Raising the Levels of Modular Representations

Kenneth A. Ribet Personal ver-

sion of William Stein.

1 Introduction

Let ℓ be a prime number, and let F be an algebraic closure of the prime field F_{ℓ}. Suppose that

ρ : Gal(\overline{Q}/Q) → GL(2, F)

is an irreducible (continuous) representation. We say that ρ is modular of level N, for an integer N ≥ 1, if ρ arises from cusp forms of weight 2 and trivial character on Γ_{0}(N).

The term “arises from” may be interpreted in several equivalent ways. For our present purposes, it is simplest to work with maximal ideals of the Hecke algebra for weight-2 cusp forms on Γ_{0}(N). Namely, let S(N) be the C-vector space consisting of such forms, and for each n ≥ 1 let T_{n} ∈ End S(N) be the nth Hecke operator. Let T = T_{N} be the subring of End S(N) generated by these operators. As is well known ([3], Th. 6.7 and [7], §5), for each maximal ideal m of T, there is a semisimple representation

ρ_{m} : Gal(\overline{Q}/Q) → GL(2, T/m),

unique up to isomorphism, satisfying

tr ρ_{m}(Frob_{r}) = T_{r} (mod m), det ρ_{m}(Frob_{r}) = r (mod m)

for almost all primes r. (Here Frob_{r} is a Frobenius element in Gal(\overline{Q}/Q) for the prime r.) This representation is in fact unramified at every prime r prime to ℓN, and the indicated relations hold for all such primes. We understand that ρ is modular of level N if there is a maximal ideal m of T, together with an inclusion ω: T/m ↪ F, so that the representations ρ and ρ_{m} ⊗_{ω} F are isomorphic. (Cf. [7], §5.)

The representations ρ_{m} are nothing other than the Galois representations attached to mod ℓ eigenforms of weight 2 on Γ_{0}(N). Indeed, let L be the space of forms in S(N) which have rational integral q-expansions. As is well known, L is a lattice in S(N), cf. [3], Proposition 2.7. The space \overline{L} = L/ℓL is the space of mod ℓ cusp forms on Γ_{0}(N). The F_{ℓ}-algebra A generated by the Hecke operators T_{n} in End \overline{L} may be identified with T/ℓT (see, for example, [7], §5). To give a pair (m, ω) as above is to give a character (i.e., homomorphism)

ϵ : A → F.

If f is a non-zero element of \overline{L} ⊗_{F_{ℓ}} F which is an eigenvector for all T_{n}, the action of A on the line generated by f defines such a character ϵ. It is an elementary fact that all characters ϵ arise in this manner.

Assume now that ρ is modular of level Np, where p is a prime number not dividing N. We say that ρ is p-new (of level pN) if ρ arises in a similar manner from the p-new subspace S(pN)_{p−new} of S(pN). Recall that there are two natural inclusions (or degeneracy maps) S(N) \overrightarrow{→} S(pN) and dually two trace maps S(pN) \overrightarrow{→} S(N). (See [1] for the former maps.) The two maps S(N) \overrightarrow{→} S(pN) combine to give an inclusion S(N) ⊕ S(N) ↪ S(pN), whose image is known as the p-old subspace S(pN)_{p−old} of S(pN). The space S(pN)_{p−new} is defined as the orthogonal complement to S(pN)_{p−old} in S(pN), under the Petersson inner product on S(pN). It may also be characterized algebraically as the intersection of the kernels of the two trace maps; this definition is due to Serre. The space S(pN)_{p−new} is T_{pN}-stable.

The image of T_{pN} in End S(pN)_{p−new} is the p-new quotient

\overline{T}_{pN} = T_{pN/p−new}

of T_{pN}. We say that ρ is p-new if m ⊂ T_{pN} and ω may be found, as above, in such a way that the maximal ideal m of T_{pN} is the inverse image of a maximal ideal of \overline{T}_{pN}, under the canonical quotient map T_{pN} → \overline{T}_{pN}. On a concrete level, this means that the character

ϵ : T_{pN} → F

coming from (m, ω) is defined by an eigenform in the mod ℓ reduction of the space S(pN)_{p−new}, i.e., in the F-vector space Λ ⊗_{Z} F, where Λ is the lattice in S(pN)_{p−new} consisting of forms with rational integral coefficients.

Theorem 1 Let ρ be modular of level N. Let p |̸ℓN be a prime satisfying one or both of the identities

tr ρ(Frob_{p}) = ±(p + 1) (mod ℓ).  (1)

Then ρ is p-new of level pN.

