ρ. If r ≥ α is large enough so that ℓ^{r} is divisible by the square of the conductor of ξ, then f ⊗ ξ is again of level Nℓ^{r}.

In what follows, we fix an r > 0 so that ρ arises from Γ_{o}(ℓ^{r}) ∩ Γ_{1}(ℓN).

Step 2. ρ arises from Γ_{o}(ℓ^{r}) ∩ Γ_{1}(N).

We assume as above that ρ arises from an eigenform of type (k, ϵω^{i}) on the group Γ_{o}(ℓ^{r})∩Γ_{1}(ℓN). Take the exponent i to be a positive integer. The Eisenstein series

G := L(1−i, ω^{−i})/2 + ∑_{n=1}^{∞} (∑_{d|n} ω^{−i}(d)d^{i−1}) q^{n}

is then of type (i, ω^{−i}) on Γ_{o}(ℓ), as proved (for example) in [46, Lemme 10]. Thus fG is on the group Γ_{o}(ℓ^{r}) ∩ Γ_{1}(N), if r is positive. On the other hand, it is well known that the order at v of L(1−i, ω^{−i}) is negative: this follows from the Kummer congruences and the Von Staudt theorem. (See, e.g., [22, §3.4].) In other words, if c is the constant term of G, then E := c^{−1}G is a form with v-integral coefficients such that E ≡ 1 mod v. Consequently, the product fE, viewed mod v, is a nonzero eigenform with the same eigenvalues as f. At the same time, fE is on the group Γ_{o}(ℓ^{r}) ∩ Γ_{1}(N). By a well known lemma [12, Lemme 6.11], we may find an eigenform on Γ_{o}(ℓ^{r}) ∩ Γ_{1}(N) whose eigenvalues are congruent to those of f.

Step 3. ρ arises from Γ_{o}(ℓ) ∩ Γ_{1}(N).

Assume that ρ arises from an eigenform f = ∑ a_{n}q^{n} on Γ_{o}(ℓ^{r}) ∩ Γ_{1}(N), with r > 1. Let K be a finite Galois extension of Q containing the a_{n}, and let σ ∈ Gal(K/Q) be a Frobenius element for v. Thus σa ≡ a^{ℓ} mod v for all elements a of the ring of integers of K. The series ∑ σ^{−1}a_{n}q^{n} is the Fourier expansion of a normalized eigenform σ^{−1}f of the same weight as f. We wish to show that f is congruent mod v to a cusp form of some weight on Γ_{o}(ℓ^{r−1}) ∩ Γ_{1}(N). Consider g := (σ^{−1}f)^{ℓ}|U, where U is the ℓ^{th} Hecke operator T_{ℓ}. On the one hand, g is a form on Γ_{o}(ℓ^{r−1}) ∩ Γ_{1}(N) (see, e.g., [29, Lemma 1]). On the other hand, the Fourier expansion of g is congruent mod v to ∑(σ^{−1}a_{n})^{ℓ}q^{n}, which in turn is congruent to the Fourier expansion of f.

Step 4. ρ arises from Γ_{1}(N).

To prove this, we use the argument given by Serre in [46, §3.2], replacing the operators W, U, V, … by their analogues for forms on Γ_{1}(N) ∩ Γ_{o}(ℓ). More precisely, we let U be the ℓ^{th} Hecke operator, as above, and let V be the operator ∑ a_{n}q^{n} ↦ ∑ a_{n}q^{ℓn}, which takes forms on Γ_{1}(N) to forms of the same weight on Γ_{1}(N) ∩ Γ_{o}(ℓ). Further, we let W be the operator V_{ℓ}^{Nℓ} as defined in §1 of [29].

Thus W is given by the matrix ( \begin{matrix}ℓx & y \\ Nℓz & ℓ \end{matrix} ), where x, y, and z are integers and we

have ℓx − Nyz = 1. This operator is the inverse of the operator denoted W_{Q} in [1]. Indeed, by [1, Proposition 1.1], one has W = ϵ(ℓ)W_{Q} on the space of forms with character ϵ on Γ_{1}(N)∩Γ_{o}(ℓ). (We consider ϵ as a Dirichlet character mod N, which permits us to evaluate it at ℓ.) At the same time, W^{2} is multiplication by ϵ(ℓ), according to Lemma 2 of [29].

Next, suppose that F is a form of weight w and character ϵ on Γ_{1}(N) ∩ Γ_{o}(ℓ). Then [29, Lemma 3] implies that the form

Tr(F) := F + ϵ^{−1}(ℓ)ℓ^{1−w/2}F |W |U

is a form of weight w on Γ_{1}(N). (Apply [29, Lemma 3] to F |W.) Further, if G is a form of weight w and character ϵ on Γ_{1}(N), one has G|W = ℓ^{w/2}ϵ(ℓ)G|V [1, Prop. 1.5].

If ℓ > 3, let E be the normalized Eisenstein series of weight ℓ−1 on SL(2, Z), so that we have E ≡ 1 mod ℓ. Similarly, if ℓ = 3 let E be the normalized Eisenstein series E_{4} of weight four. Let a denote the weight of E. As in [46], we introduce the form

g := E − ℓ^{a/2}E|W = E − ℓ^{a}E|V.

One has g ≡ 1 mod ℓ; furthermore, g|W is divisible by a positive power of ℓ (in fact, by ℓ^{1+a/2}).

Let f be an eigenform giving rise to ρ, as above, and consider Tr(fg^{i}), where i is a positive integer. The modular form Tr(fg^{i}) is a form on Γ_{1}(N). On the other hand, a calculation similar to that of §3.2 of [46] shows that Tr(fg^{i}) ≡ f mod ℓ for sufficiently large i. ■

A partial converse to (2.1) is the following result:

(2.2) Theorem. Suppose that ρ arises from S_{k}(Γ_{1}(N)), with N prime to ℓ and 2 ≤ k ≤ ℓ+1. Assume that ℓ > 3 or that N > 3. Then ρ arises from S_{2}(Γ_{1}(Nℓ)).

For ℓ ≥ 5, the theorem is proved by Ash and Stevens [2, Theorem 3.5a] by methods involving parabolic cohomology. Another approach, which originated with Serre, is worked out in Gross’s article [17, Proposition 9.3]. Although Gross’s prime p (which corresponds to our ℓ) is arbitrary, he assumes that the level N satisfies N ≥ 4. According to comments in §10 of [17], Proposition 9.3 of [17] extends to the case where N ≤ 2 and k is even.

3. Adjustment of the weight.

Let ρ: Gal(\overline{Q}/Q) → GL(2, F) be an irreducible representation, and let k(ρ) be the “weight” attached to ρ by Serre [49, §2]. Edixhoven [15, §4] proved that if ρ arises from a cusp form of some weight, then ρ arises from a cusp form of weight k(ρ). More precisely, [15, Th. 4.3] gives the following result:

(3.1) Theorem. Let N be prime to ℓ, and assume that ρ arises from an eigenform f ∈ S_{k}(Γ_{1}(N)) with character ϵ. Then ρ arises from an eigenform f′ ∈ S_{k(ρ)}(Γ_{1}(N)) with character ϵ. The integers k and k(ρ) are congruent mod ℓ−1; moreover, we have k ≥ k(ρ) provided that ℓ is odd.

The proof of (3.1) relies on two points which merit independent discussion. First, let D_{ℓ} be a decomposition group for ℓ in Gal(\overline{Q}/Q) and let I_{ℓ} be the inertia subgroup of D_{ℓ}. Suppose that

ρ: Gal(\overline{Q}/Q) → Aut V ≈ GL(2, F)

arises from an eigenform f = ∑ a_{n}q^{n} of weight k and character ϵ on Γ_{1}(N), with N prime to ℓ. The proof of (3.1) requires information about the restriction of ρ to D_{ℓ}, in the case where k satisfies 2 ≤ k ≤ ℓ.

Specifically, suppose that this inequality is satisfied, and that f is ordinary in the sense that a_{ℓ} ≢ 0 mod v. A theorem of Deligne (proved in [17]) states that V has an unramified quotient V/V_{0} on which Frob_{ℓ} acts by multiplication by a_{ℓ} (or, more precisely, by the image of a_{ℓ} in F). The action of D_{ℓ} on V_{0} is determined by this information, since the determinant of ρ may be computed from k and ϵ. On the other hand, suppose that f is supersingular, i.e., not ordinary. Let ψ and ψ′ be the two fundamental characters I_{ℓ} → F of level 2, as defined in [44, §1.7]. A theorem of Fontaine states that the restriction of ρ to I_{ℓ} is the direct sum of two one-dimensional representations, on which I_{ℓ} operates by the k−1^{st} powers of ψ and ψ′. This theorem was proved by Fontaine in an exchange of letters with Serre in 1979; a new proof was given by Edixhoven in [15].

Secondly, Edixhoven’s proof requires B. H. Gross’s result about companion forms [17], which conversely may be viewed as a special case of (3.1). Before stating Gross’s result, we recall that there is an operator θ = q \frac{d}{dq} on mod ℓ modular forms for Γ_{1}(N) when N is prime to ℓ. This operator maps forms of weight k to forms of weight k+ℓ+1. It was considered initially by Serre and Swinnerton-Dyer [45], [53] for forms on SL(2, Z), and then constructed by Katz [26] for forms of full level N, where N ≥ 3. It is introduced in [17,§4] in the case of forms on Γ_{1}(N), with N ≥ 4. According to Edixhoven [15, §2.1], mod ℓ modular forms of weight k on Γ_{1}(N) are GL(2, Z/nZ)-invariant sections of the line bundle ω^{k} on the curve X. Here n ≥ 3 is prime to ℓ, and X is the modular curve parametrizing elliptic curves with a (Γ_{1}(N), Γ(n))-structure. The integer N is assumed only to be prime to ℓ. Using this fact, Edixhoven constructs θ for modular forms of weight k on Γ_{1}(N).

Consider now an ordinary form f as above, whose weight k satisfies 2 ≤ k ≤ ℓ−1. Suppose in addition that the exact sequence of F[D_{ℓ}]-modules

0 → V_{0} → V → V/V_{0} → 0

is split. The inertia group I_{ℓ} acts on V_{0} via the character χ^{k−1}, since Gal(\overline{Q}/Q) acts on det V via the product ϵχ^{k−1} and since I_{ℓ} acts trivially on V/V_{0}. Consider the representation ρ′ := ρ ⊗ χ^{1−k} of Gal(\overline{Q}/Q). Locally at ℓ, it is the direct sum of V_{0} ⊗ χ^{1−k} and (V/V_{0}) ⊗ χ^{1−k}. In particular, it is an extension of the unramified line V_{0} ⊗ χ^{1−k} by a line on which I_{ℓ} acts via the character χ^{1−k} = χ^{ℓ+1−k−1}. This suggests that ρ′ arises from an eigenform of weight k′ := ℓ+1−k and character ϵ on Γ_{1}(N); and indeed Serre’s recipe for weights sets k(ρ′) = k′. It is clear that ρ′ is modular of some weight, because twisting representations by χ corresponds to applying the operator θ on modular forms. Hence (3.1) asserts, in particular, that ρ′ arises from an eigenform f′ of weight k′. The construction of the “companion” f′ to f is the main result of [17]. (If k = ℓ and ρ is split locally at ℓ, then Gross still produces a companion form, of weight 1, in certain cases. For ℓ odd, the remaining cases of weight 1 are treated in a recent article by Coleman and Voloch [9].) This completes the discussion of Theorem 3.1.

As above, we shall use the phrase “ρ arises from Γ_{1}(N)” to indicate that ρ arises from an eigenform in S_{k}(Γ_{1}(N)) for some k ≥ 2.

(3.2) Corollary. Suppose that ρ arises from Γ_{1}(N), with N prime to ℓ. Then there exists a power χ^{i} of the mod ℓ cyclotomic character χ such that ρ ⊗ χ^{i} arises from S_{k}(Γ_{1}(N)) for some k ≤ ℓ+1.

