The weight in Serre’s conjectures on modular forms.

Bas Edixhoven^{∗} December 24, 1998

Contents

1 Introduction. 1 2 Modular forms mod p and Galois representations. 3 3 Tate’s θ-cycles. 7 4 The weight in Serre’s conjectures. 9 5 Group schemes of type (p, …, p) when e = p−1. 12 6 The local representation in the supersingular case. 16 7 Reduction to weight at most p+1. 23 8 Finiteness at p. 26 9 Multiplicity one results. 28

1 Introduction.

Let p be a prime number and let f be a modular form of level N with p |̸N, weight k and character ε, with coefficients in \overline{F}_{p}. Suppose that f is an eigenform for all Hecke operators T_{l}^{∗}, l prime, say with eigenvalues a_{l} ∈ \overline{F}_{p}. Then there exists a unique 2-dimensional semi-simple continuous representation ρ_{f}: G_{Q} = Gal(\overline{Q}/Q) → GL_{2}(\overline{F}_{p}), which is unramified outside pN and has the property that trace(ρ_{f}(Frob_{l})) = a_{l} and det(ρ_{f}(Frob_{l})) = ε(l)l^{k−1} for all l |̸pN. For a historical account of this result see [24], §6, see also [3]. It follows from the identities det(ρ_{f}(Frob_{l})) = ε(l)l^{k−1} that ρ_{f} is odd, i.e., det(ρ_{f}(c)) = −1 for c ∈ G_{Q} a complex conjugation.

Serre conjectured in 1973 that every continuous semi-simple odd representation ρ: G_{Q} → GL_{2}(\overline{F}_{p}) arises in this way. In his article [26] he stated a more precise version of this conjecture for irreducible ρ: ρ should arise from a form of level N(ρ), weight k_{ρ} and character ε(ρ), where N(ρ), k_{ρ} and ε(ρ) are described in terms of ρ; N(ρ) and k_{ρ} are meant to be as small as possible. The aim of this paper is to show that if ρ comes from a modular form at all, say of some type (N, k, ε), then ρ also comes from a modular form of type (N, k_{ρ}, ε) and k_{ρ} is (almost) minimal. For a precise statement see Thm. 4.5. For results on N(ρ) we refer to [22] and [11].

∗Supported by the Netherlands Organisation for Scientific Research (N.W.O.)

Let us now briefly describe the proof of Thm. 4.5. Suppose that ρ ≅ ρ_{g}, with g of type (N, k, ε). According to a well known result there exists an eigenform f_{1} of some type (N, k_{1}, ε) with k_{1} ≤ p+1 such that ρ_{g} ≅ ρ_{f_{1}} ⊗ χ^{a} for some a ∈ Z/(p−1)Z; here χ is the p-cyclotomic character (see Thm. 3.4). Let G_{p} be a decomposition group of G_{Q} at p. Since k_{1} ≤ p + 1, a lot is known about the local representation ρ_{f_{1}}|_{G_{p}}, see Thm. 2.5–2.9. One finds k_{1} and a in terms of ρ|_{G_{p}}. The desired form f of weight k_{ρ} is then found by “untwisting” the form f_{1}, i.e., one applies the operator θ (see §3) a times and then divides as many times as possible by the Hasse-invariant.

The main complication in the proof is that k_{1} is not necessarily unique. This can arise in two ways: there might exist a “companion form” f′_{1} of weight p+1−k_{1}, or f_{1} might be a multiple of the Hasse-invariant (this can only happen if k_{1} = p or k_{1} = p+1). For k_{1} = p these two ways are essentially the same. What one needs is a criterion in terms of ρ whether or not this non-uniqueness occurs. In both cases Serre conjectured such a criterion. The one concerning the “companion form” was proved by Gross in [10] (with some exception, however, see 2.9). For k = p+1 the problem, which has to do with “finiteness at p”, has been settled by Mazur, see Thm. 2.8. It should be clear to the reader that Serre must have known the proof of Thm. 4.5, modulo Thm.’s 2.8 and 2.9, in order to give the recipe for computing k_{ρ} in [26].

As explained above, our results depend on Gross’ article [10], in which several isomorphisms between different p-adic cohomology theories have not been checked to be compatible with Hecke operators (cf. [10], Introduction). R. Coleman has checked that the map η in (16.6) of [10] commutes with the action of Hecke operators; this result can be extracted from [4]. The remaining compatibilities (pages 504 and 505 of [10]) have not been checked.

Let us now make some remarks on forms of weight 1. In [26], Serre considers modular forms mod p as reductions of forms in characteristic 0. Since forms mod p of weight 1 cannot necessarily be lifted to characteristic 0 they had to be avoided. Later, for another reason ([27]), Serre replaced his definition of mod p modular forms by Katz’s (we will use Katz’s definition, see §2). Now one can allow forms of weight 1 without problem. As a consequence we have to change the definition of k_{ρ} if we want the conjecture to specify the minimal possible weight k(ρ) of a form f with ρ ≅ ρ_{f} (see Def. 4.3 and Rem. 4.4). Serre’s conjectures then suggest that k(ρ) should be 1 precisely when ρ is unramified at p (in [26] k_{ρ} is defined to be p if ρ is unramified at p). If f is an eigenform of type (N, p, ε), with T_{p}^{∗}f = a_{p}f, such that ρ is unramified at p and a_{p}^{2} ≠ ε(p), then Gross’s Thm. 2.9 shows that ρ_{f} comes from weight 1. It is not known if this remains true without the assumption a_{p}^{2} ≠ ε(p).

Let us end with some technicalities. The reader will note that most formulas concerning the modular curves X_{1}(N) differ from those in [10]. The reason for this is that we work with points of exact order N in the terminology of [16] whereas Gross works with embeddings of μ_{N}.

We apply the “Drinfeldian” moduli problems of [16] to the generalized elliptic curves of [8]; in the cases we consider this gives algebraic stacks. We denote the Hecke operators U_{l}^{∗} for l|N by T_{l}^{∗}.

Finally, it is a pleasure to thank R.F. Coleman, B.H. Gross, J.-P. Serre, D.L. Ulmer and especially K.A. Ribet for conversations on this subject and their comments on this article.

After this article was written R.F. Coleman and J.F. Voloch have obtained new results concerning companion forms [5]. Their main result, that does not depend on unproven compatibilities, states that a cuspidal eigenform f over \overline{F}_{p} of type (N, k, ε) with T_{p}^{∗}f ≠ 0 and 2 < k ≤ p has a companion form if and only if ρ_{f} is tamely ramified at p (compare Thm. 2.9). The consequences for this article are:

1. For p > 2 our results do not depend anymore on the unproven compatibilities mentioned above.

2. For p > 2 and f a weight 1 cuspidal eigenform over \overline{F}_{p} the representation ρ_{f} is unramified at p (compare Prop. 2.7).

3. All statements in Thm. 4.5 are true for p > 2 without the condition “ρ is not exceptional”.

2 Modular forms mod p and Galois representations.

2.1 Modular forms. Let p be a prime number and let N ≥ 1 be prime to p, let k be an integer, and let ε : (Z/NZ)^{∗} → \overline{F}_{p}^{∗} be a character. By a modular form of type (N, k) over \overline{F}_{p} we will mean a section of the invertible sheaf \underline{ω}^{⊗k} over the algebraic stack [Γ_{1}(N)]\overline{F}_{p} (denoted M_{Γ_{00}(N)} ⊗_{Z} \overline{F}_{p} in [8]). The objects of this algebraic stack are pairs (E/S/\overline{F}_{p}, α), where E is a generalized elliptic curve over S and α : (Z/NZ)_{S} ↪ E is an embedding of group schemes whose image meets all irreducible components of all geometric fibres. The invertible sheaf \underline{ω} associates to each pair (E/S/\overline{F}_{p}, α) the invertible O_{S}-module \underline{ω}_{E/S} = 0^{∗}Ω_{E/S}^{1}. In more downto-earth terms this means that a modular form f of type (N, k) over \overline{F}_{p} is a rule, compatible with cartesian squares, that associates to each (E/S/\overline{F}_{p}, α) a section f(E/S, α) of \underline{ω}_{E/S}^{⊗k}. If n ≥ 3 is prime to p then the stack [Γ_{1}(N), Γ(n)]\overline{F}_{p} is represented by a non-connected smooth proper curve over \overline{F}_{p}, say X, on which GL_{2}(Z/nZ) acts. Modular forms of type (N, k) are then GL_{2}(Z/nZ)-invariant sections of \underline{ω}^{⊗k} over X. The proof of this goes as follows. It is clear that a form f of type (N, k) gives an invariant section on X. Suppose now that we have an invariant section on X. Let (E/S/\overline{F}_{p}, α) be an object of [Γ_{1}(N)]\overline{F}_{p} such that the induced morphism S → [Γ_{1}(N)]\overline{F}_{p} is etale. Let S′ → S be the universal S-scheme over which E/S acquires a Γ(n)-structure. Then S′ → S is finite and flat and it is the quotient morphism for the action of GL_{2}(Z/nZ). Let E′ be the pullback of E to S′. Then \underline{ω}_{E′/S′} is the pullback to S′ of \underline{ω}_{E/S}. Since S′ → S is the quotient for the action of GL_{2}(Z/nZ) and \underline{ω}_{E/S} is a locally free O_{S}-module, invariant sections descend.