Remarks.

1. In the Theorem, and in the discussion below, we assume that ρ is irreducible, as above.

2. A slightly stronger conclusion may be obtained if one assumes that ρ is q-new of level N, where q is a prime number which divides N, but not N/q. Under this hypothesis, plus the hypothesis of Theorem 1, one may show that ρ is pq-new of level pN, in a sense which is easy to make precise as above. (See [7], §7, where a theorem to this effect is proved, under the superfluous additional hypothesis p ≡ −1 (mod ℓ).) The interest of Theorem 1 is that no hypothesis is made about the existence of a prime number q.

3. The case p = ℓ can be included in the Theorem if its hypothesis (1) is reformulated. Namely, (1) tacitly relies on the fact that ρ is unramified outside the primes dividing ℓN. Choose a maximal ideal m for ρ as in the definition of “modular of level N.” Then (1) may be re-written as the congruence

T_{p} ≡ ±(p + 1) (mod m).

Assuming simply that p is prime to N, but permitting the case p = ℓ, one proves that ρ is p-new of level pN if this congruence is satisfied (with at least one choice of ±).

Corollary Let ρ be modular of level N. Then there are infinitely many primes p, prime to ℓN, such that ρ is p-new of level pN.

Indeed, suppose that p is prime to ℓN. Then p is unramified in ρ, so that a Frobenius element Frob_{p} is well defined, up to conjugation, in the image of ρ. By the Cebotarev Density Theorem, there are infinitely many such p such that Frob_{p} is conjugate to ρ(c), where c is a complex conjugation in Gal(\overline{Q}/Q). Both sides of the congruence (1) are then 0, so that (1) is satisfied. (Cf. [7], Lemma 7.1.)

Our corollary is stated (in terms of mod ℓ eigenforms) as “Théorème (A)” in a recent preprint of Carayol [2]. Carayol describes his Théorème (A) as having been proved in preliminary versions of [7], as an application of results in [6]. In later versions of [7], Théorème (A) was replaced by a theorem involving pq-new forms (alluded to above), which is proved by methods involving Shimura curves. The aim of this present note is to resurrect Théorème (A).

Our derivation of Theorem 1 is based on the results of [6]. Although we couch our results in the language of Jacobians of modular curves, it should be clear to the reader that we use no fine arithmetic properties of these Jacobians: the argument is entirely cohomological. As F. Diamond has recently shown [4], an elaboration of these methods leads to results for cusp forms of weight k ≥ 2.

2 Summary of [6]

First let N be a positive integer, and consider the modular curve X_{0}(N)_{C}, along with its Jacobian J_{0}(N) = Pic^{0}(X_{0}(N)). The curve X_{0}(N) comes equipped with standard Hecke correspondences T_{n}, which induce endomorphisms of J_{0}(N) by Pic functoriality (cf. [7], §3). These endomorphisms, in turn, act on the space of holomorphic differentials on the abelian variety dual to J_{0}(N), which is the Albanese variety of X_{0}(N). This space of differentials is canonically identified with S(N), and via this identification the endomorphism T_{n} of J_{0}(N) acts on the space of differentials as the usual Hecke operator T_{n} of S(N). Since the action of End(J_{0}(N)) on S(N) is faithful, it follows that the subring of End(J_{0}(N)) generated by the T_{n} is “nothing other” than the ring T_{N}.

We now choose a prime p prime to N and consider X_{0}(pN) and J_{0}(pN), to which the same remarks apply. The two curves X_{0}(pN) and X_{0}(N) are linked by a pair of natural degeneracy maps δ_{1}, δ_{p}: X_{0}(pN) \overrightarrow{→} X_{0}(N), with the following (naive) modular interpretation. The curve X_{0}(pN) is associated to the moduli problem of classifying elliptic curves E which are furnished with cyclic subgroups C_{N} and C_{p} of order N and p, respectively. Similarly, X_{0}(N) classifies elliptic curves with cyclic subgroups of order N. The degeneracy map δ_{1} maps (E, C_{N}, C_{p}) to (E, C_{N}), while δ_{p} maps (E, C_{N}, C_{p}) to (E/C_{p}, C′_{N}), where C′_{N} is the image of C_{N} on E/C_{p}.