Proof. Replacement of ρ by any of its twists ρ ⊗ χ^{i} (i = 0, 1, 2, …) corresponds to applying the operator θ = q \frac{d}{dq} on modular forms i times. Hence, ρ arises from a given Γ_{1}(N) if and only if ρ_{i} := ρ ⊗ χ^{i} arises from the same group. The definition of k(ρ) is such that a suitable twist ρ_{i} satisfies the inequality

2 ≤ k(ρ_{i}) ≤ ℓ+1.

(See [15, Theorem 3.4], and the discussion following the statement of that theorem.) By Theorem 3.1, ρ_{i} arises from S_{k(ρ_{i})}(Γ_{1}(N)). ■

Let ρ: Gal(\overline{Q}/Q) → GL(2, F) be an irreducible representation. Consider the following four sets of positive integers:

N_{1} := { N prime to ℓ | ρ arises from S_{k(ρ)}(Γ_{1}(N)) };
N_{2} := { N prime to ℓ | ρ arises from Γ_{1}(N) };
N_{3} := { N prime to ℓ | ρ arises from Γ_{1}(Nℓ^{α}) for some α ≥ 0 };
N_{4} := { N prime to ℓ | ρ arises from S_{2}(Γ_{1}(Nℓ^{2})) }.

(3.3) Theorem. If ℓ ≥ 5, then the four sets of integers N_{i}(ρ) are equal.

Proof. The equality of N_{1} and N_{2} is guaranteed by Theorem 3.1. That N_{2} and N_{3} are equal follows from Theorem 2.1. Since it is clear that N_{4} ⊆ N_{3}, it remains only to show that N_{3} ⊆ N_{4}.

We will show, equivalently, that N_{2} ⊆ N_{4}. Assume that ρ arises from Γ_{1}(N), and choose i ≥ 0 such that ρ ⊗ χ^{i} arises from S_{k}(Γ_{1}(N)), with 2 ≤ k ≤ ℓ+1. By (2.2), ρ ⊗ χ^{i} arises from S_{2}(Γ_{1}(Nℓ)). To see that N ∈ N_{4}, we view the twisting operator ⊗χ^{−i} as a a Dirichlet twist on modular forms. Such a twist changes the level of a modular form, but not its weight. From [1, Proposition 3.1], we find that ρ = (ρ⊗χ^{i}) ⊗ χ^{−i} arises from S_{2}(Γ_{1}(Nℓ^{2})). ■

In case ℓ ≥ 5, we will write simply N (ρ) for the common value of the four sets N_{i}(ρ).

4. The levels of ρ and of f.

Suppose that an irreducible representation ρ: Gal(\overline{Q}/Q) → GL(2, F) is “modular” in the sense that it arises from an eigenform of some weight and some level. Problem (1.4) requires that we find an eigenform of weight k(ρ) and level N(ρ) which gives rise to ρ. Because of Edixhoven’s theorem (Theorem 3.1 above), the weight aspect of (1.4) disappears. In fact, suppose that ℓ ≥ 5. Then Theorem 3.3 translates (1.4) into the folllowing question:

N(ρ) ∈^{?} N (ρ). (4.1)

In other words, our goal is to show that ρ is modular of level Nℓ^{α}, for some α ≥ 0. With (4.1) in mind, we are led to examine the set N (ρ) more closely. The following theorem was proved by Carayol [7] and, independently, by Livné [30, Proposition 0.1]:

(4.2) Theorem. Suppose that ρ arises from Γ_{1}(N). Then Serre’s invariant N(ρ) divides N.

In attacking problem (4.1), one begins with an eigenform f giving rise to ρ. If N is the level of f, we have N(ρ)|N. Since N(ρ) is, by definition, prime to ℓ, we have N(ρ)|N′, where N′ is the prime-to-ℓ part of N. In case this divisibility is strict, the aim is to “lower” N by replacing it by a divisor. For the most part, the strategy is to do this “locally”: we consider a prime p ≠ ℓ which divides N/N(ρ) and seek to replace N by N/p^{i} for some positive integer i.

As a preliminary step, we should certainly replace f by the newform (or primitive form) associated to f. This is an eigenform whose system of λ-adic representations coincides with the system (ρ_{λ}) attached to f, and whose level is minimal among all such eigenforms. The level of the newform attached to f is a divisor of N. (For the theory of newforms, see for instance the account in [29], or [34, §4.6].) According to a theorem of Deligne, Langlands [28], and Carayol [6], the conductor of the system (ρ_{λ}) coincides with N once this replacement is made.

Assume, then, that f is a newform. Theorem 4.2 is actually a consequence of a much more refined comparison of N(ρ) with (the prime-to-ℓ part of) N. This comparison is again due to Carayol [7] and Livné [30]. We will discuss the analysis of Carayol and Livné, under the simplifying assumption ℓ ≥ 5. In view of (1.3), this assumption permits us to assume that the character ϵ which is associated to f has prime-to-ℓ order, and thereby facilitates the discussion.

Consider that representation ρ_{λ} which is attached to f and the prime λ induced by the chosen place v of \overline{Q}. Then ρ is the reduction of ρ_{λ} in the sense which is outlined above. Since ρ is irreducible, ρ is in fact the “naive” reduction of any model of ρ_{λ} over O_{λ}: no semisimplification is required.

Fix a prime p ≠ ℓ, and let D_{p} ⊂ Gal(\overline{Q}/Q) again be a decomposition group for p. Let I_{p} be the inertia subgroup of D_{p}, and let e = ord_{p}(N/N(ρ)). Thus, e is a non-negative integer, according to (4.2).

The inequality e ≥ 0 can be interpreted conceptually. Consider ρ_{λ} and ρ as homomorphisms

Gal(\overline{Q}/Q) → Aut(V_{λ}), Gal(\overline{Q}/Q) → Aut V,

where V_{λ} and V are two dimensional vector spaces over E_{λ} and F respectively. Then one has e = dim_{F} V^{I_{p}} − dim_{E_{λ}} V_{λ}^{I_{p}}, (4.3)

where V^{I_{p}} is the space of I_{p}-invariants in V and V_{λ}^{I_{p}} is space of I_{p}-invariants in V_{λ}. In particular, we have e > 0 if and only if

dim_{F} V^{I_{p}} > dim_{E_{λ}} V_{λ}^{I_{p}}.

Carayol and Livné have classified the situations in which we have e > 0. We will now describe their classification. For each situation which arises in the classification, we will summarize what is known about the question of showing that ρ is modular of level (dividing) N/p, i.e., that ρ arises from Γ_{1}(N/p). Our summary appeals to a number of results which are proved only later in this article (for example, Theorems 5.1, 1.5, and 8.1). The material in this § may thus be viewed as a motivation for the theorems in §§5–8.

The first step in the classification is to distinguish cases according to the behavior of the component at p of the automorphic representation attached to f. Recall that the newform f is associated to an automorphic representation π_{A} of GL(2, A), where A is the adèle ring of Q. Let π be the component of π_{A} at p, so that π is an admissible representation of GL(2, Q_{p}). This representation may be classified as a principal series representation, a special representation, or a cuspidal representation of GL(2, Q_{p}).

The Langlands correspondence attaches to π a λ-adic representation ρ_{λ,π} of the Weil group of Q_{p}. (Strictly speaking, ρ_{λ,π} is defined only up to isomorphism. See [54], §4 for a discussion of the relation between λ-adic representations of the Weil group of Q_{p} and complex representations of the corresponding Weil-Deligne group.) By the main result of [6], ρ_{λ,π} is somorphic to the restriction of ρ_{λ} to the Weil group of Q_{p}. Therefore, properties of π are mirrored by the local behavior of ρ_{λ} at p. The restriction of ρ_{λ} to D_{p} is: reducible and semisimple when π is a principal series representation; reducible, but not semisimple when π is special; and irreducible when π is cuspidal.

We now discuss the inequality e > 0 in each of these three cases.

The principal series case.

In this case, which was treated by Carayol, the representation π arises from an (unordered) pair of Grossencharacters of Q_{p}^{∗}, say α and β. Identify these characters with characters of D_{p}, using the reciprocity map of local classfield theory. Then ρ_{λ}|_{D_{p}} is the direct sum α ⊕ β.

(4.4) Proposition [7]. Assume that e > 0. Then there is a Dirichlet character ϕ of conductor p and ℓ-power order such that the newform associated to f ⊗ ϕ has level dividing N/p. In particular, ρ is modular of level N/p.

Proof. One of α and β is ramified, but has unramified reduction mod λ. After possibly permuting α and β, we may assume that the restriction of α to I_{p} is nontrivial, but has ℓ-power order. On the other hand, because of formula (1.1), and the assumption that the order of ϵ is prime to ℓ, the restriction of det ρ_{λ} to I_{p} has primeto-ℓ order. (Note that χ̃ is ramified only at ℓ.) Thus αβ|_{I_{p}} has prime-to-ℓ order. It follows from this that, on I_{p}, α is a power of β, since β is the product α^{−1} · αβ of characters of relatively prime orders. Thus the kernel of α^{−1}|_{I_{p}} contains that of β|_{I_{p}}, and it follows that the conductor of α^{−1}β is at most that of β. (Incidentally, we will write the conductor multiplicatively: the conductor of a character of D_{p} is a power of p, and the conductor of the trivial character of D_{p} is 1 = p^{0}.)

Since α|_{I_{p}} has prime-to-p order, α coincides on I_{p} with some power ω^{i} of the Teichmüller character ω which appeared in the proof of (2.1). Note that ω is a character which is defined naturally on Gal(\overline{Q}/Q). Let ϕ = ω^{−i}, and consider the twist f ⊗ ϕ, which corresponds to the λ-adic representation ρ_{λ} ⊗ ϕ. Since ϕ has ℓ-power order, the reductions of ρ_{λ} ⊗ ϕ and ρ_{λ} coincide. On the other hand, (ρ_{λ} ⊗ ϕ)|_{D_{p}} is a direct sum of the unramified αϕ and the character βϕ. This latter character coincides with βα^{−1} on I_{p}. Therefore, its conductor is a divisor of the conductor of β. At the same time, the conductor of αϕ, which is 1, is a strict divisor of the conductor of α. Therefore, the power of p dividing the conductor of ρ_{λ}ϕ is smaller than the power of p dividing the conductor of ρ_{λ}.

In the language of cusp forms, this means that the level of the newform associated to f ⊗ ϕ is divisible by a smaller power of p than the level N of f. The two levels differ only at p, since the character ϕ is ramified only at p. ■

The special case.

Here, we have π = α ⊗ sp, where sp is a “standard” special representation, and α is a Grossencharacter of Q_{p}^{∗}, which we identify as above with a character of D_{p}. We consider that sp has been normalized so that ρ_{λ}|_{D_{p}} is an extension of the onedimensional representation with character α by the one-dimensional representation with character αχ̃. (The fact that ρ_{λ}|_{D_{p}} has this form, when π is special, was proved by Langlands [28].) The semisimplification of ρ|_{D_{p}} then has the form ᾱ⊕ᾱχ, where ᾱ is the reduction of α mod λ. Also, ϵ = α^{2}χ̃^{2−k}. Finally, the conductor of α ⊗ sp is the square of the conductor of α if α is ramified, and p^{1} if not.

(4.5) Proposition [7]. Assume that e > 0 and that α is ramified. Then there is a Dirichlet character ϕ of conductor p and ℓ-power order such that the newform associated to f ⊗ ϕ has level dividing N/p. In particular, ρ is modular of level N/p.