One obtains the q-expansions of f at the various cusps of [Γ_{1}(N)]\overline{F}_{p} by evaluating f on pairs (Tate(q^{d}), α), where Tate(q^{d}) is the Tate curve G_{m}/q^{dZ} over \overline{F}_{p}[[q]](q^{−1}) and d|N. Explicitly: the q-expansion f_{d,α}(q) of f at the cusp (Tate(q^{d}), α) is the power series f(Tate(q^{d}), α)/(dt/t)^{⊗k}

in \overline{F}_{p}[[q]].

If N > 4 then the stack [Γ_{1}(N)]\overline{F}_{p} is represented by the curve X_{1}(N)\overline{F}_{p} and modular forms of type (N, k) are just global sections of the invertible sheaf \underline{ω}^{⊗k} on X_{1}(N)\overline{F}_{p}.

If N is 3 or 4 then there is up to isomorphism exactly one object in [Γ_{1}(N)]\overline{F}_{p}(\overline{F}_{p}) with non-trivial automorphisms. For N = 4 this object is the generalized elliptic curve with two components equipped with a point of order 4; its automorphism group has order 2. For N = 3 this object is the elliptic curve with j-invariant 0 equipped with a point of order 3 that is fixed by the automorphisms of order 3 of the curve. In these cases modular forms of type (N, k) can be identified with sections of \underline{ω}^{⊗k} over the complement of these points in the curve X_{1}(N)\overline{F}_{p} that extend after suitable ramified coverings (compare [17], p.73–74).

If N equals 1 or 2 then every pair (E/S, α) has the automorphism −1, which implies that (−1)^{k}f = f. It follows that in these cases f = 0 unless k is even or p = 2. In this case too modular forms can be identified with sections of \underline{ω}^{⊗k} over the complement of the points with extra automorphisms in the curve X_{1}(N)\overline{F}_{p} that extend after suitable ramified coverings.

From the fact that there exists a modular form Δ of type (1, 12), with a simple zero at infinity, it follows that there are no non-zero forms with k < 0. We let A denote the Hasse invariant ([16], §12.4); it is the modular form of type (1, p−1) which has q-expansion 1 at all cusps.

Let M(N, k) denote the \overline{F}_{p}-vector space of modular forms over \overline{F}_{p} of type (N, k) and let M^{0}(N, k) be its subspace of cusp forms, i.e. the space of forms that have value zero on degenerate curves.

The group (Z/NZ)^{∗} acts on [Γ_{1}(N)]\overline{F}_{p} by:

⟨a⟩ : (E/S, α) ↦ (E/S, aα),

for a ∈ (Z/NZ)^{∗}. This gives an action by (Z/NZ)^{∗} on modular forms:

(⟨a⟩^{∗}f)(E/S, α) = f(E/S, aα).

A modular form f of type (N, k) is said to be of type (N, k, ε) if ⟨a⟩^{∗}f = ε(a)f for all a ∈ (Z/NZ)^{∗}. Of course, f can be non-zero only if ε(−1) = (−1)^{k}.

Let R be any commutative ring in which N is invertible. By replacing “\overline{F}_{p}” above by “R” one gets the definition of a modular form over R. For more details concerning modular forms in this setting see [14], Chapter 1 or [8], VII, §3.

2.2 Hecke operators. The \overline{F}_{p}-vector spaces of modular forms of type (N, k, ε) are equipped with certain endomorphisms, called Hecke operators. For each prime number l we have an operator T_{l}^{∗}. For the construction of these operators we refer to [10]. Let us just mention that for l |̸pN and (E/\overline{F}_{p}, α), with E a genuine (i.e., not generalized) elliptic curve, we have:

(T_{l}^{∗}f)(E/\overline{F}_{p}, α) = \frac{1}{l} ∑_{ϕ} ϕ^{∗}(f(E′/\overline{F}_{p}, ϕ^{∘}α)),  (2.2.1)

where the sum is over the l + 1 isogenies ϕ : E → E′ of degree l and source E. For l|N the operator T_{l}^{∗} is given by the same formula, but with the sum restricted to those ϕ such that ϕ^{∘}α is injective. The construction of T_{p}^{∗}, especially for k = 1, is somewhat harder. The operators T_{l}^{∗} all commute with each other; the common eigenspaces are one dimensional since the eigenvalues determine the q-expansions up to a constant factor.

2.3 Galois representations. Let f be of type (N, k, ε) and suppose that f is a common eigenvector for all the T_{l}^{∗} (l |̸pN) with eigenvalues a_{l} ∈ \overline{F}_{p}. Let G_{Q} = Gal(\overline{Q}/Q) denote the absolute Galois group of Q. According to a theorem of Deligne ([9], Theorem 6.7), there exists a unique (up to isomorphism) continuous semi-simple representation ρ_{f} : G_{Q} → GL(V), with V a 2-dimensional \overline{F}_{p}-vector space, which is unramified outside pN and has the property that trace(ρ_{f}(Frob_{l})) = a_{l} and det(ρ_{f}(Frob_{l})) = ε(l)l^{k−1} for all l |̸pN. In this last equality ρ_{f}(Frob_{l}) denotes a Frobenius element at l (unique up to conjugation). If f is not a cusp form then ρ_{f} is reducible.

2.4 The local representation at p. Let \overline{Z} be the integral closure of Z in \overline{Q}. Let us choose a surjection \overline{Z} → \overline{F}_{p} (these are permuted transitively by G_{Q}). Then we have a sequence of subgroups:

I_{p} ⊲ I ⊲ G_{p} ⊂ G_{Q},

where G_{p} is the decomposition subgroup, I the ramification subgoup, and I_{p} the wild ramification subgroup. We can identify G_{p} with Gal(\overline{Q}_{p}/Q_{p}), G_{p}/I with Gal(\overline{F}_{p}/F_{p}), and I_{t} := I/I_{p} (the tame ramification group) with Gal(Q_{p}^{tr}/Q_{p}^{unr}) and with lim_{⟵} F_{p^{n}}^{∗}. A character ϕ : I_{t} → \overline{F}_{p}^{∗} is said to be of level n if n is the smallest integer such that ϕ factors through F_{p^{n}}^{∗}. The n characters I_{t} → F_{p^{n}}^{∗} → \overline{F}_{p}^{∗} that are induced by embeddings of fields F_{p^{n}} → \overline{F}_{p} are called the fundamental characters of level n. We let χ : G_{Q} → \overline{F}_{p}^{∗} denote the cyclotomic character giving the action on the p^{th} roots of unity: σ(ζ) = ζ^{χ(σ)}.

Let ρ_{p} : G_{p} → GL(V) be any continuous 2-dimensional representation on a \overline{F}_{p}-vector space. Let V^{ss} be the semi-simplification of V with respect to the action of G_{p}. Then I_{p} acts trivially on V^{ss} and the action of I_{t} on V^{ss} is given by two characters ϕ, ϕ′ : I_{t} → \overline{F}_{p}^{∗}. Since Gal(\overline{F}_{p}/F_{p}) acts by conjugation on I_{t} it follows that {ϕ^{p}, ϕ^{′p}} = {ϕ, ϕ′}. This means that there are two cases:

2.4.1 ϕ and ϕ′ are both of level 1, ρ_{p} is reducible.

2.4.2 ϕ and ϕ′ are both of level 2, ϕ^{p} = ϕ′, ϕ^{′p} = ϕ, and ρ_{p} is irreducible.

Now suppose that f, ρ_{f} and V are as in (2.3). Then we have the following results on ρ_{f,p} := ρ_{f}|_{G_{p}}. For any a ∈ \overline{F}_{p}^{∗} let λ(a) denote the unramified character G_{p} → \overline{F}_{p}^{∗} such that λ(Frob_{p}) = a.

2.5 Theorem. (Deligne) Let f be a cusp form of type (N, k, ε) with 2 ≤ k ≤ p+1 which is an eigenform for all T_{l}^{∗} with eigenvalues a_{l}. Suppose that a_{p} ≠ 0. Then ρ_{f,p} is reducible and

ρ_{f,p} = ( \begin{matrix}χ^{k−1}λ(ε(p)/a_{p}) & ∗ \\ 0 & λ(a_{p}) \end{matrix} ) .

2.6 Theorem. (Fontaine) Let f be a cusp form of type (N, k, ε) with 2 ≤ k ≤ p + 1 which is an eigenform for all T_{l}^{∗} with eigenvalues a_{l}. Suppose that a_{p} = 0. Then ρ_{f,p} is irreducible and

ρ_{f}|_{I} = ( \begin{matrix}ψ^{k−1} & 0 \\ 0 & ψ^{′k−1} \end{matrix} ) ,

where ψ and ψ′ are the two fundamental characters of level 2.