In a similar vein, we recall the modular interpretation of the correspondences T_{p} on X_{0}(N) and on X_{0}(pN). First, for X_{0}(N) we have

T_{p} : (E, C_{N}) ↦ ∑_{D}(E/D, (C_{N} ⊕ D)/D),

where the sum is taken over the (p + 1) different subgroups D of order p in E. For X_{0}(pN), we have a sum of p terms

T_{p} : (E, C_{N}, C_{p}) ↦ ∑_{D≠C_{p}}(E/D, (C_{N} ⊕ D)/D, E[p]/D),

where E[p] is the group of p-division points on E. (This latter group is the direct sum C_{p} ⊕D.) These formulas lead immediately to the relations among correspondences

δ_{1}∘T_{p} = T_{p}∘δ_{1} − δ_{p},  δ_{p}∘T_{p} = p · δ_{1}.  (2)

The maps δ_{1} and δ_{p} combine to induce a map on Jacobians

α : J_{0}(N) × J_{0}(N) → J_{0}(pN), (x, y) ↦ δ_{1}^{∗}(x) + δ_{p}^{∗}(y).

The image of this map is by definition the p-old subvariety A of J_{0}(pN); the kernel of α is a certain finite group which is calculated in [6].

Namely, let Sh be the Shimura subgroup of J_{0}(N), i.e., the kernel of the map J_{0}(N) → J_{1}(N) which is induced by the covering of modular curves X_{1}(N) → X_{0}(N). The group Sh is a finite group which may be calculated in the following way: Consider the maximal unramified subcovering X → X_{0}(N) of X_{1}(N) → X_{0}(N), and let G be the covering group of this subcovering. Then G and Sh are canonically G_{m}-dual.

Let Σ ⊂ J_{0}(N) × J_{0}(N) be the image of Sh under the antidiagonal embedding

J_{0}(N) → J_{0}(N) × J_{0}(N), x ↦ (x, −x).

According to [6], Theorem 4.3, we have This prop. im-

plies that the map J_{0}(N) → J_{0}(Np) is 2torsion!!?

Proposition 1 The kernel of α is the group Σ.

The map α is equivariant with respect to Hecke operators T_{n} with (n, p) =

1. Namely, we have α∘T_{n} = T_{n}∘α for all n prime to p, with the understanding that the endomorphism T_{n} of J_{0}(N) acts diagonally on the product J_{0}(N) × J_{0}(N). On the other hand, this formula must be modified when n is replaced by p, as one sees from (2).

Before recording the correct formula for T_{p}, we introduce the notational device of reserving the symbol T_{p} for the p^{th} Hecke operator at level N, and the symbol U_{p} for the p^{th} Hecke operator at level pN. With this notation, we have (as a consequence of (2)) the formula

U_{p}∘α = α∘ ( \begin{matrix}T_{p} & p \\ −1 & 0 \end{matrix} ) ,  (3)

in which the matrix refers to the natural left action of M(2, T_{N}) on the product J_{0}(N) × J_{0}(N).

[[But I think the matrix should be ( \begin{matrix}T_{p} & p \\ 1 & 0 \end{matrix} ), since

U_{p}δ_{1}(x) + U_{p}δ_{p}(x) = δ_{p}(x) + δ_{1}T_{p}(x) + pδ_{1}(y) = δ_{1}(T_{p}(x) + py) + δ_{p}(x).]]

Concerning the behavior of Sh and Σ under Hecke operators, the following (easy) result is noted briefly in [6] and proved in detail in [8].

Proposition 2 The Shimura subgroup Sh of J_{0}(N) is annihilated by the endomorphisms

η_{r} = T_{r} − (r + 1)

of J_{0}(N) for all primes r |̸N.

Corollary The subgroup Σ of J_{0}(N) × J_{0}(N) lies in the kernel of the

endomorphism ( \begin{matrix}1 + p & T_{p} \\ T_{p} & 1 + p \end{matrix} ) of J_{0}(N) × J_{0}(N). It is annihilated by the

operators T_{r} − (r + 1) for all prime numbers r not dividing pN.

The significance of the endomorphism introduced in the corollary appears

when we note the formula β∘α = ( \begin{matrix}1 + p & T_{p} \\ T_{p} & 1 + p \end{matrix} ), in which β : J_{0}(Np) →

J_{0}(N) × J_{0}(N) is the map induced by the two degeneracy maps X_{0}(Np) \overrightarrow{→} X_{0}(N) and Albanese functoriality of the Jacobian. (The map β becomes the dual of α when we use “autoduality of the Jacobian” to identify the Jacobians with their own duals.) The formula results from the fact that the two degeneracy maps are each of degree p + 1, and from the usual definition of T_{p} as a correspondence in terms of degeneracy maps.