Proof. Since e > 0, we have dim_{F} V^{I_{p}} > 0, so that ᾱ is unramified. Thus α|_{I_{p}} has ℓ-power order, and coincides with some power ω^{i} of ω. It follows that the conductor of α is p. As above, we set ϕ = ω^{−i}. The representation π ⊗ϕ then has conductor p, whereas π itself has conductor p^{2}. Hence replacing f by the newform associated to f ⊗ ϕ replaces N by N/p. ■

In the situation of Proposition 4.5, ϵ̄ is unramified at p, since the determinant of ρ is unramified at p. Since we have assumed that the order of ϵ is prime to ℓ, this implies that ϵ is unramified at p. Hence f is in fact a newform on the group Γ_{1}(M) ∩ Γ_{o}(p^{2}), where M = N/p^{2} is prime to p, and the character ϕ is quadratic. In the case where π is special but Proposition 4.5 does not apply, α is an unramified character, so that ϵ, in particular, is unramified. The newform f then has level Mp, with M prime to p, and f is a cusp form on the group Γ_{1}(M) ∩ Γ_{o}(p). We have e > 0 if and only if the representation ρ is unramified. When this is the case, N(ρ) is prime to p, and one must show that ρ is represented by an eigenform of level M. Here is a statement of the problem to be solved (cf. [7, Conjecture B]):

(4.6) Problem. Suppose that ℓ ≥ 5 and that p is a prime distinct from ℓ. Assume that ρ arises from an eigenform f of weight k on Γ_{1}(M) ∩ Γ_{o}(p), where M is prime to p and that ρ is unramified at p. Show that ρ arises from an eigenform on Γ_{1}(M).

The representation π is automatically special in all non-trivial cases of (4.6). Indeed, let f be an eigenform on Γ_{1}(M) ∩ Γ_{o}(p). If the newform associated to f has level dividing M, then in particular ρ arises from an eigenform on Γ_{1}(M). If, to the contrary, the level of the newform associated to f is divisible by p, then π is special.

We now describe the extent to which (4.6) has been treated. A first remark, which is a consequence of the discussion concerning the four sets N_{i}(ρ), is that it suffices to solve (4.6) in either of the following situations: a. k = 2 and M is of the form Nℓ^{α} with N prime to ℓ; b. M is prime to ℓ and 2 ≤ k ≤ ℓ+1. One can assume α ≤ 2 in case (a), but this does not seem to be helpful.

(4.7) Mazur’s Principle. Suppose that ℓ ≥ 5 and that p ≠ ℓ is a prime such that p ≢ 1 mod ℓ. Assume that ρ arises from an eigenform f of weight k on Γ_{1}(M) ∩ Γ_{o}(p), with M prime to p, and that ρ is unramified at p. Then ρ arises from an eigenform on Γ_{1}(M).

This theorem was proved by Mazur in a letter to J-F. Mestre [31] in the special case where k = 2 and f has trivial character, i.e., is a cusp form on Γ_{o}(Mp). Mazur’s proof was presented by the author in [41, §6]. In proving the theorem more generally, we may apply the analysis above to work either in case (a) or in case (b). We choose to work in situation (a), i.e., in the case of forms of weight k = 2. This is essentially the case we have already treated in [41, §6]; the main difference is that we now allow the Nebentypus character of f to be non-trivial. (This character is naturally defined mod M, because f is a form on Γ_{1}(M) ∩ Γ_{o}(p).) For the convenience of the reader, we establish Mazur’s principle, in case (a), in §8 below. (In §8, we require only that ℓ be odd: we do not exclude the case ℓ = 3. Also, we use slightly different notation: the prime q of §8 plays the role of p in 4.7.)

After Mazur formulated his principle in 1985, problem 4.6 was studied extensively when k = 2 and f is a cusp form on Γ_{o}(Mp). Here, one wishes to show that ρ arises from a weight-two eigenform on Γ_{o}(M). In [41], the author proved this result when M is prime to ℓ. Next, in a joint article [32], Mazur and the author established a “multiplicity one” theorem which implies, by the techniques of [41], that Theorem 1.5 holds when ℓ, but not ℓ^{2}, divides M. Then, in a recent note [43], the author proved Theorem 1.5 when M is “exactly divisible” by a prime q, different from p and ℓ, at which ρ is ramified. The argument given in [43] relies heavily on a recent theorem of N. Boston, H.W. Lenstra, Jr., and the author [4], but avoids appeal to the “multiplicity one” principle which appears in [41]. Theorem 1.5, which will be proved in §7, may be viewed as the most recent link in this chain.

Theorem 1.5 falls short of solving (4.6) because of the requirement that the character of f be trivial. However, as already mentioned in §1, the author expects that the proof of (1.5) will extend without difficulty to the case where the character associated to f is an arbitrary character on (Z/MZ)^{∗}.

Problem 4.6 has been studied from perspective (b) by Jordan and Livné [24], who expect to treat this case (at least) in the case where f has trivial character and the weight k satisfies the inequality 2 ≤ k < ℓ. Their arguments are an adaptation to weight k of arguments given in [41]. In particular, the arguments of Jordan and Livné rely on a weight-k “multiplicity one” principle, which will be treated in a forthcoming article of Faltings and Jordan [16]. It is quite possible that the arguments of §7 can be adapted to their situation, thereby obviating the necessity of using a weight-k multiplicity-one principle.

The cuspidal case.

In this case, the restriction of ρ_{λ} to D_{p} is irreducible. Assume that e > 0. Then, according to Carayol [7, §§1.1–1.2], ρ_{λ}|_{D_{p}} is the two-dimensional representation induced from an abelian character ξ: H → \overline{Q}^{∗}, where H is the subgroup of index two in D_{p} which corresponds to the unramified quadratic extension of Q_{p}. Moreover, the reduction of ξ mod v is unramified. The level N is divisible exactly by p^{2}; i.e., it is divisible by p^{2}, but not by p^{3}. The Serre invariant N(ρ) is divisible at most by p. One wishes to show that ρ arises from an eigenform on Γ_{1}(N/p) of some weight, i.e., that ρ arises from Γ_{1}(N/p).

For this, we can (and will) assume that the weight k of f satisfies 2 ≤ k ≤ ℓ+1. By Theorem 5.1 below, there are infinitely many prime numbers q, prime to N

and congruent to −1 mod ℓ, such that: ρ arises from a weight-k eigenform on the group Γ_{1}(N)∩Γ_{o}(q) for which the associated newform has level divisible by q. (One takes q to be an “auxiliary prime” as defined in §5.) Fix such a prime, together with an eigenform F satisfying the given condition. Let π_{A} be the representation of GL(2, A) associated with F. In the language of automorphic representations, the condition on F ensures that the component at q of π_{A} is a special representation.

Consider the twisted form of GL(2) which arises from the quaternion algebra over Q which is ramified exactly at p and at q. Since the components at both p and q of π_{A} are discrete series representations, we can conclude that π_{A} arises by the Jacquet-Langlands correspondence from an automorphic representation on this twisted form. Using this correspondence, and a lemma which generalizes (1.3), Carayol proves that ρ arises from an eigenform on Γ_{1}(N/p) ∩ Γ_{o}(q). Note that, since ℓ is odd, the congruence q ≡ −1 mod ℓ implies that one has q ≢ +1 mod ℓ. Using this fact, together with Mazur’s Principle (4.7), we conclude, as desired, that ρ is modular of level N/p.

5. Diamond’s theorem.

This § concerns the result of F. Diamond [13] which was alluded to above. We establish an analogue (Theorem 5.1) of a result proved by the author [38], [42] in the case of weight-two forms on Γ_{o}(N). This analogue is a simple variant of Theorem 1 of [13]. More precisely, we shall focus attention on some intermediate results obtained by Diamond during the course of the proof of [13, Theorem 1] and then derive Theorem 5.1 from these.

Results like Theorem 5.1 have been useful both in “level lowering” (as in§4 above) and in other contexts. For example, [5] applies the theorem of [42] to the study of deformations of mod ℓ representations arising from weight-two modular forms.

Let ρ: Gal(\overline{Q}/Q) → GL(2, F) be an irreducible mod ℓ representation arising from S_{k}(Γ_{1}(N)), with N prime to ℓ. Consider the following two conditions on a prime number q, assumed to be prime to Nℓ: I. The representation ρ arises from a weight-k eigenform on Γ_{1}(N)∩Γ_{o}(q) for which

the associated newform has level divisible by q. II. The characteristic polynomial of ρ(Frob_{q}) is of the form (T − a)(T − qa), with

a ∈ F^{∗}.

(5.1) Theorem (Diamond). Assume that k satisfies 2 ≤ k ≤ ℓ+1. Then conditions I and II are equivalent.

We first make some comments concerning the two conditions. Let S be the space S_{k}(Γ_{1}(N) ∩ Γ_{o}(q)), and let S^{q−new} be the “q-new” subspace of S, defined, for example, as the kernel of the natural trace map from S to the direct sum of two copies of S_{k}(Γ_{1}(N)). Condition I means simply that ρ arises from an eigenform in the space S^{q−new}.

Next, we should point out that Condition II is satisfied in the case where q is an auxiliary prime. Let σ be the three-dimensional representation ρ × χ, where the mod ℓ cyclotomic character χ is regarded as a one-dimensional representation of Gal(\overline{Q}/Q) over F. An auxiliary prime is a prime number q, prime to ℓN, such that σ(Frob_{q}) is conjugate to the matrices σ(c), with c ∈ Gal(\overline{Q}/Q) a complex conjugation. If q is such a prime, the characteristic polynomial of ρ(Frob_{q}) coincides with that of the ρ(c). Since ρ is an odd representation, ρ(c) is conjugate

to ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), and thus has characteristic polynomial (T − 1)(T + 1). Because

χ(Frob_{q}) = χ(c) = −1, we have q ≡ −1 (mod ℓ). Thus the characteristic polynomial of ρ(Frob_{q}) may be written (T − a)(T − qa), where a can be taken to be either +1 or −1, so that condition II of §5 is satisfied.

By the Čebotarev Density Theorem, the set of prime numbers for which c and Frob_{q} map to conjugate elements in the image of ρ × χ has positive density. In particular, there are an infinite number of auxiliary primes.

Also, we should point out that the implication “Condition I ⇒ Condition II” is a direct consequence of Langlands’s theorem ([28], see also [6]). Indeed, suppose that I is satisfied, and let f be a newform in S^{q−new} which gives rise to ρ. Let π be the admissible representation of GL(2, Q_{q}) which is associated to f. The fact that f is associated with Γ_{1}(D) ∩ Γ_{o}(q), for some D|N, implies that π is a special representation α ⊗ sp, with α unramified. As discussed above, a theorem of Langlands states that ρ_{λ}|_{D_{q}} is an extension of the one-dimensional representation with character α by the one-dimensional representation with character αχ̃. The semisimplification of ρ|_{D_{q}} has the form ᾱ ⊕ ᾱχ, where ᾱ is the reduction of α

mod λ. Applying ᾱ ⊕ ᾱχ to Frob_{q} ∈ D_{q}, we obtain the matrix ( \begin{matrix}a & 0 \\ 0 & qa \end{matrix} ), with

a = α(Frob_{q}). The characteristic polynomial of this matrix is (T − a)(T − qa). Hence condition I implies condition II.

We now turn to the implication “Condition II ⇒ Condition I,” which is the essential content of Theorem 5.1. We first compare eigenforms in S_{k}(Γ_{1}(N)) and in S. Suppose that ρ arises from a weight-k eigenform f = ∑ c_{n}e^{2πinτ} on Γ_{1}(N), and let ϵ be the character of (Z/NZ)^{∗} associated with f. (For obvious reasons, we must refrain from writing q = e^{2πiτ}.) The forms f and f′ := ∑ c_{n}(e^{2πinτ})^{q} may be considered as elements of S; indeed, this construction defines the standard inclusion of S_{k}(Γ_{1}(N)) ⊕ S_{k}(Γ_{1}(N)) into S as a space of oldforms. Both forms f and f′ in S are eigenvectors for the diamond operators ⟨d⟩ with d ∈ (Z/NZ)^{∗} and the Hecke operators T_{n} with (n, q) = 1; the eigenvalues are the same as those for f when considered as an element of S_{k}(Γ_{1}(N)). (Some authors would write ⟨d⟩_{N} for the operator ⟨d⟩, to emphasize that only the group (Z/NZ)^{∗}, rather than (Z/NqZ)^{∗}, is operating. The group (Z/NqZ)^{∗} operates through its quotient (Z/NZ)^{∗} because we are considering cusp forms on Γ_{o}(q) ∩ Γ_{1}(N).) However, neither of the forms f, f′ is (in general) an eigenvector for the q^{th} Hecke operator U := T_{q} on S.