For the proof of Theorem 2.5 we refer to [10]. Theorem 2.6 was proved by Fontaine in 1979 in two letters to Serre, but that proof has never been published; we will give a proof in §6.

2.7 Proposition. Let f be a cuspidal eigenform of type (N, 1, ε) with T_{p}^{∗}f = a_{p}f. Suppose that the polynomial X^{2} − a_{p}X + ε(p) has two distinct roots in \overline{F}_{p}, or that f is the reduction of a characteristic zero eigenform. Then ρ_{f} is unramified at p.

Proof. Let f_{d,α}(q) = ∑ b_{n}q^{n} be the q-expansion of f at some cusp (Tate(q^{d}), α). Let V_{p}^{∗}f be the form of type (N, p, ε) with (V_{p}^{∗}f)_{d,α}(q) = ∑ b_{n}q^{pn}. Then Af and V_{p}^{∗}f are eigenforms for all T_{l}, l ≠ p, with the same eigenvalues as f. The action of T_{p}^{∗} on the two dimensional subspace of M^{0}(N, p, ε) spanned by Af and V_{p}^{∗}f is given by: T_{p}^{∗}(Af) = a_{p}Af − ε(p)V_{p}^{∗}f, T_{p}^{∗}(V_{p}^{∗}f) = Af, where T_{p}^{∗}f = a_{p}f (see formula 4.7 in [10]). It follows that the characteristic polynomial of T_{p}^{∗}, acting on this two-dimensional vector space, equals X^{2} − a_{p}X + ε(p); let a and a′ be the two roots, and let g and g′ be the corresponding two normalized eigenforms. The representations ρ_{f}, ρ_{g} and ρ_{g′} are isomorphic, since f, g and g′ have the same eigenvalues for all T_{l}^{∗}, l ≠ p. According to Theorem 2.5 we have:

ρ_{g,p} = ( \begin{matrix}λ(a′) & ∗ \\ 0 & λ(a) \end{matrix} ) , ρ_{g′,p} = ( \begin{matrix}λ(a) & ∗ \\ 0 & λ(a′) \end{matrix} ) .

From this we see that ρ_{f} is unramified at p if a ≠ a′. One also knows that ρ_{f} is unramified at p if f is the reduction mod p of a weight 1 form of level N in characteristic zero (see Theorem 4.1 in [9]). It is not known whether ρ_{f} is unramified or not if a = a′ and f is not the reduction of a characteristic zero eigenform. □

2.8 Theorem. (Mazur) Let f be a cuspidal eigenform of type (N, p+1, ε). Suppose that ρ_{f} is irreducible and that there exists no form g of type (N, 2) such that f = Ag. Then a_{p}^{2} = ε(p), and the extension in Theorem 2.5 is not “finite at p” ([26], §2.8). I.e., the corresponding group scheme over Q_{p} corresponding to a realization of ρ_{f} over a finite field does not admit a finite extension over Z_{p}.

See [22], §6 for a proof in the case where ε is trivial and p > 2. That proof generalizes without problems to arbitrary ε. For p = 2 one argues as follows. Lemma 6.2 of [22] and its proof only require a slight adaptation: if V is unramified then the result is immediate, so we may assume that V^{l} and V^{et} are 1-dimensional k-vector space schemes; V_{Q_{2}}^{l} has finite Zariski closure in J by the Néronian property (since p=2); this Zariski closure is multiplicative since it is contained in the toric part. As in the proof of Lemma 6.3 of [22] one finds that V lands in the toric part of J, but this is a contradiction with the 1-dimensionality of U^{t} in the proof of Prop. 18 in [18].

In §8 we will give a proof that the notions “finite at p” and “peu ramifié” ([26], 2.4.7) are equivalent for representations ρ_{f} where f is a cuspidal eigenform of type (N, p+1, ε). Let us also remark that it follows from the construction of ρ_{f} that ρ_{f} is finite at p if f has weight 2.

The following theorem, which gives a criterion for ρ_{f,p} to be tamely ramified, was conjectured by Serre and recently proved by Gross ([10], Theorem 13.10).

2.9 Theorem. (Gross) Let f be as in Theorem 2.5, and suppose that a_{p}^{2} ≠ ε(p) if k = p. Then ρ_{f,p} is tamely ramified (i.e., its restriction to the inertia group is diagonalizable) if and only if there exists a cuspidal eigenform f′ of type (N, k′, ε), with k′ = p+1−k, such that la′_{l} = l^{k′}a_{l} for all l.

The form f′ is called a “companion form” of f. The hard part of the proof of the theorem is to show the existence of a companion f′ if ρ_{f,p} is tamely ramified. It seems not to be known whether a similar statement is still true if k = p and a_{p}^{2} = ε(p).

3 Tate’s θ-cycles.

3.1 The operator θ on modular forms mod p. Let M(N, k) denote the \overline{F}_{p}-vector space of modular forms over \overline{F}_{p} of type (N, k) and let M(N) = ⊕_{k≥0}M(N, k). Then M(N) is a graded \overline{F}_{p}-algebra. Each M(N, k) has a descending filtration consisting of the subspaces A^{i}M(N, k − i(p − 1)), i ≥ 0, where A is the Hasse-invariant. The filtration w(f) of f ∈ M(N, k) is defined by:

w(f) = min{k − i(p − 1) | f ∈ A^{i}M(N, k − i(p − 1))}.  (3.1.1)

Equivalently, w(f) is the smallest integer k′ for which there exists a form in M(N, k′) which, at some cusp, has the same q-expansion that f does. This last equivalence follows from the fact that for any cusp (d, α) the kernel of the q-expansion map M(N) → \overline{F}_{p}[[q]] is the ideal generated by A−1 (see [15], §1). Note that if f ∈ M(N, k) and f′ ∈ M(N, k′) have the same q-expansion at some cusp then k′ ≡ k (mod p−1). Taking into account our desription of modular forms in §2.1 Katz proves in [15] that there exists a derivation

θ : M(N) → M(N)

that increases degrees by p+1 and whose effect on q-expansions (at all cusps) is qd/dq: if f_{d,α}(q) = ∑ a_{n}q^{n}, then (θf)_{d,α}(q) = ∑ na_{n}q^{n}. If f ∈ M(N, k) has filtration k, and p |̸k, then θf has filtration k+p+1. If f ∈ M(N, pk) and θf = 0, then f = g^{p} for a unique g ∈ M(N, k). The commutation relations of θ with the Hecke operators are:

T_{l}^{∗}(θf) = l θ(T_{l}^{∗}f).

In particular, T_{p}^{∗}(θf) = 0. Note that θ preserves the subspace of cusp forms.

If f is an eigenform of type (N, k, ε) with eigenvalues a_{l}, then θf is an eigenform of type (N, k+p+1, ε) with eigenvalues la_{l}. This implies that

ρ_{θf} = ρ_{f} ⊗ χ.

3.2 θ-cycles. Let f be a homogeneous element in M(N) such that θ(f) ≠ 0. The θ-cycle of f is by definition the sequence of p−1 integers

(w(θf), …, w(θ^{p−1}f)).

Note that always w(θ^{p}f) = w(θf), but w(θ^{p−1}f) = w(f) only if the q-expansion of f at some cusp has a_{pn} = 0 for all n. It is straightforward but surprising that up to a cyclic permutation a θ-cycle is always of one of the forms

\begin{matrix}(k, k + p + 1, .., k + (p − 2)(p + 1)) & k ≡ 2 (p) \\ (k, k + p + 1, .., k + (p − k_{0})(p + 1), k_{1}, k_{1} + p + 1, .., k_{1} + (k_{0} − 3)(p + 1)) & k ≡ k_{0} (p), \end{matrix}

where 3 ≤ k_{0} ≤ (p + 3)/2 and k_{1} = k + p + 3 − 2k_{0}, at least for p > 3. A proof of this fact for level 1 can be found in [13], §7. For p = 2 the only possibility is (k) with k ≡ 0 (2). For p = 3 the possibilities are (again up to cyclic permutation): (k, k + 4) with k ≡ 2 (3) and (k, k) with k ≡ 0 (3).