Let Δ ⊂ J_{0}(N) × J_{0}(N) be the kernel of ( \begin{matrix}1 + p & T_{p} \\ T_{p} & 1 + p \end{matrix} ). Then Δ is a

finite subgroup of J_{0}(N) × J_{0}(N). Indeed, Δ differs only by 2-torsion from the direct sums of the kernels of T_{p} ± (p + 1) on J_{0}(N). These latter kernels are finite because neither number ±(p + 1) can be an eigenvalue of T_{p} on S(N), in view of Weil’s Riemann hypothesis, which bounds T_{p}’s eigenvalues by 2\sqrt{p}. Further, the group Δ comes equipped with a perfect G_{m}-valued skew-symmetric pairing, in view of its interpretation as the kernel K(L) of a How does this polarization map pairing work?

ϕ_{L} : J_{0}(N) × J_{0}(N) → (J_{0}(N) × J_{0}(N))ˇ .

(One takes L to be the pullback by α of the “theta divisor” on the Jacobian J_{0}(pN).)

The subgroup Σ of Δ is self-orthogonal under the pairing on Δ. In other words, if we let Σ^{⊥} be the annihilator of Σ in the pairing, we have a chain of groups

Δ ⊃ Σ^{⊥} ⊃ Σ.

Note also that Δ/Σ is naturally a subgroup of the abelian variety A, since A and Σ are the image and kernel of α, respectively. Thus the subquotient Σ^{⊥}/Σ of Δ is in particular a subgroup of A.

On the other hand, the quotient Δ/Σ^{⊥} is canonically the Cartier (i.e., G_{m}) dual Σ^{∗} of Σ. It is naturally a subgroup of Aˇ . Indeed, Σ is the kernel of the isogeny J_{0}(N) × J_{0}(N) → A induced by α. The kernel of the dual homomorphism Aˇ → (J_{0}(N) × J_{0}(N))ˇ may be identified with Σ^{∗}.

To state the final result that we need, we introduce the p-new abelian subvariety B of J_{0}(pN). To define it, consider the map

J_{0}(pN)ˇ → Aˇ

which is dual to the inclusion A ↪ J_{0}(pN). Its kernel is an abelian subvariety Z of J_{0}(pN)ˇ . Using the autoduality of J_{0}(pN) to transport Z back to J_{0}(pN), we obtain B. This subvariety of J_{0}(pN) is a complement to A in the sense that J_{0}(pN) = A + B and A ∩ B is finite. It is p-new in that T_{pN} stabilizes B and acts on B through its p-new quotient \overline{T}_{pN} (which acts faithfully on B). The following main result of [6] is a formal consequence of Proposition 2:

Theorem 2 The finite groups A ∩ B and Σ^{⊥}/Σ are equal.

In the notation of [6], A ∩ B is the group Ω, which can be described directly in terms of Δ and the kernel of α ([6], pp. 508–509). Once this kernel is identified, the description of Theorem 2 is immediate.

3 Proof of Theorem 1

We assume from now on that ρ is modular of level N, and choose an ideal m of T_{N}, plus an embedding ω: T_{N}/m ↪ F as in the definition of “modular of level N.” Assuming that one of the two congruences (1) is satisfied, we will construct

1. A maximal ideal M of \overline{T}_{pN}, and

2. An isomorphism T_{N}/m ≈ \overline{T}_{pN}/M which takes T_{r} to T_{r} for all primes r ≠ p.

This is enough to prove the theorem, since the representations ρ_{m} and ρ_{M} will necessarily be isomorphic, in view of the T_{r}-compatible isomorphism between the residue fields of m and M. Our procedure is to construct M first as a maximal ideal of T_{Np} and then to verify that M in fact arises by pullback from a maximal ideal of \overline{T}_{pN}.

It might be worth pointing out explicitly that our construction of M depends on the sign ± in (1). If p ≢ −1 (mod ℓ), then there is a unique sign ± which makes (1) true, under the hypothesis of the theorem, and our construction proceeds in a mechanical way. In case p ≡ −1 (mod ℓ), both congruences (1) are satisfied under the hypothesis of the theorem, and the construction requires us to decide whether (1) should read 0 ≡ +0 or 0 ≡ −0. The two choices of sign lead to different ideals M, at least when ℓ is odd, since our construction shows that U_{p} ≡ ±1 (mod M), with the same sign ± as in (1).