In order to obtain a true eigenform in S, we consider the two roots α and β of the polynomial T^{2} − c_{q}T + q^{k−1}ϵ(q). A computation shows that the two forms f − αf′ and f − βf′ are eigenvectors for U on S, with eigenvalues β and α, respectively.

Consider the Hecke algebra T = T_{S} associated with S, i.e., the subring of End S generated by the T_{n} with n ≥ 1. We have ⟨d⟩ ∈ T for all d ∈ (Z/NZ)^{∗}. Indeed, if r is a prime number which is prime to qN, then T_{r}^{2} − T_{r^{2}} = ⟨r⟩r^{k−1}, so that r^{k−1}⟨r⟩ ∈ T. By taking two distinct r’s which map to d mod N, we find ⟨r⟩ ∈ T.

The action of T on g := f − βf′ is given by a homomorphism φ: T → C, which is characterized by the formulas φ(⟨d⟩) = ϵ(d), φ(U) = α, and φ(T_{n}) = c_{n}, the latter valid for n prime to q. We may view φ as a homomorphism T → O, where O is the integer ring of a suitable number field. Our fixed place v|ℓ of \overline{Q} induces a prime λ|ℓ of O, together with an inclusion O/λ ↪ F. On composing φ with the map O → F, we obtain a ring homomorphism

φ̄: T → F.

This map depends on f, and also on (α, β) as an ordered pair.

Assume now that condition II is satisfied and fix a ∈ F^{∗} as in the condition. Then T^{2} − c_{q}T + q^{k−1}ϵ(q) ≡ (T − a)(T − qa) mod λ. After permuting the roots α and β if necessary, we have α ≡ a mod λ. Setting

η = U^{2} − q^{k−2}⟨q⟩,

we find from condition II that φ̄(η) = 0. Indeed, one has φ̄(qη) = qa^{2} − ϵ̄(q)q^{k−1}, but ϵ(q)q^{k−1} is congruent to qa^{2} in view of condition II. In other words, the kernel 𝔪 of φ̄ is a maximal ideal of T which contains η.

The subspace S^{q−new} of S is stable under all Hecke operators T_{n} and ⟨d⟩. Thus S^{q−new} cuts out a quotient T_{q−new} of T: the image of T in End S^{q−new}. One verifies condition I by proving that 𝔪 arises by pullback from a maximal ideal \overline{𝔪} of T_{q−new}, i.e., that 𝔪 lies in the support of the T-module S^{q−new}. It is well known that this property of 𝔪 implies that ρ arises from an eigenform in S; see [17, Proposition 9.3] and [12, Lemma 6.11], and the proofs of these results. Therefore, to prove that condition II implies condition I, it suffices to verify

(5.2) Proposition. Suppose that ℓ is prime to N and that k satisfies 2 ≤ k ≤ ℓ+1. Let 𝔪 be a maximal ideal of T associated with S = S_{k}(Γ_{1}(N) ∩ Γ_{o}(q)), where q is a prime number not dividing Nℓ. Assume that 𝔪 divides ℓ (i.e., 𝔪 contains ℓ) and that 𝔪 contains η = U^{2} − q^{k−2}⟨q⟩. Then 𝔪 arises by pullback from the q-new quotient of T.

This proposition is implicit in Diamond’s work, although it is not explicitly stated in [13].

To prove the proposition, we recall some results from [13], using somewhat different notation. Let L be the parabolic cohomology group with Z_{ℓ}-coefficients which is associated with S, so that L is a free Z_{ℓ}-module of rank 2 dim_{C} S. To construct L, we let Γ be the image of Γ_{1}(N) ∩ Γ_{o}(q) in PSL(2, Z), and consider the parabolic cohomology group H_{par}^{1}(Γ, Sym^{k−2}(Z_{ℓ} ⊕ Z_{ℓ})). This group is a finitely-generated Z_{ℓ}-module M which is not necessarily torsion free; L is defined as the image of M in M ⊗ Q_{ℓ}, i.e., the largest torsion free quotient of M. We let X = X_{1} ⊕ X_{2} be the direct sum of two copies of the corresponding parabolic cohomology group made with Γ_{1}(N) in place of Γ_{1}(N) ∩ Γ_{o}(q). Diamond considers the degeneracy map X ↪ L which corresponds to the inclusion of S_{k}(Γ_{1}(N)) ⊕ S_{k}(Γ_{1}(N)) in S on the level of modular forms. We will denote this map by α. (There is no further need to refer to the roots α and β which occurred in the discussion above.) The map α becomes T-equivariant when one endows X with its natural action of T. (In this action, the T_{n} with n prime to q act “diagonally” on X, while U acts as

the two-by-two matrix ( \begin{matrix}T_{q} & q^{k−1} \\ −⟨q⟩ & 0 \end{matrix} ).) In particular, the image of α is stable

under T.

Using results of Ihara, and the hypotheses on ℓ, Diamond proves that the cokernel Y of α is torsion free. The tautological exact sequence

0 → X → L → Y → 0 (5.3)

is thus an exact sequence of free Z_{ℓ}-modules. The quotient of T ⊗ Z_{ℓ} cut out by Y is the ℓ-adic completion of the q-new quotient of T, while the quotient of T ⊗ Z_{ℓ} cut out by X is the completion of the analogous q-old quotient of T; this is the same quotient which is defined by the action of T on S_{k}(Γ_{1}(N)) ⊕ S_{k}(Γ_{1}(N)).

In particular, let 𝔪 be a maximal ideal of T with residue characteristic ℓ. Because T acts faithfully on L, 𝔪 lies in the support of L. In view of (5.3), it follows that 𝔪 lies either in Supp X or in Supp Y, and perhaps in both. The statement to be proved may be rephrased as follows: if 𝔪 ∈ Supp X and if η ∈ 𝔪, then 𝔪 ∈ Supp Y. Note that the two hypotheses on 𝔪 may be combined into the statement that 𝔪 belongs to the support of X/ηX. Indeed, if 𝔪 ∈ Supp X, then X/𝔪X is non-zero, by Nakayama’s lemma. If furthermore we have η ∈ 𝔪, then X/𝔪X is a quotient of X/ηX. Hence 𝔪 belongs to the support of X/ηX, since 𝔪 belongs to the support of X/𝔪X. Conversely, if 𝔪 lies in the support of X/ηX, it is clear that η is an element of 𝔪 and that 𝔪 ∈ Supp X. Thus one must prove

(5.4). If 𝔪 ∈ Supp(X/ηX), then 𝔪 ∈ Supp Y.

Diamond proves (5.4) by constructing a T-equivariant surjection Y → X/ηX. To construct this map, we first dualize (5.3), thereby obtaining an exact sequence

0 → Hom(Y, Z_{ℓ}) → Hom(L, Z_{ℓ}) → Hom(X, Z_{ℓ}) → 0.

As Diamond recalls, perfect pairings constructed by Hida [19, Theorem 3.2] identify Hom(L, Z_{ℓ}) with L and Hom(X, Z_{ℓ}) with X. Using these pairings, we obtain an exact sequence

0 → Hom(Y, Z_{ℓ}) → L →^{β} X → 0. (5.5)

The map β is “almost” T-equivariant: it intertwines the maps labeled T_{n} on L and on X whenever (n, q) = 1, but intertwines the q^{th} Hecke operator U of L with the

endomorphism ( \begin{matrix}0 & q \\ −⟨q⟩q^{k−2} & T_{q} \end{matrix} ) of X (cf. [19, Prop. 3.3]).

Consider the automorphism ω = ( \begin{matrix}−q^{k−2}⟨q⟩ & T_{q} \\ 0 & −⟨q⟩ \end{matrix} ) of X. A computation

shows that the composite ω^{∘}β: L → X is T-equivariant. Moreover, (ω^{∘}β)^{∘}α is the endomorphism η of X. It follows that ω^{∘}β induces a surjection L/αX → X/ηX. Since Y is, by definition, L/αX, we obtain the desired map.

6. Character groups and component groups.

In this §, we recall some material from [41] concerning the component groups and character groups associated with bad reductions of ordinary modular curves and Shimura curves. In addition, we explore in further detail a relation that was first found by Jordan and Livné [23] and then deepened by [41, Theorem 4.3].

For each N ≥ 1, we let X_{o}(N) be the classical modular curve (over Q) which classifies elliptic curves which are endowed with cyclic subgroups of order N. The space of holomorphic differentials on X_{o}(N)_{/C} may be identified with S_{2}(Γ_{o}(N)) in a canonical fashion. The curve X_{o}(N) comes equipped with a family of correspondences T_{n} (n ≥ 1); the correspondence T_{n} induces on S_{2}(Γ_{o}(N)) the endomorphism T_{n}. The Jacobian Pic^{o} X_{o}(N) will be denoted J_{o}(N), as usual; we write again simply T_{n} for the endomorphism of J_{o}(N) which is induced by the correspondence T_{n} of X_{o}(N).

Consider first a prime number q and a positive integer N prime to q. We are interested in the modular curves X_{o}(qN) and X_{o}(N), and in their Jacobians J_{o}(qN) and J_{o}(N). The two standard degeneracy coverings X_{o}(qN) \overrightarrow{→} X_{o}(N) induce, by pullback, a map δ: J_{o}(N) × J_{o}(N) → J_{o}(qN). The kernel of δ is finite; it is the “antidiagonal” image in J_{o}(N) × J_{o}(N) of the kernel of the pullback of the natural covering X_{1}(N) → X_{o}(N) [38]. (The curve X_{1}(N) classifies elliptic curves which are furnished with a point of order N.)

For each n ≥ 1, a Hecke operator labeled T_{n} acts diagonally on the product J_{o}(N) × J_{o}(N), and a similarly-named operator acts on the target J_{o}(qN) of δ. When n is prime to q, δ is compatible with the two operators T_{n}. However, when n = q, the situation is more subtle. Reserve the symbol T_{q} for the q^{th} Hecke operator on J_{o}(N) and let U denote the q^{th} Hecke operator on J_{o}(qN). Then U^{∘}δ =

δ^{∘}U, where the latter operator U denotes the matrix ( \begin{matrix}T_{q} & q \\ −1 & 0 \end{matrix} ) of endomorphisms

of J_{o}(N). (We view this matrix as an endomorphism of J_{o}(N)×J_{o}(N).) Let T = T_{qN} be the subring of End J_{o}(qN) generated by the T_{n} for n ≥ 1, i.e., by the T_{n} for n prime to q and the operator U = T_{q}. We endow J_{o}(N) × J_{o}(N) with the operation of T such that the elements U and T_{n} of T (with n prime to q) act as the operators of J_{o}(N) × J_{o}(N) with the same names. This is the unique operation for which δ is T-equivariant.

We note in passing that the endomorphisms T_{n} and U and the homomorphism δ are defined over Q. Also, in analogy with the situation of §5, we let η = U^{2} − 1 in T.

(6.1) Theorem. There is a unique homomorphism of abelian varieties

σ: J_{o}(qN) → J_{o}(N) × J_{o}(N)

such that σ^{∘}δ = η. This homomorphism is T-equivariant.

To construct σ, we consider the map δ′: J_{o}(qN) → J_{o}(N)×J_{o}(N) which is induced by the degeneracy coverings X_{o}(qN) \overrightarrow{→} X_{o}(N) using Albanese (i.e., covariant) functoriality of the Jacobian. The definition of T_{q} as a correspondence on X_{o}(N)

shows that δ′^{∘}δ = ( \begin{matrix}q + 1 & T_{q} \\ T_{q} & q + 1 \end{matrix} ). Define σ = ( \begin{matrix}−1 & T_{q} \\ 0 & −1 \end{matrix} )^{∘}δ′. A computation

shows that σ^{∘}δ coincides with η as a matrix of endomorphisms of J_{o}(N). For the unicity and the T-equivariance, we observe that

Hom(Q, J_{o}(N)×J_{o}(N)) = 0,

where Q = J_{o}(qN)/δ(J_{o}(N)×J_{o}(N)). This fact may be seen arithmetically, by noting that Q has purely toric reduction at the prime q [11], whereas J_{o}(N) has good reduction at q. It proves the unicity, since the difference between two operators σ which satisfy σ^{∘}δ = η is an operator which vanishes on the image of δ. Similarly, let T ∈ T, and consider σ^{∘}T −T^{∘}σ, which is a priori a homomorphism h: J_{o}(qN) → J_{o}(N)×J_{o}(N). We have h^{∘}δ = ηT − T η = 0, so that h factors through Q; as a consequence, h = 0. ■

Now specialize to the case where N = pM, where p is a prime number which does not divide M. The level qN becomes the product pqM in which the two prime numbers p and q play symmetrical roles. The space S of weight-two cusp forms on Γ_{o}(pqM) is then analogous to the space denoted S in §5, in the special case N = pM, k = 2. The main difference is that we have replaced Γ_{1}(N) ∩ Γ_{o}(q) by Γ_{o}(qN).