3.3 Proposition. Let f be a cuspidal eigenform of type (N, k, ε) with 1 ≤ k ≤ p+1, with eigenvalues a_{l} and with w(f) = k. Then the θ-cycle of f is given as follows:

a_{p} = 0 \{ \begin{matrix}k = 1 & (p + 2, .., p + 2 + (p − 2)(p + 1)) \\ k = 2 & (2 + p + 1, .., 2 + (p − 2)(p + 1), 2) \\ 3 ≤ k ≤ p − 1 & (k + p + 1, .., k + (p − k)(p + 1), k_{1}, .., k_{1} + (k − 3)(p + 1), k) \\ k = p & (3, .., 3 + (p − 3)(p + 1), p) \\ k = p + 1 & does not occur, \end{matrix}

a_{p} ≠ 0 \{ \begin{matrix}k = 1 & (p + 2, .., p + 2 + (p − 2)(p + 1)) \\ 2 ≤ k ≤ p − 1 & (k + p + 1, .., k + (p − k)(p + 1), k′ + p + 1, .., k′ + (k − 1)(p + 1)) \\ k = p & (p + 2, .., p + 2 + (p − 2)(p + 1)) \\ k = p + 1 & (2p + 2, .., 2p + 2 + (p − 2)(p + 1)), \end{matrix}

where k_{1} = p+3−k and k′ = p+1−k. The correct interpretation of these formulas for p = 2 is:

a_{2} = 0 \{ \begin{matrix}k = 1 & (4) \\ k = 2 & (2) \\ k = 3 & does not occur, \end{matrix} a_{2} ≠ 0 \{ \begin{matrix}k = 1 & (4) \\ k = 2 & (4) \\ k = 3 & (6) \end{matrix}

and for p = 3:

a_{3} = 0 \{ \begin{matrix}k = 1 & (5, 9) \\ k = 2 & (6, 2) \\ k = 3 & (3, 3) \\ k = 4 & does not occur, \end{matrix} a_{3} ≠ 0 \{ \begin{matrix}k = 1 & (5, 9) \\ k = 2 & (6, 6) \\ k = 3 & (5, 9) \\ k = 4 & (8, 12) \end{matrix}

Proof. This follows from the classification above, except for the case a_{p} ≠ 0 and k = p. In that case one has to show that w(θf) ≠ 3. But if w(θf) = 3 then w(θ^{p−1}f) = p = w(f), f−θ^{p−1}f = V_{p}g for some non-zero g of weight 1, f is a linear combination of Ag and V_{p}g, w(θf) = p+2, contradiction. See also [13], §7. □

The reader will probably have noticed by now that we have been gathering results concerning forms of weight at most p+1. Fortunately, there is the following result.

3.4 Theorem. Let f be an eigenform of some type (N, k, ε), then there exist integers i and k′ with 0 ≤ i ≤ p−1, k′ ≤ p + 1 and an eigenform g of type (N, k′, ε) such that f and θ^{i}g have the same eigenvalues for all T_{l}^{∗} (l ≠ p).

The fact that eigenforms of weight at most p+1 give, up to twist, all systems of eigenvalues, has been known to be true for a long time. A possible proof, which has never been published, uses the construction of ρ_{f} in terms of the étale cohomology of the locally constant sheaf Sym^{k−2}F on the curve X = X_{1}(N)_{Q}−cusps, where F is the p-torsion of the universal elliptic curve. Locally constant sheaves on X correspond to representations of the fundamental group of X (for some chosen base point). The mod p representation theory of GL_{2}(F_{p}) then shows that Sym^{k−2}F has a Jordan-Hölder sequence with successive quotients of the form (Sym^{i}F) ⊗ (det F)^{⊗j} with 0 ≤ i ≤ p−1 and 0 ≤ j ≤ p−2. For example, let V = F_{p} ⊕ F_{p} be the standard representation of GL_{2}(F_{p}). Then Sym^{p}(V) has a subrepresentation V ↪ Sym^{p}(V) by sending v to v^{p}. The quotient is isomorphic to Sym^{p−2}(V) ⊗ det. This induces a short exact sequence

0 ⟶ F ⟶ Sym^{p}(F) ⟶ Sym^{p−2}(F) ⊗ det(F) ⟶ 0.

The first published proof of Thm. 3.4, for p≥5, is due to Ash and Stevens in [1], Thm. 3.4, 3.5. We will give another, modular form theoretic proof for arbitrary p in §7.

4 The weight in Serre’s conjectures.

4.1 Let ρ : G_{Q} → GL_{2}(\overline{F}_{p}) be a continuous representation of Gal(\overline{Q}/Q). Suppose that ρ is irreducible and that det(ρ) is odd: det(ρ(c)) = −1, where c ∈ G_{Q} is the complex conjugation induced by an embedding of \overline{Q} into C. In [26], §1–2 Serre attaches a triple (N(ρ), k_{ρ}, ε(ρ)) to ρ, where N(ρ) is a positive integer which is prime to p; k_{ρ} is a positive integer and ε(ρ) : (Z/N(ρ)Z)^{∗} → \overline{F}_{p}^{∗} a character. By definition, N(ρ) is the prime to p part of the Artin conductor of ρ (defined as in characteristic zero). To define k_{ρ} mod p−1 and ε(ρ) one considers the character det(ρ) : G_{Q} → \overline{F}_{p}^{∗}. Since the conductor of det(ρ) divides pN(ρ) one can identify det(ρ) with a homomorphism (Z/pN(ρ)Z)^{∗} → \overline{F}_{p}^{∗}, or equivalently, with a couple of homomorphisms ε(ρ) : (Z/N(ρ)Z)^{∗} → \overline{F}_{p}^{∗} and χ^{k_{ρ}−1} : (Z/pZ)^{∗} → \overline{F}_{p}^{∗}, where χ denotes the cyclotomic character as in §2.4. For the complete definition of k_{ρ}, which depends only on ρ|_{I}, we refer to [26], §2.

4.2 Conjecture. (Serre [26], 3.2.4) Let ρ be as above. Then there exists a cusp form f of type (N(ρ), k_{ρ}, ε(ρ)), which is an eigenform for all T_{l}^{∗}, such that ρ is isomorphic to ρ_{f}.

The mod p modular form f in this conjecture was originally meant to be the reduction mod p of some modular form in characteristic zero. Because of this definition of modular form mod p, Serre avoided the possibility k_{ρ} = 1. Later, in [27], Serre replaced this definition of modular form mod p by the definition that we are using in this article (i.e., Katz’s definition). This change in definition makes it natural to allow forms of weight 1. As a consequence we have to change the definition of k_{ρ} if we want the conjecture to specify the minimal possible weight of a form f with ρ ≅ ρ_{f}. We propose the following definition.

4.3 Definition. Let ρ_{p}: G_{p} → GL(V), V^{ss}, ϕ and ϕ′ be as in §2.4. We associate an integer k(ρ_{p}) to ρ_{p} as follows.

1. Suppose that ϕ and ϕ′ are of level 2. We have:

ρ_{p}|_{I} ≅ ( \begin{matrix}ϕ & 0 \\ 0 & ϕ′ \end{matrix} )

After interchanging ϕ and ϕ′ if necessary, we have (uniquely) ϕ = ψ^{a+pb} = ψ^{a}ψ^{′b} and ϕ′ = ψ^{′a}ψ^{b} with 0 ≤ a < b ≤ p−1. We set k(ρ_{p}) = 1 + pa + b.

2. Suppose that ϕ and ϕ′ are of level 1.

(a) If ρ_{p}|_{I_{p}} is trivial then we have:

ρ_{p}|_{I} ≅ ( \begin{matrix}χ^{a} & 0 \\ 0 & χ^{b} \end{matrix} )

with 0 ≤ a ≤ b ≤ p−2. We set k(ρ_{p}) = 1 + pa + b.

(b) Suppose that ρ_{p}|_{I_{p}} is not trivial. We have:

ρ_{p}|_{I} ≅ ( \begin{matrix}χ^{β} & ∗ \\ 0 & χ^{α} \end{matrix} )

for unique α and β with 0 ≤ α ≤ p−2 and 1 ≤ β ≤ p−1. We set a = min(α, β), b = max(α, β). If χ^{β−α} = χ and ρ_{p} ⊗ χ^{−α} is not finite at p (see §8.1) then we set

k(ρ_{p}) = 1 + pa + b + p − 1; otherwise we set k(ρ_{p}) = 1 + pa + b.

4.4 Remark. 1. By comparing the definitions of k_{ρ} and k(ρ) one sees that always k(ρ) ≤ k_{ρ} and that they differ only in two cases. In both cases ϕ and ϕ′ are of level 1. In the first case ρ_{p}|_{I_{p}} is trivial and a = 0 = b: then k(ρ) = 1 and k_{ρ} = p. In the second case p = 2, ρ_{p}|_{I_{p}} is non-trivial, α = 0, β = 1 and ρ_{p} is not finite at p: then k(ρ) = 3 and k_{ρ} = 4.

2. We call ρ exceptional if ρ_{p} is isomorphic to an extension (either split or non-split) of two copies of an unramified character ε of G_{p}:

ρ_{p} ≅ ( \begin{matrix}ε & ∗ \\ 0 & ε \end{matrix} )

4.5 Theorem. Let ρ : G_{Q} → GL_{2}(\overline{F}_{p}) be continuous, irreducible and odd. Let k(ρ) and k_{ρ} be defined as in 4.3 and 4.4. Suppose that there exists a cusp form g of some type (N, k, ε) with p |̸N, which is an eigenform for all T_{l}^{∗}, such that ρ is isomorphic to ρ_{g}. Then there exists a cuspidal eigenform f of type (N, k_{ρ}, ε) which has the same eigenvalues for T_{l}^{∗} (l ≠ p) as g has, such that ρ is isomorphic to ρ_{f}. If ρ is not exceptional (see 4.4) then there exists an eigenform f of type (N, k(ρ), ε) with the same eigenvalues for T_{l}^{∗} (l ≠ p) as g has, such that ρ is isomorphic to ρ_{f}. If ρ is not exceptional then there is no eigenform of level prime to p and of weight less than k(ρ) whose associated Galois representation is isomorphic to ρ.