Before beginning the construction, we introduce the following abbreviations:

R = T_{N},  k = T_{N}/m,  T = T_{pN},  \overline{T} = \overline{T}_{pN}.

Also, let

V = J_{0}(N)[m]

be the kernel of m on J_{0}(N), i.e., the intersection of the kernels on J_{0}(N) of the various elements of m. This group is a finite k-vector space which is easily seen to be non-zero (cf. [5], or [7], Theorem 5.2). The group V × V is then a finite subgroup of J_{0}(N)×J_{0}(N). This subgroup has zero intersection with Sh × Sh, in view of the irreducibility of ρ_{m}, Proposition 2 above, and [7], Theorem 5.2(c). In particular, α maps V × V isomorphically into A. Therefore, we can (and will) regard V × V as a subgroup of that abelian variety.

We now assume that one of the two congruences (1) is satisfied. To fix ideas we will treat only the case

tr ρ(Frob_{p}) ≡ −(p + 1) (mod ℓ).

Using the isomorphism between ρ and ρ_{m} ⊗_{ω} F, we restate this congruence in the form

T_{p} ≡ −(p + 1) (mod m).  (4)

(The left-hand side of (4) is the trace of ρ_{m}(Frob_{p}).) We embed V in V ×V via the diagonal embedding; the antidiagonal embedding would be used instead if T_{p} were p + 1 modulo m. We have

V ↪ V × V ↪ A.

Lemma 1 The subgroup V of A is stable under T. The action of T on V is summarized by a homomorphism γ : T → k which takes T_{n} to T_{n} modulo m for (n, p) = 1 and takes U_{p} to −1.

Proof . That T_{n} ∈ T acts on V in the indicated way, for n prime to p, follows from the equivariance of α with respect to such T_{n}. The statement relative to U_{p} then follows from (3) and (4).

Define M = ker γ, so that we have an inclusion T/M ↪ k = R/m. This map is in fact an isomorphism since k is generated by the images of the T_{n} with n prime to p. Indeed, T_{p} lies in the prime field F_{ℓ} of k because of (4).

To conclude our proof of Theorem 1, we must show that the maximal ideal M of T arises by pullback from \overline{T}. For this, it suffices to show that T acts on V through its quotient \overline{T}. This fact follows from

Lemma 2 The subgroup V of A lies in the intersection A ∩ B.

Proof . We first note that V, considered diagonally as a subgroup of J_{0}(N)× J_{0}(N), lies in the group Δ. Indeed, V ⊂ J_{0}(N) is killed by T_{p} + p + 1 by virtue of (4). The isomorphic image of V in J_{0}(pN) therefore lies in Δ/Σ.

To prove the lemma, we must show that this image lies in the subgroup A ∩ B = Σ^{⊥}/Σ of Δ/Σ. In other words, we must show that the image of V in Δ/Σ^{⊥} is 0.

A somewhat painless way to see this is to view the varieties J_{0}(N), J_{0}(Np), A, … as being defined over Q. The group Δ/Σ^{⊥} is canonically the G_{m}-dual of Σ, which may identified Gal(\overline{Q}/Q)-equivariantly with the Shimura subgroup Sh of J_{0}(N). This latter group is in turn the G_{m}-dual of the covering group G introduced above. It follows that the action of Gal(\overline{Q}/Q) on Δ/Σ^{⊥} is trivial. (We note in passing that the action of Gal(\overline{Q}/Q) on Sh is given by the cyclotomic character Gal(\overline{Q}/Q) → Ẑ^{∗}.) Hence if V maps non-trivially to Δ/Σ^{⊥}, the semisimplification of V (as a F_{ℓ}[Gal(\overline{Q}/Q)]-module) contains the trivial representation. This semisimplification may be constructed by the following recipe: find the semisimplification W on V as a k[Gal(\overline{Q}/Q)]-module, and consider W as an F_{ℓ}-module. (A simple representation over k remains semisimple after “restriction of scalars” from k to F_{ℓ}.) Hence W contains Gal(\overline{Q}/Q)-invariant vectors, if V maps non-trivially to Δ/Σ^{⊥}. This conclusion is absurd, since W is the direct sum of a number of copies of the k-simple 2-dimensional representation ρ_{m} ([5], Chapter II, Proposition 14.2).

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K. A. Ribet

Mathematics Department University of California Berkeley CA 94720 U.S.A.