Let J_{o}(pqM)_{/F_{p}} be the fiber at p of the Néron model of J_{o}(pqM). This group variety is an extension of a finite “component group” Θ_{p} by a connected group J_{o}(pqM)_{/F_{p}}^{o}, which is in turn an extension of the product of two copies of J_{o}(qM) by a torus T over F_{p}. Let L_{p} be the character group of this torus, and let L_{q} be the character group of the analogous torus for J_{o}(pqM)_{/F_{q}}. Similarly, let X_{p} be the analogue of L_{p} for (J_{o}(pM)×J_{o}(pM))_{/F_{p}} and let X_{q} be the analogue of X_{p} with p replaced by q. The groups X_{p} and X_{q} are naturally direct sums of two copies of the character groups coming from J_{o}(pM)_{/F_{p}} and J_{o}(qM)_{/F_{q}}, respectively; thus, we may represent endomorphisms of X_{p}, say, as two-by-two matrices of endomorphisms of the character group associated to J_{o}(pM)_{/F_{p}}. The endomorphisms T_{n} (and U) of J_{o}(pqM), J_{o}(pM)×J_{o}(pM) and J_{o}(qM)×J_{o}(qM) induce maps on the four character groups L_{p}, X_{p}, L_{q}, and L_{p}. In what the author hopes is an acceptable abuse of notation, these maps will be denoted simply T_{n} and U. In particular,

the endomorphism U of X_{p} is the matrix of endomorphisms ( \begin{matrix}T_{q} & −1 \\ q & 0 \end{matrix} ).

Continuing the abuse of notation, we will write simply δ for the map L_{p} → X_{p} induced by the degeneracy map δ: J_{o}(Mp)×J_{o}(Mp) → J_{o}(Mpq). The surjectivity of δ: L_{p} → X_{p} was established in [41, Th. 3.15]. (In [41], the author wrote “X” for the character group associated to a single copy of J_{o}(pM) or J_{o}(qM).) Theorem 4.1 of [41] identifies the kernel of δ with an analogue of L_{q} in which X_{o}(pqM) is replaced by an appropriate Shimura curve. Namely, let C be the Shimura curve made with a quaternion algebra of discriminant pq and Γ_{o}(M)-type level structure (cf. [41, §4]). Let J be the Jacobian of C, and let Y_{p} and Y_{q} be the analogues of L_{p} and L_{q} for J. The groups J_{/F_{p}} and J_{/F_{q}} are extensions of their component groups Ψ_{p} and Ψ_{q} by the tori whose character groups are Y_{p} and Y_{q}. (As was proved by Cerednik [8] and Drinfeld [14], all components of C_{/F_{p}} and C_{/F_{q}} have genus zero.) In other words, the reductions of J at p and q are “semiabelian,” with trivial abelian variety parts. Then theorem 4.1 of [41] provides an exact sequence

0 → Y_{q} →^{ι} L_{p} →^{δ} X_{p} → 0 (6.2)

and an analogue 0 → Y_{p} → L_{q} → X_{q} → 0 (6.3)

in which the roles of p and q have been permuted. These sequences are compatible with the Hecke operators labeled T_{n} on J and on J_{o}(pqM); note, incidentally, that the operators T_{p} and T_{q} on J are in fact involutions. (See [41, Theorem 4.1] and also the results proved in [40].) The sequences (6.2) and (6.3) seem to play a role analogous to that played by (5.3) and its Z_{ℓ}-dual in the discussion of §5.

One deduces from (6.2) or (6.3) that the Hecke ring Z[…, T_{n}, …] ⊆ End J associated to J is naturally a quotient of the ring T associated to J_{o}(pqM). Thus, we may speak of the action of T on J.

Let us introduce the notation ′ to denote a dual abelian variety; thus J_{o}(pqM)′, for example, will be the abelian variety dual to J_{o}(pqM). We use T′_{n} and U′ to denote the endomorphisms of the dual objects which are induced functorially by the operators T_{n} and U. Thus, for instance, there is a natural action of the ring T on J′, in which the element of T labeled T_{n} acts on J′ as the endomorphism T′_{n}. We let Y′_{q}, L′_{p}, and so on, represent the character groups of the tori arising from the dual abelian varieties. The operator T_{n} (say) in T then induces an endomorphism of L′_{q} which will be denoted T′_{n}; this endomorphism is induced by the endomorphism T′_{n} of J_{o}(pqM)′. Similar remarks apply to the component groups which we have introduced: Θ_{p} and Θ_{q} in the case of J_{o}(pqM), and Ψ_{p} and Ψ_{q} in the case of J.

Next, we discuss monodromy pairings. To fix ideas we will first consider L_{p}. The pairing associated with L_{p} is a bilinear map ⟨ , ⟩ : L_{p} × L′_{p} → Z, which induces an injection L′_{p} ↪ Hom(L_{p}, Z) with finite cokernel. According to [18, §11], the cokernel of this injection is canonically isomorphic to Θ_{p}. The exact sequence

0 → L′_{p} → Hom(L_{p}, Z) → Θ_{p} → 0

is T-equivariant when the elements of T act in the natural way, i.e., when T_{n} ∈ T acts as T_{n} on Θ_{p}, as T′_{n} on L′_{p}, and as Hom(T_{n}, Z) on Hom(L_{p}, Z). In a similar way, we have monodromy pairings

Y_{q} × Y′_{q} → Z, X_{p} × X′_{p} → Z.

One potential source of confusion is that the abelian varieties under discussion are Jacobians of modular curves, or else products of two Jacobians. It follows that they are naturally self-dual. After using the autoduality of the Jacobian to identify J′ and J, for example, we find that the lattices Y_{q} and Y′_{q} are equal, or at least canonically isomorphic. In point of fact, it is often fruitful to identify both Y_{q} and Y′_{q} with H_{1}(Γ, Z), where Γ is the “dual graph” attached to the mod q reduction of C. The group H_{1}(Γ, Z) thereby acquires two actions of End J; these differ by the Rosati involution of End J which arises from the theta divisor on C. Similarly, the group X_{p} has two natural actions of T, which differ by the Rosati involution

( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}a′ & c′ \\ b′ & d′ \end{matrix} ) on End J_{o}(pM)×J_{o}(pM), where ′ is the Rosati involution

on End J_{o}(pM). For example, the operator U, which acts on X_{p} as ( \begin{matrix}T_{q} & −1 \\ q & 0 \end{matrix} ) in

its standard action, becomes ( \begin{matrix}T′_{q} & q \\ −1 & 0 \end{matrix} ) in the other action. (It might be worth

pointing out that the endomorphisms T_{q} and T′_{q} of J_{o}(N) are equal.) As a mnemonic device, we will attempt whenever possible to use the ′ notation when the Hecke operators in T act as T′_{n} and U′.

These examples show that the natural autoduality J_{o}(N) ≈ J_{o}(N)′ does not necessarily intertwine the actions of T on these two abelian varieties. Nonetheless, it is a fact that are natural T-equivariant isomorphisms J ≈ J′ and J_{o}(N) ≈ J_{o}(N)′ for each N. In the case of J_{o}(N), the isomorphism is provided by the involution w = w_{N} on J_{o}(N) which is induced by the Atkin-Lehner involution (E, C) ↦ (E/C, E[N]/C) on the set of elliptic curves with cyclic subgroups of order N (see [52, Proposition 3.54]). In the case of J, there is an analogous involution w on the set of “fake elliptic curves” which are classified by C. These T-equivariant isomorphisms are occasionally useful.

Once we identify L_{p} with L′_{p}, the monodromy pairings associated to L_{p} and Y_{q} become maps L_{p} × L_{p} → Z and Y_{q} × Y_{q} → Z, respectively. Both pairings are symmetric; they are given by explicit formulas involving the dual graphs of the mod p reduction of X_{o}(pqM) and the mod q reduction of C. According to [41, Th. 4.1], the restriction to Y_{q} × Y_{q} of the monodromy pairing L_{p} × L_{p} → Z agrees with the monodromy pairing on Y_{q}. Here, one uses ι to embed Y_{q} in L_{p}.

(6.4) Proposition. The embedding ι: Y_{q} ↪ L_{p} remains T-equivariant when it is regarded as a map Y′_{q} → L′_{p}.

We regard L′_{p} and Y′_{q} as the same physical groups as L_{p} and Y_{q}, only with different actions of T. Fix T ∈ T, and use the symbols T_{Y} and T_{L} to denote the endomorphisms of Y_{q} and of L_{p} induced by T. Write T′_{Y} and T′_{L} for the endomorphisms of these groups which T induces in its “dual” actions. The endomorphism T′_{L} (say) is the adjoint of T_{L} with respect to the monodromy pairing ⟨ , ⟩_{L} on L_{p}. Proposition (6.4) states the formula T′_{L}^{∘}ι = ι^{∘}T′_{Y}.

Consider the homomorphism κ: J_{o}(Mp)′×J_{o}(Mp)′ → J_{o}(Mpq)′ which is dual to the map σ of (6.1). It induces a T-equivariant map κ: L′_{p} → X′_{p}; this map is adjoint to the map X_{p} → L_{p} induced by σ. The construction of σ shows that κ differs from δ by an automorphism of X_{p}. Therefore, the kernel of κ coincides with the kernel of δ, which is the image of ι. It follows that T′_{L} preserves this image, since we may write κ^{∘}T′_{L}^{∘}ι = T′_{X}^{∘}κ^{∘}ι = 0. Hence the map ι^{∘}T′_{Y} − T′_{L}^{∘}ι maps Y_{q} to ι(Y_{q}). Consequently, to show that ι^{∘}T′_{Y} = T′_{L}^{∘}ι, it suffices to verify the equality of ⟨T′_{Y} y, z⟩_{Y} and ⟨T′_{L}^{∘}ιy, ιz⟩_{L} for all y, z ∈ Y_{q}. Here, we make use of the fact that the monodromy pairing ⟨ , ⟩_{Y} on Y_{q} is the restriction to Y_{q} of the pairing ⟨ , ⟩_{L} on L_{p}.

To verify the desired equation, we note the series of equalities

⟨T′_{L}ιy, ιz⟩_{L} = ⟨ιy, T_{L}ιz⟩_{L} = ⟨ιy, ιT_{Y} z⟩_{L} = ⟨y, T_{Y} z⟩_{Y} = ⟨T′_{Y} y, z⟩_{Y} .

These follow from the adjuction relations and the equivariance of ι with respect to the standard actions of T, plus the compatibility between the two monodromy pairings. ■

(6.5) Corollary. We have an exact sequence of T-modules

0 → Y′_{q} →^{ι} L′_{p} →^{κ} X′_{p} → 0. ■

Theorem 4.3 of [41] asserts that there is an exact sequence of T-modules

0 → K → X_{p}/ηX_{p} → Ψ_{q} → C → 0.