Proof. First we will show the existence of f. The proof will distinguish between the various possible “types” for ρ_{p}. As in §2.4 let ϕ, ϕ′ denote the two characters: I_{t} → \overline{F}_{p}^{∗} associated to ρ_{p}.

Suppose that ϕ and ϕ′ are of level 2. Then ϕ ≠ ϕ^{p} = ϕ′; ρ_{p} is irreducible and

ρ|_{I} = ( \begin{matrix}ϕ & 0 \\ 0 & ϕ′ \end{matrix} ) .

Let ψ, ψ′ be the two fundamental characters of level 2. After permuting ϕ and ϕ′ if necessary we can write uniquely ϕ = ψ^{a}ψ^{′b} with 0 ≤ a < b ≤ p−1. According to Theorem 3.4 there exists at least one α ∈ Z/(p − 1)Z such that ρ ⊗ χ^{−α} is isomorphic to the Galois representation corresponding to an eigenform f_{1} of type (N, k_{1}, ε) with k_{1} ≤ p+1. We may suppose that k_{1} = w(f_{1}) and that f_{1} has eigenvalues l^{−α}a_{l} (l ≠ p), where a_{l} are the eigenvalues of g. By Thm.’s 2.5, 2.6, 2.8 and Prop. 2.7 we have k_{1} ≥ 2; T_{p}^{∗}f_{1} = 0; k_{1} ≤ p. Since χ = ψψ′ it follows from Theorem 2.6 that the only possible values for (α, k_{1}) are:

\{ \begin{matrix}(a, 1 + b − a), (b − 1, p + 2 + a − b) & if b − a ≠ 1 \\ (a, 2) & if b − a = 1. \end{matrix}

Because of Prop. 3.3 these possible values all occur. Let f_{1} be an eigenform of type (N, 1+b−a, ε) with eigenvalues l^{−a}a_{l} for all T_{l}^{∗} (l ≠ p) (this f_{1} is unique up to multiplication by a scalar). Then we can take f := θ^{a}f_{1}. We have

w(f) = 1 + b − a + (p + 1)a = 1 + pa + b = k(ρ) = k_{ρ}.

We can now suppose that ϕ and ϕ′ are of level 1. Then we have

ρ_{p} = ( \begin{matrix}χ^{β}ε_{1} & ∗ \\ 0 & χ^{α}ε_{2} \end{matrix} ) ,

where ε_{1} and ε_{2} are unramified characters G_{p} → \overline{F}_{p}^{∗}. Let us first suppose that ρ_{p} is not tamely ramified, i.e., “∗ ≠ 0”. In that case there exists exactly one twist of ρ by a power of χ that comes from an eigenform f_{1} of type (N, k_{1}, ε) with 2 ≤ k_{1} ≤ p+1, namely ρ⊗χ^{−α} (see Thm. 2.5 and Prop. 2.7). We may suppose that f_{1} has eigenvalues l^{−α}a_{l} (l ≠ p) and that k_{1} = w(f_{1}) (even if k_{1} = p, see Prop. 2.7). We normalize α and β as Serre does:

0 ≤ α ≤ p − 2 and 1 ≤ β ≤ p − 1.

By Thm. 2.5 we have the congruence k_{1}−1 ≡ β−α (mod p−1); together with Thm. 2.8 this determines k_{1}. Let us deal with the case p = 2 first. Then α = 0, β = 1, k_{1} = 2 if ρ_{p} is “peu ramifié” and k_{1} = 3 otherwise. In this case f_{1} is the form f of weight k(ρ) we are looking for; clearly the desired form of weight k_{ρ} exists too. Now suppose p > 2. If α < β then k_{1} = 1+β−α+p−1 if β = α+1 and ρ is “très ramifié” and k_{1} = 1+β−α otherwise. It follows from Prop. 3.3 that

w(θ^{α}f_{1}) = k_{1} + (p + 1)α = k(ρ) = k_{ρ}.

If α ≥ β then k_{1} = β−α+p and Prop. 3.3 shows:

w(θ^{α}f_{1}) = w(θ^{β−1}θ^{α−β+1}f_{1}) = 1 + pβ + α = k(ρ) = k_{ρ}.

Finally we have the case where “∗ = 0”. Then we can normalize α and β such that 0 ≤ α ≤ β ≤ p−2. According to Thm.’s 3.4, 2.5, 2.6, 2.8, 2.9 andthe proof of Prop. 2.7 ρ ⊗ χ^{−α} comes from a form f_{1} of type (N, k_{1}, ε) with 2 ≤ k_{1} < p+1. As above we have k_{1} ≡ 1+β−α (mod p−1). If α < β then we find the desired form f of type (N, k(ρ), ε) with k(ρ) = 1+pα+β = k_{ρ} as above. If α = β ≠ 0 then k_{1} = p and θ^{α}f_{1} has weight k(ρ) = k_{ρ}. If α = β = 0 then, since we are assuming that ρ is not exceptional, the desired form f of weight k(ρ) = 1 exists by Thm. 2.9.

Now we prove the minimality statement in the theorem. Suppose that ρ is not exceptional and that g is an eigenform of some type (M, k, ε′) with p |̸M and ρ ≅ ρ_{g}. Then the preceding part of the proof gives us a form f of type (M, k(ρ), ε′) with ρ ≅ ρ_{f} such that f has the same eigenvalues a_{l} for all l ≠ p as g has. If both T_{p}^{∗}f = 0 and T_{p}^{∗}g = 0 then the q-expansions of f and g at a fixed cusp differ by a constant factor and we have k(ρ) = w(f) = w(g) ≤ k. If T_{p}^{∗}g ≠ 0 then k ≤ p+1 by Proposition 4.12 in [10] and a “case by case” check shows that k(ρ) ≤ k. □

5 Group schemes of type (p, …, p) when e = p−1.

5.1 Raynaud’s classification. Let p be a prime number, let r ≥ 1 and let F be a field with q = p^{r} elements. As in [21], a F-vector space scheme over a scheme S is a commutative group scheme G/S given with a homomorphism of F to the ring of S-endomorphisms of G. Let D be a mixed characteristic strictly henselian discrete valuation ring with residue field k of characteristic p. Let v : D → Z be the valuation and let e = v(p) be the absolute ramification index. If S is a D-scheme and G is affine over S we can decompose O_{G} = O_{S} ⊕ (⊕_{χ}I_{χ}) as a direct sum of O_{S}-modules, where χ: F^{∗} → D^{∗} is a character and I_{χ} is the subsheaf of the augmentation ideal I consisting of sections f such that for all λ ∈ F^{∗} one has λ^{#}f = χ(λ)f. Here λ^{#} denotes the pull-back map on functions induced by the multiplication by λ on G (note that q−1 is invertible in D and that D contains the (q−1)^{th} roots of unity). The F-vector space scheme G/S is then said to satisfy condition (∗∗) ([21], §1.5) if all q−1 O_{S}-modules I_{χ} are invertible. Note that this condition is automatically satisfied if G/S is finite locally free of rank q and S is integral with fraction field of characteristic 0 ([21], 1.2.2). Note that if S is the spectrum of an algebraically closed field of characteristic 0 and G/S is finite, then G/S is a constant F-vector space scheme and (∗∗) means that it has dimension 1. A complete classification up to isomorphism of the G/S/D satisfying (∗∗) is given in [21], Theorem 1.4.1 and [21], §1.5 gives more information for more specific S. The results contained here in sections 5.1, 5.2 and 5.3 can be found in greater detail in [21].

A character χ: F^{∗} → μ_{q−1}(D) ⊂ D^{∗} is called fundamental if, when we define χ(0) = 0, the composition χ: F → D → k is a homomorphism of rings. The r fundamental characters play a key role in the classification mentioned above. If χ is a fundamental character then the r fundamental characters are the χ^{p^{i}}, with i ∈ Z/rZ. Let us fix a fundamental character χ_{0}, and let χ_{i} := χ_{0}^{p^{i}}. The fundamental characters χ: F^{∗} → μ_{q−1}(D) are essentially the inverses of the fundamental characters I_{t} → \overline{F}_{p}^{∗} of §2.4.

5.2 The classification over K. Let K be the field of fractions of D. The classification for S = Spec(K) is as follows. Each F-vector space scheme G over K can be described by r equations in r variables X_{i}, i ∈ Z/rZ

X_{i}^{p} = δ_{i}X_{i+1}, i ∈ Z/rZ, 0 ≠ δ_{i} ∈ K.  (5.2.1)

Here, for all i, X_{i} is a generator of I_{χ_{i}}. Let G′/K be given by equations X_{i}^{′p} = δ′_{i}X′_{i+1}. Any morphism of F-vector space schemes f: G → G′ is necessarily of the form f^{#}: X′_{i} ↦ u_{i}X_{i}, with u_{i} ∈ K. Such a f is an isomorphism if and only if u_{i} ∈ K^{∗} for all i. In that case the δ′_{i} and the u_{i} have to satisfy

δ′_{i} = u_{i}^{p}δ_{i}u_{i+1}^{−1}, i ∈ Z/rZ.  (5.2.2)

It follows that the map

(δ_{0}, …, δ_{r−1}) ↦ (v(δ_{0}), …, v(δ_{r−1}))  (5.2.3)

induces a bijection between the set of isomorphism classes of G/K satisfying (∗∗) and the set Z^{r}/Λ, where Λ is the lattice

Λ = Z(p, 0, …, 0, −1) ⊕ Z(−1, p, 0, 0, …, 0) ⊕ ⋯ ⊕ Z(0, …, 0, −1, p).  (5.2.4)

The set

{(a_{0}, …, a_{r−1}) ∈ Z^{r} | 0 ≤ a_{i} ≤ p − 1, (a_{0}, …, a_{r−1}) ≠ (p − 1, …, p − 1)}  (5.2.5)

is a set of representatives and hence corresponds bijectively to the set of isomorphism classes of G/K satisfying (∗∗).

Of course, a finite group scheme G/K is determined by the action of Gal(\overline{K}/K) on G(\overline{K}). Let G/K be the F-vector space scheme given by equations 5.2.1 with a_{i} := v(δ_{i}). Then the action of Gal(\overline{K}/K) on G(\overline{K}) is given by a character Gal(\overline{K}/K) → F^{∗}. As in 2.4 we identify the tame ramification group I_{t} of Gal(\overline{K}/K) with lim_{⟵} μ_{n}(K). The character above is then identified with a character ϕ: μ_{q−1}(K) → F^{∗}. For all i ∈ Z/rZ let ψ_{i}: μ_{q−1}(K) → F^{∗} denote the inverse mapping of χ_{i}: F^{∗} → μ_{q−1}(K); the ψ_{i} are the fundamental characters of level r in the sense of 2.4. Note that ψ_{i}^{p} = ψ_{i−1}.

Elimination of the X_{i}, i ≠ 0, from the equations 5.2.1 defining G/K gives the equation

X_{0}^{q} = δX_{0}, where δ = δ_{0}^{p^{r−1}}δ_{1}^{p^{r−2}} ⋯ δ_{r−1}.  (5.2.6)

We can now identify G(\overline{K}) with the roots of 5.2.6 in \overline{K}. Then for all λ ∈ F^{∗}, multiplication by λ on G(\overline{K}) corresponds to multiplication by χ_{0}(λ) on these roots (since λ^{#}X_{0} = χ_{0}(λ)X_{0}). An element ζ ∈ μ_{q−1}(K), viewed as an element of the Galois group of the degree q − 1 extension of K, acts on the roots by multiplication by ζ^{v(δ)}. From v(δ) = p^{r−1}a_{0} + ⋯ + pa_{r−2} + a_{r−1} it follows that

ϕ = ψ_{1}^{a_{0}}ψ_{2}^{a_{1}} ⋯ ψ_{r}^{a_{r−1}}.  (5.2.7)

For more details see [21], §3.4.

5.3 The classification over D. Each F-vector space scheme G over D satisfying (∗∗) is given by equations

X_{i}^{p} = δ_{i}X_{i+1}, i ∈ Z/rZ, δ_{i} ∈ D, 0 ≤ v(δ_{i}) ≤ e.  (5.3.1)

The reason the conditions “v(δ_{i}) ≤ e” show up is that the p/δ_{i} are needed to define the group structure. If G′/D is given by equations X_{i}^{′p} = δ′_{i}X′_{i+1} then G and G′ are isomorphic if and only if there exist units u_{i} ∈ D^{∗} satisfying the equations 5.2.2. As above, the map 5.2.3 induces a bijection between the set of isomorphism classes of F-vector space schemes over D satisfying (∗∗) and the set

{(a_{0}, …, a_{r−1}) ∈ Z^{r} | 0 ≤ a_{i} ≤ e}.  (5.3.2)

Note that in the case e = p−1 this shows that every F-vector space scheme G over K satisfying (∗∗) can be extended over D; that any two such extensions are uniquely isomorphic except when G corresponds to (0, …, 0) in the set 5.2.5 (i.e., G is constant); if G is constant then there are exactly two extensions corresponding to (0, …, 0) and (p−1, …, p−1) which are étale and multiplicative, respectively.

5.4 From e = 1 to e′ = p−1. Let D := W (\overline{F}_{p}) be the ring of Witt vectors of \overline{F}_{p}, i.e., D is a complete discrete valuation ring with residue field \overline{F}_{p} and uniformizer p. Let D′ := D[ζ_{p}] where ζ_{p} is a primitive p^{th} root of unity. Note that D and D′ are strictly henselian discrete valuation rings with absolute ramification indices e = 1 and e′ = p−1, respectively; let K and K′ denote their fields of fractions and let v and v′ denote their valuations.

Let G_{K} be a F-vector space scheme given by equations 5.2.1 in the isomorphism class corresponding to an element a = (a_{0}, …, a_{r−1}) in the set 5.2.5. Let G_{K′} be the pull-back to Spec(K′) of G_{K}. Of course, G_{K′} is described by the same equations as G_{K}, but note that v′(δ_{i}) = (p − 1)v(δ_{i}) for all i. As remarked above in §5.3, G_{K′} can be extended (uniquely in all cases except one) to a F-vector space scheme G_{D′} over D′ satisfying (∗∗). Let G_{D′} be such an extension given by equations X_{i}^{′p} = δ′_{i}X′_{i+1}; let a′_{i} = v′(δ′_{i}) and let a′ = (a′_{0}, …, a′_{r−1}). Note that 0 ≤ a′_{i} ≤ p−1 for all i and that (p−1)a−a′ ∈ Λ with Λ as in 5.2.4. Let

f: G_{K′} ˜ ⟶ G_{D′,K′} f^{#}: X′_{i} ↦ u_{i}X_{i} (u_{i} ∈ K^{′∗})  (5.4.1)

be an isomorphism (unique up to F^{∗}). Let b_{i} = −v′(u_{i}) (note the sign). The b_{i} are independent of the choice of f and from 5.2.2 it follows that the b_{i} are the solution to the set of equations

a′_{i} = −pb_{i} + (p − 1)a_{i} + b_{i+1} (i ∈ Z/rZ).  (5.4.2)

In order to describe the action (yet to be defined) of Gal(\overline{K}/K) on the special fibre G\overline{F}_{p} of G_{D′} in terms of the a_{i} we need to have a more explicit description of the a′_{i} and the b_{i}. In §5.2, after identifying I_{t} with lim_{⟵} μ_{n}(K), it was shown that the action of Gal(\overline{K}/K) on G_{K}(\overline{K}) is given by the character ϕ in 5.2.7. Let I′_{t} denote the tame ramification group of Gal(\overline{K}/K′). Just as for I_{t}, we identify I′_{t} with lim_{⟵} μ_{n}(K′) = lim_{⟵} μ_{n}(\overline{F}_{p}) = lim_{⟵} μ_{n}(K). Then the canonical injection I′_{t} ⊂ I_{t} corresponds to multiplication by p−1 on lim_{⟵} μ_{n}(K). It follows that the F-vector space scheme G_{K′} corresponds to the character ϕ′ = ϕ^{p−1}. From equation 5.2.7 for both ϕ and ϕ′ it the follows that

p^{r−1}a′_{0} + ⋯ + pa′_{r−2} + a′_{r−1} ≡ (p − 1)(p^{r−1}a_{0} + ⋯ + pa_{r−2} + a_{r−1}) (mod p^{r} − 1). (5.4.3)

For any set of x_{i}, i ∈ Z/rZ, with 0 ≤ x_{i} ≤ p − 1 let [x_{r−1} ⋯ x_{1}x_{0}]_{p} denote the number x_{r−1}p^{r−1} + ⋯ + x_{1}p + x_{0}, written in base p. Then the equation 5.4.3 means that one obtains [a′_{0} ⋯ a′_{r−2}a′_{r−1}]_{p} by summing [a_{1} ⋯ a_{r−1}a_{0}]_{p} and [(p−1−a_{0}) ⋯ (p−1−a_{r−1})]_{p} with a “cyclic carry”, i.e., one adds them as if they were numbers written out in base p but the carry that shows up on the left must be added on the right to the result.

Let us now discuss how to find the unique solution of equation 5.4.2 for the b_{i}. One sees immediately that b_{i} ≡ a′_{i−1} + a_{i−1} (mod p) for all i. If j is such that |b_{j}| is maximal among the |b_{i}| we see from 5.4.2 that |pb_{j}| ≤ (p − 1)^{2} + |b_{j}|, i.e., |b_{i}| ≤ p − 1 for all i. Combining equation 5.4.2 and the fact that b_{i} ≥ −(p−1) for all i shows that b_{i} ≥ −1 for all i. Now remark that b_{i} = −1 implies that a_{i} = 0, a′_{i} = p − 1 and b_{i+1} = −1. So we can conclude that

b_{i} ≡ a′_{i−1} + a_{i−1} (mod p) and 0 ≤ b_{i} ≤ p − 1 for all i, or
a_{i} = 0, a′_{i} = p − 1, and b_{i} = −1 for all i.