Here, K and C are each closely related to Θ_{p} and to the component group of the group variety (J_{o}(pM)×J_{o}(pM))_{/F_{p}}; in particular, they are Eisenstein T-modules in the sense that they are annihilated by T_{r} −(r+1) for almost all prime numbers r, cf. [41, Th. 3.12], or [39]. (This allows us to neglect C and K in practice.) The author now believes that he was not sufficiently attentive to the distinction between the two actions of T and that the exact sequence of [41, Th. 4.3] should read

0 → K′ → X_{p}/ηX_{p} → Ψ′_{q} → C′ → 0, (6.6) where C′ and K′ are variants of K and C, which are again Eisenstein. Note, however, that Ψ_{q} and Ψ′_{q} are isomorphic as T-modules, in view of the Atkin-Lehner automorphism w: J ≈ J′ which intertwines the two actions of T. Hence [41, Th. 4.3] can be used as stated.

To derive a version of (6.6), we consider the monodromy pairings on L_{p} and X_{p} as injections

X_{p} ↪ Hom(X′_{p}, Z) and L_{p} ↪ Hom(L′_{p}, Z),

respectively. Their cokernels are the component groups Φ′_{p} and Θ′_{p} associated with the mod p reductions of J_{o}(pM)×J_{o}(pM) and J_{o}(pqM). Using ι, we regard Y_{q} as a submodule of L_{p}. We consider the T-equivariant maps σ: X_{p} → L_{p} and κ: L′_{p} → X′_{p} induced by the homomorphism σ: J_{o}(pqM) → J_{o}(pM)×J_{o}(pM), and use the abbreviation κ^{∗} to refer to the map Hom(κ, Z) which is the Z-linear dual of κ. We have a commutative square

L_{p} → Hom(L′_{p}, Z)
↑ σ  ↑ κ^{∗}
X_{p} → Hom(X′_{p}, Z)

in which the horizontal maps are the injections coming from the two monodromy pairings. On dividing L_{p} by its submodule Y_{q}, we obtain a commutative diagram

0 → L_{p}/Y_{q} → Hom(L′_{p}, Z)/Y_{q} → Θ′_{p} → 0
↑ σ ↑ κ^{∗} ↑
0 → X_{p} → Hom(X′_{p}, Z) → Φ′_{p} → 0

in which the rows are exact. The map δ: L_{p} → X_{p} identifies L_{p}/Y_{q} with X_{p}. Also, we have on X_{p} the identity δ^{∘}σ = η, in view of the equation σ^{∘}δ = η involving homomorphisms of abelian varieties. Hence the diagram may be rewritten

0 → X_{p} → Hom(L′_{p}, Z)/Y_{q} → Θ′_{p} → 0
↑ η ↑ κ^{∗} ↑
0 → X_{p} → Hom(X′_{p}, Z) → Φ′_{p} → 0.

In view of (6.5), the cokernel of κ^{∗} is Hom(Y′_{q}, Z). Therefore, the cokernel of the middle vertical map is the finite group Hom(Y′_{q}, Z)/Y_{q} = Ψ′_{q}. By a dimension count, this shows, in particular, that the middle vertical map is injective. From the snake lemma, we deduce a version of (6.6) in which the groups K′ and C′ are the kernel and cokernel of the right-hand vertical map.

(6.7) Proposition. The T-modules Ψ_{q} and Ψ′_{q} are mutually Q/Z-dual.

Proof. The indicated duality between the two component groups may be read directly from the recipe for making Ψ_{q} and Ψ′_{q} from the monodromy pairing on Y_{q}. For more details, see [18,§11.4], which treats the more general case of an abelian variety with semistable reduction, and also [18,§11.3], which discusses the primeto-p parts of component groups associated to abelian varieties over a local field of residue characteristic p which do not necessarily have semistable reduction. ■

As O. Gabber and R. Livné explained to the author, Grothendieck conjectured in [18, 1.2.1] that there is a perfect Q/Z-duality between the component groups associated to two dual abelian varieties over a local field. In [18], Grothendieck did not treat the p-primary parts of component groups associated to arbitrary abelian varieties over local fields of residue characteristic p. This gap was filled, in the case of a perfect residue field, by L. Bégueri as Th. 8.3.3 of [3]. Another proof, for an abelian variety over the fraction field of a discrete valuation ring with finite residue field, is given in [33].

In the statement of the following corollary, we introduce Mazur’s notation “[𝔪]” for a kernel: if 𝔪 is an ideal of T, and M is a T-module, then M[𝔪] denotes the submodule of M consisting of those element of M which are annihilated by all elements of 𝔪.

(6.8) Corollary. Let 𝔪 ⊂ T be a maximal ideal. Then 𝔪 lies in the support of Ψ_{q} if and only if 𝔪 lies in the support of Ψ′_{q}. Moreover, the T/𝔪-dimensions of Ψ_{q}/𝔪Ψ_{q} and Ψ′[𝔪] are equal; similarly, the dimensions of Ψ′_{q}/𝔪Ψ′_{q} and Ψ_{q}[𝔪] coincide.

Proof. The two equalities of dimension result directly from (6.7). Suppose now that 𝔪 lies in the support of Ψ_{q}. By Nakayama’s Lemma, Ψ_{q}/𝔪Ψ_{q} is non-zero. According to (6.7), however, Ψ_{q}/𝔪Ψ_{q} and Ψ′_{q}[𝔪] are duals of each other. Thus Ψ′_{q}[𝔪] is non-zero. This non-vanishing implies immediately that 𝔪 lies in the support of Ψ′_{q}. Similarly, the support of Ψ′_{q} is contained in the support of Ψ_{q}. ■

(6.9) Theorem. Let 𝔪 be a maximal ideal of T for which the associated representation of Gal(\overline{Q}/Q) is irreducible. Then 𝔪 lies in the support of Ψ_{q} if and only if 𝔪 contains η and lies in the support of X_{p}. Moreover, in this case we have the equalities of T/𝔪-dimensions

dim X_{p}/𝔪X_{p} = dim Ψ′_{q}/mΨ′_{q} = dim Ψ_{q}[𝔪].

Proof. Assume that the representation of Gal(\overline{Q}/Q) associated with 𝔪 is irreducible. By [41, 5.2c], this hypothesis implies that 𝔪 does not lie in the support of either of the modules K′ or C′ which appear in (6.6). Accordingly, X_{p}/ηX_{p} and Ψ′_{q} are isomorphic locally at 𝔪. Thus 𝔪 lies in the support of Ψ′_{q} if and only if it lies in the support of X_{p}/ηX_{p}. By (6.8), the supports of Ψ_{q} and Ψ′_{q} coincide. Also, the support of X_{p}/ηX_{p} clearly consists of those 𝔪 in the support of X_{p} which contain η. Finally, whenever 𝔪 is prime to the flanking modules in (6.6) and η ∈ 𝔪, we have an isomorphism between X_{p}/𝔪X_{p} and Ψ′_{q}/mΨ′_{q}. This observation, together with (6.8), gives the assertion concerning dimensions. ■

7. Proof of Theorem 1.5.

Let ρ, ℓ, M, and p be as in the statement of Theorem 1.5. In other words, we assume that ℓ ≥ 3 is a prime, that ρ: Gal(\overline{Q}/Q) → GL(2, F) is an irreducible representation arising from an eigenform f of weight two on Γ_{o}(Mp), and that p is prime to ℓM. Suppose that ρ is unramified at p. We shall prove that ρ arises from a weight-two eigenform on Γ_{o}(M).

The first step in the proof is the introduction of an “auxiliary prime” into the level (cf. the “cuspidal case” discussion in §4 above and [41, §7]). Choose a prime number q, prime to pℓM, such that ρ(Frob_{q}) is conjugate to each matrix ρ(c), with c ∈ Gal(\overline{Q}/Q) a complex conjugation. The characteristic polynomial of ρ(Frob_{q}) then coincides with that of the ρ(c); since ρ is an odd repre-

sentation, ρ(c) is conjugate to ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ), and thus has characteristic polynomial (T − 1)(T + 1). Since the determinant of ρ is the mod ℓ cyclotomic character χ, we have q ≡ −1 (mod ℓ). (Because ℓ is odd, this congruence gives q ≢ +1 mod ℓ.) Thus (T −1)(T +1) = (T −a)(T −qa), where a can be taken to be either +1 or −1, so that condition II of §5 is satisfied.

An antecedent of Theorem 5.1 implies that ρ arises from a weight-two newform on Γ_{o}(pqM) whose level is divisible by q. (See [38], [42], and [41, §7].) Curiously, we will not use this fact in what follows, although it will appear implicitly. Instead, we consider the Hecke algebra T = T_{pqM} which appeared in §6 and mimic the construction of the maximal ideal 𝔪 of §5. Namely, let U again be the q^{th} Hecke operator in T, and let η = U^{2}−1. The construction of §5 enables one to find a maximal ideal 𝔪 of T such that 𝔪 contains η, together with an embedding φ: T/𝔪 ↪ F, such that T_{r} is mapped to trace(ρ(Frob_{r})) for almost all prime numbers r. Fix such a maximal ideal in what follows. (For the purposes of the proof of Theorem 1.5, T could be defined, more sparsely, as the subring of End S_{2}(Γ_{o}(pqM)) which is generated by the T_{n} with n prime to pqM, along with the operator U = T_{q}. Since no “multiplicity-one” theorems appear in the following argument, it is possible, and perhaps desirable, to work with this more economical Hecke algebra.)

Our next step is to replace ρ by a model ρ_{𝔪}: Gal(\overline{Q}/Q) → GL(2, T/𝔪) of ρ over the finite field T/𝔪. Note that ρ is an irreducible finite-dimensional representation of Gal(\overline{Q}/Q) over F and that the characteristic polynomials of ρ(g) lie in T/𝔪 for each g ∈ Gal(\overline{Q}/Q). It is well known that this necessary condition for descending ρ to T/𝔪 is also sufficient.

To descend ρ directly, we write K for T/𝔪, and observe that ρ descends, in any case, to some finite extension L of K. (Only a finite number of matrices are required to describe ρ.) Let W be an L[Gal(\overline{Q}/Q)]-module which affords a model of ρ over L, and let V be the vector space W, viewed as a module over K[Gal(\overline{Q}/Q)]. (Thus V is the “restriction of scalars” of W from L to K.) Let X be a minimal (non-zero) K[Gal(\overline{Q}/Q)]-submodule of W. The commutant E := End_{K[Gal(\overline{Q}/Q)]} X of X is a division algebra by Schur’s lemma; since E has finite cardinality, it is a finite commutative field. It follows that the L[Gal(\overline{Q}/Q)]-module X⊗_{K} L has commutant E ⊗_{K} L. Because this commutant is commutative, X ⊗_{K} L must be multiplicity free.

On the other hand, X ⊗_{K} L is a submodule of V ⊗_{K} L. This latter module is a direct sum ⊕_{γ}^{γ}W, where γ runs through Gal(L/K) and where ^{γ}W represents the twist of W by γ, i.e., the tensor product W ⊗_{L} L where L is viewed as a base extension of itself via γ: L → L. The hypothesis on characteristic polynomials ensures that the various ^{γ}W are isomorphic to W (Brauer-Nesbitt theorem). Hence V ⊗_{K} L is isomorphic to a sum of copies of W, and it follows in particular that X ⊗_{K} L is a sum of copies of W. Since X ⊗_{K} L is multiplicity free, there can be only one summand. This means that X ⊗_{K} L is isomorphic to W, so that X affords a model of ρ over K.

Now that ρ_{𝔪} has been constructed, we will abuse notation by writing simply ρ for this representation.

We will prove Theorem 1.5 by an indirect reasoning: we will assume that ρ is not modular of level qM, and obtain a contradiction from this assumption. This reasoning will show that ρ is modular of level qM, but then Mazur’s Principle (4.7) will enable us to deduce the desired conclusion that ρ is modular of level M.

We continue to work with the series of abelian varieties which appeared in §6:

J_{o}(pqM), J_{o}(pM)^{2} = J_{o}(pM)×J_{o}(pM), J_{o}(qM)^{2} = J_{o}(qM)×J_{o}(qM), and the Jacobian J = Pic^{o} C of the Shimura curve C. We attach “multiplicities” λ and μ to J_{o}(pqM) and J in the following manner. Consider the kernel J[𝔪] of 𝔪 on J(\overline{Q}). The main theorem of [4] implies that this T/𝔪[Gal(\overline{Q}/Q)]-module is isomorphic to a direct sum V ⊕ ⋯ ⊕ V, where V is a two-dimensional T/𝔪-vector space with a Gal(\overline{Q}/Q)-action equivalent to ρ. We define μ to be the number of factors, and refer to it as the multiplicity of ρ in J. The multiplicity λ is defined analogously, by considering J_{o}(pqM)[𝔪] in place of J[𝔪].