(5.4.4)

For later use we give the following explicit formulas in the case r = 2:

\begin{aligned}& a′_{0} & a′_{1} & b_{0} & b_{1} \\ a_{0} > a_{1} & p + a_{1} − a_{0} & a_{0} − a_{1} − 1 & a_{0} − 1 & a_{1} \\ a_{0} < a_{1} & a_{1} − a_{0} − 1 & p + a_{0} − a_{1} & a_{0} & a_{1} − 1 \\ a_{0} = a_{1} & 0 & 0 & a_{1} & a_{0} \\ & p − 1 & p − 1 & a_{1} − 1 & a_{0} − 1 \end{aligned}  (5.4.5)

5.5 Action of inertia. The notation is as in §5.4. The cyclotomic character χ (see 2.4) gives an isomorphism Gal(K′/K) → F_{p}^{∗}. For σ ∈ Gal(K′/K) we let Spec(σ) denote the automorphisms of Spec(K′) and Spec(D′) that it induces. We let g(σ) be the automorphism id_{G} × Spec(σ) of G_{K′}. Then there is a unique automorphism g(σ) of G_{D′,K′}, such that the isomorphism f in 5.4.1 commutes with Spec(σ). Then we have:

f^{#}(g(σ)^{#}X′_{i}) = g(σ)^{#}f^{#}X′_{i} = g(σ)^{#}(u_{i}X_{i}) = σ(u_{i})X_{i} = f^{#}(σ(u_{i})u_{i}^{−1}X′_{i}),

from which we conclude that

g(σ)^{#}X′_{i} = σ(u_{i})u_{i}^{−1}X′_{i} = χ(σ)^{−b_{i}}X′_{i} (i ∈ Z/rZ).  (5.5.1)

From this formula it follows that the automorphisms g(σ) extend (uniquely, of course) to automorphisms g(σ) of G_{D′}. Restricting these to the special fibre then gives us an action of Gal(K′/K) on G\overline{F}_{p} by \overline{F}_{p}-automorphisms g(σ). We are interested in the action by Gal(K′/K) on Cot(G\overline{F}_{p}), the cotangent space at 0 of G\overline{F}_{p}. The \overline{F}_{p}-vector space Cot(G\overline{F}_{p}) has the following presentation:

Cot(G\overline{F}_{p}) = ( ⨁_{i∈Z/rZ} \overline{F}_{p} · dX′_{i} )/( ⨁_{i∈Z/rZ} \overline{F}_{p} · δ̄′_{i}dX′_{i+1}) = ⨁_{a′_{i−1}>0^{i∈Z/rZ}} \overline{F}_{p} · dX′_{i},  (5.5.2)

where δ̄′_{i} denotes the image of δ′_{i} in \overline{F}_{p}. From 5.5.1 we see that Gal(K′/K) acts on the generator dX′_{i} via the character χ^{−b_{i}}.

6 The local representation in the supersingular case.

6.1 The aim of this section is to give a proof of Theorem 2.6. So we let f be a cusp form of type (N, k, ε) over \overline{F}_{p} which is an eigenform for all T_{l}^{∗} with eigenvalues a_{l}; we suppose that 2 ≤ k ≤ p+1 and that a_{p} = 0. By Theorem 2.8 we may suppose that 2 ≤ k ≤ p. In order to prove Theorem 2.6 we may suppose, after replacing N by a multiple of N, that the curve X_{1}(N) over Z[1/N] represents the stack [Γ_{1}(N)]_{Z[1/N]}. For k > 2 we will use a construction of ρ_{f} using models of the modular curve X_{1}(pN) over various rings. All details of this construction can be found in [10]. Unfortunately Gross [10] works with embeddings of μ_{pN} into elliptic curves whereas we prefer to work with points of exact order pN in the terminology of [16], since then we can apply all the results of [16] without change. This explains that most formulas in [10] differ from ours. The construction, very briefly, goes as follows. First one constructs a differential form ω(f) on the so-called Igusa curve over X_{1}(N)\overline{F}_{p} which has the same eigenvalues for all T_{l}^{∗} as f has. The Igusa curve occurs in the characteristic p fibre of the stable model of X_{1}(pN) over Z_{p}[ζ_{p}], where ζ_{p} is a p^{th} root of unity. This allows one to lift ω(f) to a characteristic 0 eigenform on X_{1}(pN)\overline{Q}. Let J_{Q} denote the jacobian of X_{1}(pN)_{Q}. Then ρ_{f} is obtained as the “reduction mod p” of one of the factors of the Q_{p}[G_{Q}]-module (lim_{⟵} J_{Q}[p^{n}](\overline{Q})) ⊗_{Z_{p}} Q_{p}.

6.2 The reduction mod p of X_{1}(pN). Let [Γ_{1}(p), Γ_{1}(N)] denote the algebraic stack over Z classifying triples (E/S, ϕ, α) where E/S is a generalized elliptic curve, where ϕ: F_{p} → E(S) and α: Z/NZ → E(S) are a F_{p}-structure and a Z/NZ-structure (see [16], §1.5) such that the group generated by the images of ϕ and α meets all irreducible components of all geometric fibres.

Let [Γ(p), Γ_{1}(N)] denote the algebraic stack over Z classifying triples (E/S, ϕ, α), where E/S and α are as above and where ϕ: F_{p} × F_{p} → E[p] is a Drinfeld basis ([16], §1.4), such that the group generated by the images of ϕ and α meets all irreducible components of all geometric fibres. The group GL_{2}(F_{p}) acts on [Γ(p), Γ_{1}(N)] from the right by:

g ∈ GL_{2}(F_{p}): (E/S, ϕ, α) ↦ (E/S, ϕ^{∘}g, α).

The map (E/S, ϕ, α) ↦ (E′/S, ϕ|_{F_{p}×{0}}, α), where E′/S is obtained from E/S by contractions (see [8], IV, §1), from [Γ(p), Γ_{1}(N)] to [Γ_{1}(p), Γ_{1}(N)] is the quotient, in the sense of [16], §7.1, by the action of the group Γ := {g ∈ GL_{2}(F_{p}) | g(1, 0) = (1, 0)} ([16], Theorem 7.4.2).

For a scheme or an algebraic stack X over Z let X ˜⊗_{Z}Z[ζ_{p}] denote the normalization of X ⊗_{Z} Z[ζ_{p}]. Then we get an isomorphism (see [16], §9 and §5.1.1):

( ∐_{σ∈Gal(Q(ζ_{p})/Q)} [Γ(p)^{σ(ζ_{p})−can}, Γ_{1}(N)])/Γ ˜ ⟶ [Γ_{1}(p), Γ_{1}(N)] ˜⊗_{Z}Z[ζ_{p}].

Since Γ acts transitively on the terms in the disjoint union we get an isomorphism:

[Γ(p)^{ζ_{p}−can}, Γ_{1}(N)]/({1 \atop 0} {∗ \atop 1}) → [Γ_{1}(p), Γ_{1}(N)] ˜⊗_{Z}Z[ζ_{p}].  (6.2.1)

Let X/Z_{p}[ζ_{p}] be the moduli scheme over Z_{p}[ζ_{p}] representing [Γ_{1}(p), Γ_{1}(N)] ˜⊗_{Z}Z[ζ_{p}]. Using the results from [16], Chapter 13 it is not hard to obtain the following description of X.

6.2.2 X is a regular stable model of X_{1}(pN) in the sense of [7],

6.2.3 X_{F_{p}} = I ∪ E, where I ≅ \overline{M}([Ig(p), Γ_{1}(N)]) and E ≅ \overline{M}([ExIg(p, 1), Γ_{1}(N)]) (these are isomorphisms over X_{1}(N)_{F_{p}}; they are not unique),

6.2.4 I ∩ E is the set of supersingular points.

Here [Ig(p)] is the F_{p}-stack classifying pairs (E/S, P) where P ∈ E^{(p)}(S) generates ker V: E^{(p)} → E, and [ExIg(p, 1)] classifies pairs (E/S, P) where P ∈ E(S) is such that (P, 0) is a Drinfeld basis of E[p]. The curves I and E are smooth, proper and geometrically irreducible over F_{p}; the assignment

(E/S, P, Q) ↦ (E^{(p)}/S, P, Q^{(p)})  (6.2.5)

induces an “exotic” ([16], §11) isomorphism [Ig(p), Γ_{1}(N)] → [ExIg(p, 1), Γ_{1}(N)].