Assume from now on that ρ is not modular of level qM. Using this assumption, we shall prove that both multiplicities λ and μ are zero. This will give the desired contradiction, since λ is positive. Indeed, T, by definition, operates faithfully on J_{o}(pqM), and it is well known that this implies the inequality λ > 0.

To prove that λ and μ are both zero, we will establish a series of relations among the dimensions of the T/𝔪-vector spaces obtained as the mod 𝔪 reductions of the character groups which appear in (6.2) and (6.3). The hypothesis that ρ is not modular of level qM implies that 𝔪 generates the unit ideal in the “p-old” quotient of T. This implies, for example, that 𝔪 does not lie in the support of the T-module X_{q}, since T operates on X_{q} through its p-old quotient.

(7.1) Proposition. There are exact sequences of T-modules

Y_{p}/𝔪Y_{p} ≈ L_{q}/𝔪L_{q}; (7.2)

(7.3)  ⋯ →Y_{q}/𝔪Y_{q} → L_{p}/𝔪L_{p} → X_{p}/𝔪X_{p} → 0.

Proof. The isomorphism (7.2) and the exact sequence (7.3) are obtained from (6.2) and (6.3) by first localizing at 𝔪 and then reducing mod 𝔪. The localization at 𝔪 of X_{q} vanishes, as remarked above. ■

Proposition 7.1 implies the following relations among T/𝔪-dimensions:

(7.4)  dim Y_{p}/𝔪Y_{p} = dim L_{q}/𝔪L_{q};
(7.5)  dim L_{p}/𝔪L_{p} ≤ dim Y_{q}/𝔪Y_{q} + dim X_{p}/𝔪X_{p}.

(7.6) Proposition. We have

dim Y_{p}/𝔪Y_{p} = 2μ, dim L_{p}/𝔪L_{p} = 2λ.

As in [43], these equalities are obtained by considering actions of a decomposition group D_{p} for p in Gal(\overline{Q}/Q). By the main theorem of [4], the kernels of 𝔪 on J_{o}(pqM), and J are direct sums of copies of ρ, first as representations of Gal(\overline{Q}/Q), and then (by restriction) as representations of D_{p}. In particular, they are unramified at p.

By a well known theorem of Serre-Tate [51], the kernel of 𝔪 in the mod p reduction of J may be identified with the group of I_{p}-invariants in J[𝔪], where I_{p} is the inertia subgroup of D_{p}. Since J[𝔪] is unramified at p, the group of I_{p}-invariants is all of J[𝔪], and therefore J_{/F_{p}}(\overline{F}_{p})[𝔪] is of dimension 2μ.

Now, as we recalled above, J_{/F_{p}} is an extension of its “group of components” Ψ_{p} by its toric part. Also, 𝔪 does not lie in the support of X_{q}. By (6.9), or more precisely the variant of (6.9) gotten by interchanging p and q, the T-module Ψ_{p} is prime to 𝔪. In view of this fact, we may deduce that the toric part of J_{\overline{F}_{p}}(\overline{F}_{p})[𝔪] has dimension 2μ. This gives dim Y_{p}/𝔪Y_{p} = 2μ, which was the first of the two formulas to be proved.

The formula involving L_{p} is proved by an analogous argument in which we consider the mod p reduction of J_{o}(pqM). Let A be the fiber at p of the Néron model of J_{o}(pqM), and let B be the analogue of A for J_{o}(qM) × J_{o}(qM). Thus B is an abelian variety over F_{p}. As in the discussion involving J, we know that A(\overline{F}_{p})[𝔪] is of dimension 2λ. In contrast to the situation for J, however, the group of components of A is “Eisenstein” ([41, Th. 3.12], or [39]), and therefore prime to 𝔪 [41, Th. 5.2c]. Thus, if A^{o} is the connected component of 0 in A, A^{o}(\overline{F}_{p})[𝔪] is of dimension 2λ.

By results of Deligne-Rapoport [11] and Raynaud, there is an exact sequence

0 → T → A^{o} → B → 0,

where T is the “toric part” of A^{o}, i.e., that torus whose character group is L_{p} (cf. [41, p. 446]). Since 𝔪 is not modular of level qM, we have B(\overline{F}_{p})[𝔪] = 0. (Compare [41, Theorem 3.11], and the discussion in §8 below.) Therefore, T (\overline{F}_{p})[𝔪] has T/𝔪dimension 2λ. Consequently, L_{p}/𝔪L_{p} is of dimension 2λ. ■

(7.7) Proposition. We have

dim Y_{q}/𝔪Y_{q} ≤ μ, dim L_{q}/𝔪L_{q} ≤ λ.

This proposition is proved, analogously, by considering the action of a decomposition group D_{q} in Gal(\overline{Q}/Q) for the prime q. Pick such a group, together with a Frobenius element Frob_{q} in it. It will be useful to view D_{q} as Gal(\overline{Q}_{q}/Q_{q}).

By the choice of q, ρ is unramified at q, and ρ(Frob_{q}) has the distinct eigenvalues +1 and −1. Therefore, on J[𝔪], say, Frob_{q} has the eigenvalues +1 and −1, each with multiplicity μ, in view of the fact that J[𝔪] is a direct sum of μ copies of ρ.

By [18, §5.1] (or by the results of [51]), the T[D_{q}]-module J[𝔪] contains the submodule Hom(Y_{q}/𝔪Y_{q}, μ_{ℓ}). (Here, μ_{ℓ} denotes the group of ℓ^{th} roots of unity of \overline{Q}_{q}; the author expects that there will be no confusion with the multiplicity μ.)

We claim that the Frobenius element Frob_{q} acts on Y_{q} as the qth Hecke operator U = T_{q}. This claim implies that Frob_{q} acts on Hom(Y_{q}/𝔪Y_{q}, μ_{ℓ}) as the product qT_{q}, and therefore in particular as an element of the field T/𝔪. This scalar will be ±1, since U is an involution and since q is congruent to −1 (mod 𝔪). It follows from this that the dimension of Y_{q}/𝔪Y_{q} is bounded from above by the multiplicity μ, since each eigenvalue ±1 occurs at most μ times on J[𝔪]. In other words, we obtain the desired inequality dim Y_{q}/𝔪Y_{q} ≤ μ.

To prove the claim, we view Y_{q} as the first integral homology group of the dual graph associated with the reduced Shimura curve C_{/F_{q}}. This graph may be described explicitly in terms of certain abelian varieties in characteristic q with “quaternionic multiplication” by an order O in a rational quaternion algebra [40, §5]. The abelian varieties A which occur in the description of the graph are all “exceptional” in the language of [40, §4]. This means that the kernel of the Frobenius morphism A → A^{(q)} coincides with the kernel of multiplication by 𝔮 on A, where 𝔮 is the unique two-sided maximal ideal of O whose residue field has q^{2} elements. Since operator T_{q} has the modular description A ↦ A/A[𝔮], we may deduce that the actions of T_{q} and of the Frobenius automorphism coincide on the graph.

The inequality dim L_{q}/𝔪L_{q} ≤ λ is obtained similarly. ■

(7.8) Proposition. We have dim X_{p}/𝔪X_{p} ≤ μ.

Let J′ again denote the abelian variety dual to J and let Y′_{q} again be the analogue of Y_{q} for J′. Because J has multiplicative reduction at q, one has an exact sequence of T[Gal(\overline{Q}_{q}/Q_{q})]-modules

0 → Hom(Y_{q}/ℓY_{q}, μ_{ℓ}) → J[ℓ] → Y′_{q}/ℓY′_{q} → 0. (7.9)

Indeed, Grothendieck has constructed a “canonical filtration” [18,11.6.5] on the Gal(\overline{Q}_{q}/Q_{q})-module A[ℓ], whenever A is an abelian variety with semistable reduction over Q_{q}. This filtration has the form

0 ⊂ A[ℓ]^{t} ⊂ A[ℓ]^{f} ⊂ A[ℓ],

where A[ℓ]^{t}, the “toric part” of A[ℓ], may be written Hom(Y/ℓY, μ_{ℓ}); here, Y is the character group of the toric part of the special fiber of the Néron model of A. Analogously (and dually), the quotient A[ℓ]/A[ℓ]^{f} of Galois modules may be identified with Y′/ℓY′, where Y′ is the analogue of Y for the abelian variety dual to A ([18,11.6.6], [18,11.6.7]). In case A has purely multiplicative reduction, Y and Y′ are free of rank dim A, and the middle terms A[ℓ]^{t} and A[ℓ]^{f} of the filtration coincide. One therefore obtains an exact sequence like (7.9) whenever the reduction of A is multiplicative. (For a discussion of abelian varieties with split multiplicative reduction, the reader may consult the expository account in Chapter III of [36]. The exact sequence (7.9) appears as Lemma 3.3.1 in [36].)

Localization of (7.9) at 𝔪 yields an exact sequence of T[Gal(\overline{Q}_{q}/Q_{q})]-modules

0 → Hom(Y_{q}/ℓY_{q}, μ_{ℓ})_{𝔪} → J[ℓ]_{𝔪} → (Y′_{q}/ℓY′_{q})_{𝔪} → 0. (7.10)

This sequence splits as a sequence of T-modules. Indeed, consider again a Frobenius element Frob_{q} of Gal(\overline{Q}_{q}/Q_{q}). This operator commutes with the action of T and operates on each of Hom(Y_{q}/ℓY_{q}, μ_{ℓ})_{𝔪} and (Y′_{q}/ℓY′_{q})_{𝔪} as an element of T. Indeed, Frob_{q} operates on Y_{q} as the involution U = T_{q}; therefore it operates as qU on Hom(Y_{q}/ℓY_{q}, μ_{ℓ})_{𝔪} and as U on (Y′_{q}/ℓY′_{q})_{𝔪}. These elements of T are incongruent mod 𝔪, since q is −1 (mod ℓ). This forces the desired splitting.

Since (7.10) splits, it remains exact after we take “kernels of 𝔪.” In other words, we have a sequence of (T/𝔪)[Gal(\overline{Q}_{q}/Q_{q})]-modules

0 → Hom(Y_{q}/ℓY_{q}, μ_{ℓ})[𝔪] → J[𝔪] → (Y′_{q}/ℓY′_{q})[𝔪] → 0.

Since Frob_{q} operates on Hom(Y_{q}/ℓY_{q}, μ_{ℓ})[𝔪] and (Y′_{q}/ℓY′_{q})[𝔪] as homotheties, the dimensions of each of these modules must be μ. To see this, we observe that the eigenvalues of Frob_{q} on the two-dimensional representation V are +1 and −1. Therefore, the characteristic polynomial arising from the action of Frob_{q} on J[𝔪] is (T − 1)^{μ}(T + 1)^{μ}. Suppose that Hom(Y_{q}/ℓY_{q}, μ_{ℓ})[𝔪] has dimension d_{1} and that Frob_{q} operates on Hom(Y_{q}/ℓY_{q}, μ_{ℓ})[𝔪] as the scalar ξ_{1}. Define d_{2} and ξ_{2} similarly, using (Y′_{q}/ℓY′_{q})[𝔪] in place of Hom(Y_{q}/ℓY_{q}, μ_{ℓ})[𝔪]. Then we clearly have

(T − 1)^{μ}(T + 1)^{μ} = (T − ξ_{1})^{d_{1}}(T − ξ_{2})^{d_{2}}.

Since −1 and +1 are distinct, we have d_{1} = d_{2} = μ.

Let Ψ_{q} again be the group of components associated with the mod q reduction of J. As was recalled in §6, Ψ_{q} may be constructed from the monodromy pairing on Y_{q}. Indeed, if we view this pairing as an inclusion Y′_{q} ↪ Y_{q}^{∗}, where Y_{q}^{∗} = Hom(Y_{q}, Z), then Ψ_{q} is naturally the cokernel of this inclusion. By considering the “multiplication by ℓ” maps in the exact sequence 0 → Y′_{q} → Y_{q}^{∗} → Ψ_{q} → 0, and using the Snake Lemma, we obtain an inclusion Ψ_{q}[ℓ] ⊆ Y′_{q}/ℓY′_{q}. In particular, we have Ψ_{q}[𝔪] ⊆ (Y′_{q}/ℓY′_{q})[𝔪]. Thus, μ is bounded from below by the dimension of Ψ_{q}[𝔪]. By (6.9), this dimension coincides with that of X_{p}/𝔪X_{p}, since η lies in 𝔪. Thus we have the desired inequality dim X_{p}/𝔪X_{p} ≤ μ. ■

Using (7.4), Proposition 7.6, and Proposition 7.7, we obtain

2μ ≤ λ,

which implies μ ≤ λ, since μ is a natural number. From (7.5) and Propositions 7.6, 7.7, and 7.8, we get

2λ ≤ 2μ.

Clearly, these inequalities imply λ = μ = 0. As indicated above, this contradiction completes the proof of Theorem 1.5.

8. Mazur’s Principle

In this §, ℓ is an odd prime, and F is again a fixed algebraic closure of F_{ℓ}. Consider an odd irreducible representation ρ: Gal(\overline{Q}/Q) → GL(2, F), and a prime number q different from ℓ. (The case q = ℓ may be included if one introduces the concept of “finiteness” of a representation; see [49, p. 189] and [41, Lemma 6.2].)

The following result is a variant of (4.7):

(8.1) Theorem. Suppose that ρ arises from the space of weight-two cusp forms on Γ_{1}(N) ∩ Γ_{o}(q), where N is prime to q. Assume that ρ is unramified at q and that q ≢ 1 mod ℓ. Then ρ arises from the space of weight-two cusp forms on Γ_{1}(N).

The space S of weight-two cusp forms on Γ_{1}(N) ∩ Γ_{o}(q) contains (as its “q-old” subspace) a direct sum S_{o} of two copies of S_{2}(Γ_{1}(N)). This subspace is stable under the action of the Hecke operators T_{n} (for n ≥ 1) on S. Let T be the subring of End S generated by the T_{n}, and let T_{o} be the q-old quotient of T, i.e., the image of T in End S_{o}. These algebras contain the diamond bracket operators ⟨d⟩ for d ∈ (Z/NZ)^{∗}. The assumption that ρ arises from S implies that there is a homomorphism

φ: T → F

satisfying

φ(T_{r}) = trace(ρ(Frob_{r})), φ(⟨r⟩r) = det(ρ(Frob_{r}))

for all prime numbers r prime to ℓqN. We view φ as an embedding k ↪ F, where k := T/𝔪, and observe that there exists a model for ρ over the field k. This model is a two-dimensional k-vector space V furnished with an action of Gal(\overline{Q}/Q). The F-vector space V ⊗_{k} F affords the representation ρ. As in §5, to prove that ρ arises from S_{o}, it suffices to show that 𝔪 arises by pullback from the quotient T_{o} of T. We will sketch a proof that this is so; a more detailed proof, but with Γ_{1}(N) replaced by Γ_{o}(N), is given in [41, §6].

In the discussion below, we view V as a T-module in the obvious way: T acts through its quotient k.

Consider the modular curve C attached to the group Γ_{1}(N) ∩ Γ_{o}(q). The operators T_{n} and ⟨d⟩ act on C as correspondences. The correspondences T_{n} and ⟨d⟩ of C induce a faithful action of T on the Jacobian J := Pic^{o} C. The endomorphisms of J induced by the T_{n} and ⟨d⟩ will be denoted, as usual, by the same symbols T_{n} and ⟨d⟩. The kernel J[𝔪] of 𝔪 on J(\overline{Q}), which is non-zero, is a direct sum of copies

of V (cf. [4]). In particular, there exist T[Gal(\overline{Q}/Q)]-equivariant maps V ↪^{ι̃} J(\overline{Q}). Because V is unramified at q, and in view of the result of Serre-Tate used above, each choice ι̃ determines a map of T[Gal(\overline{Q}_{q}/Q_{q})]-modules

V ↪^{ι} \underline{J}(\overline{F}_{q}).

Here, \underline{J} represents the fiber over F_{q} of the Néron model of J. Fix one ι̃, and use the resulting map ι to view V as a subgroup of \underline{J}(\overline{F}_{q}).

By results of Deligne-Rapoport [11] and Raynaud, \underline{J} is semiabelian. In other words, \underline{J} is an extension of its component group Φ by a connected group \underline{J}^{o}, which in turn is an extension of an abelian variety A by a torus T. More precisely, A is a product of two copies of J_{1}(N)_{/F_{q}}, while the character group X of T may be interpreted as the group of degree-zero divisors on X_{1}(N)_{/F_{q}} which are concentrated at the set of supersingular points of X_{1}(N)_{/\overline{F}_{q}}. The ring T acts naturally on Φ, T and A by functoriality. To show that 𝔪 arises by pullback from T_{o}, it suffices to verify the following three points:

— The ring T acts on A through its quotient T_{o}.

— The T-module Φ is prime to 𝔪 in the sense that 𝔪 does not lie in the support of Φ.

— The submodule V of \underline{J}(\overline{Q}_{q}) cannot be contained in the torus T. For the first point, the standard degeneracy maps α, β: C \overrightarrow{→} X_{1}(N) induce (in characteristic zero) a map on Picard varieties δ: J_{1}(N)×J_{1}(N) → J which is known to be injective (cf. [13] or [38]). It is clear that T acts on the abelian subvariety J_{1}(N)×J_{1}(N) of J through its old quotient T_{o}. Indeed, if we dualize δ and then pass to cotangent spaces, we obtain the canonical inclusion S_{o} ↪ S. The map δ induces in characteristic q a map δ_{/F_{q}}: J_{1}(N)_{/F_{q}}×J_{1}(N)_{/F_{q}} → \underline{J}^{o}, and the point to be checked is that π^{∘}δ_{/F_{q}} is an isogeny, where π is the structural map \underline{J}^{o} → A. If we regard A as J_{1}(N)_{/F_{q}}×J_{1}(N)_{/F_{q}}, then π^{∘}δ_{/F_{q}} becomes a two-by-two matrix of endomorphisms of J_{1}(N)_{/F_{q}}. This matrix, as computed by Deligne-Rapoport [11, p. 287], has diagonal components equal to the identity endomorphism, and offdiagonal components equal to the Verschiebung endomorphism (dual of the Frobenius). This matrix is then visibly an isogeny, since it acts on the cotangent space of J_{1}(N)_{/F_{q}}×J_{1}(N)_{/F_{q}} as the identity map.

The following circumstance is perhaps worth stressing. In the functorial action of T on A, the Hecke operators ⟨d⟩ and T_{n} with n prime to q act “as expected.” Namely, they are induced by their namesakes on J_{1}(N), which operate diagonally on J_{1}(N)×J_{1}(N) and then by reduction (mod q) on A. However, the functorial action of T_{q} on A belongs naturally to characteristic q.

For the second point, we show that Φ is Eisenstein in the strong sense that we have on Φ the equations T_{p} = 1 + p and ⟨p⟩ = 1 for all p prime to qN. These equations imply immediately that 𝔪 is prime to the support of Φ because ρ is irreducible; cf. [41, Theorem 5.2c]. That Φ is Eisenstein was proved in [39] for the abelian variety J_{o}(qN). The result we seek then follows for N ≤ 3 because the natural map J_{o}(qN) → J is an isomorphism.

Assume that N ≥ 4. Recall that Φ is the cokernel of the map X → Hom(X, Z) coming from the monodromy pairing on X. View X as the kernel of the degree map X˜ → Z, where X˜ is the free abelian group on the set Σ of supersingular points of X_{1}(N)_{/\overline{F}_{q}}. The monodromy pairing on X is the restriction to X of a diagonal pairing on X˜ . This pairing takes the value e(σ) on (σ, σ), where e(σ) is essentially the number of automorphisms of any pair (E, P) which represents σ; we understand that E is a supersingular elliptic curve over \overline{F}_{q} and P ∈ E(\overline{F}_{q}) is a point of order N. The precise formula for (σ, σ) may be derived from [11, p. 286, Théorème 6.9]: we have

e(σ) = \{ \begin{matrix}\frac{1}{2}# Aut(E, P) & if −1 ∈ Aut σ, \\ # Aut(E, P) & if −1 ∉ Aut σ. \end{matrix}

The assumption N ≥ 4 implies that e(σ) = 1 by [27, Corollary 2.7.4]. Hence the monodromy pairing on X is just the restriction to X×X of the standard diagonal pairing on X˜ . It follows that the natural map

ξ: X˜ /X ⟶ Φ = Hom(X, Z)/X, ∑ n_{σ}σ ⟼ ⟨∑ n_{σ}σ, · ⟩

is surjective. Since X˜ /X is mapped isomorphically to Z by the degree map, Φ is a finite cyclic group.

Fix p prime to Nq, and let T denote the usual pth Hecke operator on X˜ : this is the Z-linear map of X˜ which sends (the class of) (E, P) to ∑(E/C, P mod C). Here, the sum ranges over the p+1 subgroups C of E[p] whose order is p. Let T′ denote the composite T^{∘}⟨p⟩^{−1}, where ⟨p⟩: (E, p) ↦ (E, pP). One checks the formula ⟨T x, y⟩ = ⟨x, T′y⟩, where ⟨ , ⟩ is the Euclidean pairing on X˜ . Also, it is clear that both T and T′ preserve the subgroup X of X˜ and induce the map “multiplication by p+1” on the quotient X˜ /X.

The operator T_{p} on Φ is the map on Hom(X, Z)/X induced by the operators T on X and Hom(T′, Z) on Hom(X, Z). The formula relating T, T′, and ⟨ , ⟩ shows that ξ is T_{p}-equivariant if T_{p} acts naturally on X˜ /X by multiplication by p+1. Hence T_{p} = p + 1 on Φ. In a similar manner, we find that the diamond-bracket operators are trivial on Φ.

For the last point, we must determine the action of Frob_{q} on T (\overline{F}_{q}). Let w be the automorphism of C which corresponds to the map

(E, P, C) ↦ (E/C, P mod C, E[q]/C)

on triples consisting of an elliptic curve E, a point P on E of order M, and a cyclic subgroup C of E which has order q. A short computation shows that we have w^{2} = ⟨q⟩. One checks as well that the correspondence T_{q} + w_{q} of C may be written β′^{∘}α where α and β are again the degeneracy maps C \overrightarrow{→} X_{1}(N). It follows from this that T_{q} + w_{q} induces an endomorphism of \underline{J} which kills the torus T, since this endomorphism factors through a map \underline{J} → J_{1}(N). On the other hand, w coincides with the Frobenius map on Σ but permutes the two components of C_{/F_{q}}. Therefore Frob_{q} = −w = T_{q} on X, since X is canonically the first integral homology group of the dual graph associated with C_{/F_{q}}. Thus Frob_{q} = qT_{q} on T (\overline{F}_{q}). Therefore, if V ↪ T (\overline{F}_{q}), then Frob_{q} acts on V as a scalar, i.e., as an element of k. Since the square of this scalar is q^{2}⟨q⟩, the determinant the action of Frob_{q} on V is q^{2}⟨q⟩.

On the other hand, it is clear from the Čebotarev Density Theory that the determinant of V is the character χϵ: Gal(\overline{Q}/Q) → k^{∗}, where ϵ is the Dirichlet character ⟨ ⟩, i.e., where ϵ(Frob_{r}) = ⟨r⟩ for every prime number r prime to M. In particular, the determinant of the action of Frob_{q} on V is q⟨q⟩. Therefore, we cannot have V ↪ T (\overline{F}_{q}) unless q ≡ 1 mod ℓ.

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UC Mathematics Department, Berkeley, CA 94720 USA

   E-mail address: ribet@math.berkeley.edu