6.3 The reduction mod p of J_{Q}. Let J be the Néron model over Z_{p}[ζ_{p}] of the jacobian of X_{1}(pN)_{Q_{p}(ζ_{p})}. Since X/Z_{p}[ζ_{p}] is a stable curve, Pic_{X/Z_{p}[ζ_{p}]}^{o} is semi-abelian and J^{o} = Pic_{X/Z_{p}[ζ_{p}]}^{o}. Let \overline{J} = J_{F_{p}}. The pullbacks of invertible sheaves from X_{F_{p}} to I and E induce an exact sequence:

0 ⟶ T ⟶ \overline{J}^{o} ⟶ J_{I} ⊕ J_{E} ⟶ 0,  (6.3.1)

where J_{I} and J_{E} are the jacobians of I and E, and where T is a torus whose character group X(T) = Hom\overline{F}_{p}(T\overline{F}_{p}, G_{m}\overline{F}_{p}) sits in an exact sequence:

0 ⟶ X(T) ⟶ Z^{s.s.X_{1}(N)(\overline{F}_{p})} ⟶^{+} Z ⟶ 0.  (6.3.2)

Here s.s.X_{1}(N)(\overline{F}_{p}) denotes the set of supersingular points.

6.4 The ring of Hecke operators. For each prime number l we have a correspondence T_{l} on [Γ_{1}(p), Γ_{1}(N)]:

T_{l}: (E/S, ϕ, α) ↦ ∑_{β:E→E′}(E′/S, β^{∘}ϕ, β^{∘}α),  (6.4.1)

where the sum is over the β ∈ [Γ_{0}(l)](E/S) such that (E′/S, β^{∘}ϕ, β^{∘}α) ∈ [Γ_{1}(p), Γ_{1}(N)]. Here we assume that S is irreducible, the j-invariant j: S → P_{Z}^{1} is flat at infinity and at the points with residue characteristic l, and that [Γ_{0}(l)]_{E/S} is set theoretically the union of its S-valued points; we have identified the stacks [Γ_{0}(l)] and [l-Isog].

We also have the “diamond operators” ⟨a⟩_{N}, a ∈ (Z/NZ)^{∗}, and ⟨b⟩_{p}, b ∈ F_{p}^{∗}. They act on [Γ_{1}(pN)] by automorphisms:

⟨a⟩_{N}: (E/S, ϕ, α) ↦ (E/S, ϕ, aα), ⟨b⟩_{p}: (E/S, ϕ, α) ↦ (E/S, bϕ, α).  (6.4.2)

We let the T_{l}, ⟨a⟩_{N} and ⟨b⟩_{p} act on J_{Q} via Albanese functoriality (i.e., push-forward of divisors). Let H be the subring of End(J_{Q}) generated by them; H is commutative.

We define correspondences T_{l} and automorphisms ⟨a⟩_{N} and ⟨b⟩_{p} on the stacks [Ig(p), Γ_{1}(N)] and [ExIg(p, 1), Γ_{1}(N)] by formulas analogous to 6.4.1 and 6.4.2. They induce operators T_{l}, ⟨a⟩_{N} and ⟨b⟩_{p} on J_{I} and J_{E} (independent of the choice of isomorphisms in 6.2.3).

Finally we have the automorphism w = w_{ζ_{pN}} of [Γ_{1}(p), Γ_{1}(N)]_{Z[1/pN,ζ_{pN}]} defined by

w: (E/S, ϕ, α) ↦ (E′/S, ϕ′, α′),  (6.4.3)

where β: E → E′ is the pN-isogeny with ker(β) = im(ϕ)+im(α), where im(ϕ′)+im(α′) = ker(β^{t}) and ⟨ϕ(1) + α(1), ϕ′(1) + α′(1)⟩_{β} = ζ_{pN}. Here ⟨∗, ∗⟩_{β} denotes the pairing from [16], §2.8 and β^{t}: E → E′ denotes the dual of β. The use of w is that it gives the Rosati involution on H:

∀h ∈ H ⊂ End_{Q(ζ_{pN})}(J) : h^{t} = whw^{−1}.  (6.4.4)

In fact, one can check that for all l, wT_{l}w^{−1} is the correspondence dual to T_{l}.

6.5 Action of inertia. For every σ ∈ Gal(Q(ζ_{p})/Q) the automorphism

id × Spec(σ): [Γ_{1}(p), Γ_{1}(N)] ⊗_{Z} Q(ζ_{p})

extends uniquely to an automorphism g(σ) of [Γ_{1}(p), Γ_{1}(N)] ˜⊗_{Z}Z[ζ_{p}]. This gives an automorphism g(σ) of X. Restricting the g(σ) to X_{F_{p}} gives a right-action of Gal(Q(ζ_{p})/Q) on X_{F_{p}} by F_{p}-automorphisms. The automorphism g(σ) of X induces via Albanese functoriality an automorphism g(σ) of J, whose restriction to \overline{J} we still denote by g(σ).

6.6 Some formulas. By the Néron property of J all endomorphisms of J_{Q_{p}(ζ_{p})} extend to endomorphisms of J; their restrictions to \overline{J}^{o} preserve the exact sequence 6.3.1. We write an endomorphism of J_{I}⊕J_{E} as a matrix ({a \atop c} {b \atop d}), with a ∈ End(J_{I}), b ∈ Hom(J_{E}, J_{I}), c ∈ Hom(J_{I}, J_{E}) and d ∈ End(J_{E}). Then we have the following formulas:

\begin{matrix}T_{l} = ( \begin{matrix}⟨l⟩_{p}^{−1}T_{l} & 0 \\ 0 & T_{l} \end{matrix} ) if l ≠ p,  T_{p} = ( \begin{matrix}⟨p⟩_{N}V & ∑_{a∈F_{p}^{∗}}⟨a⟩_{p}⟨p⟩_{N}ι^{−1} \\ 0 & F \end{matrix} ) , \\ ⟨a⟩_{N} = ( \begin{matrix}⟨a⟩_{N} & 0 \\ 0 & ⟨a⟩_{N} \end{matrix} ) , ⟨a⟩_{p} = ( \begin{matrix}⟨a⟩_{p}^{−1} & 0 \\ 0 & ⟨a⟩_{p} \end{matrix} ) , g(σ) = ( \begin{matrix}⟨χ(σ)⟩_{p} & 0 \\ 0 & id \end{matrix} ) . \end{matrix} (6.6.1)

where V and F denote “Verschiebung” and “Frobenius”, and where ι: J_{I} → J_{E} is induced by the isomorphism 6.2.5. For a proof of these identities (in a slightly different setting) we refer to [10]. A detailed computation of T_{p}, in a more general case, can be found in [30], §7–8.

The action of the operators above on the torus T can be described in terms of the effect of the correspondences they are induced from on the supersingular points in I (or in E). It follows from 6.3.2 that the ⟨a⟩_{p} and the g(σ) act trivially on T.

Over F_{p}(ζ_{N}) we define automorphisms w_{ζ_{N}} of [Ig(p), Γ_{1}(N)] and [ExIg(p, 1), Γ_{1}(N)] by

(E/S, Q, R) ↦ (E′/S, β^{(p)}Q, R′) and (E/S, P, R) ↦ (E′/S, βP, R′),  (6.6.2)

where β: E → E′ is the N-isogeny with kernel generated by R, where R′ generates ker(β^{t}) and where ⟨R, R′⟩_{β} = ζ_{N}. Then, for a particular choice of isomorphisms in 6.2.3, we have

w = w_{ζ_{pN}} = ( \begin{matrix}0 & w_{ζ_{N}} ⟨−1⟩_{p}⟨p⟩_{N}ι^{−1} \\ ⟨N⟩_{p}^{−1}w_{ζ_{N}} ι & 0 \end{matrix} ) .  (6.6.3)

6.7 The construction of ω(f). Let f be as in 6.1, but suppose that k > 2. Let A ∈ M(1, p−1) be the Hasse invariant. In [16], §12.8 a construction is given of a (p−1)^{th} root of A on [Ig(p)], i.e., a global section A^{1/p−1} of \underline{ω}_{E/[Ig(p)]} such that A = (A^{1/p−1})^{p−1} in \underline{ω}_{E/[Ig(p)]}^{⊗p−1}. The form A^{1/p−1} has simple zeroes at the supersingular points and for all a ∈ F_{p}^{∗} we have ⟨a⟩_{p}^{∗}A^{1/p−1} = aA^{1/p−1}.

For 2 ≤ k ≤ p we have an isomorphism of \overline{F}_{p}-vector spaces:

⨁_{k=2}^{p} M(N, k)^{0} ˜ ⟶ H^{0}([Ig(p), Γ_{1}(N)]\overline{F}_{p}, Ω^{1}),

ϕ_{k} ↦ ω(ϕ_{k}) = KS(pr^{∗}(ϕ_{k})/(A^{1/p−1})^{k−2}),

(6.7.1)

where pr: [Ig(p), Γ_{1}(N)] → [Γ_{1}(N)] is the canonical projection and where KS: \underline{ω}^{⊗2} → Ω^{1}(cusps) is the Kodaira-Spencer morphism on [Ig(p), Γ_{1}(N)] ([16], §12.8.8).

Now let ω_{X/Z_{p}[ζ_{p}]} be the dualizing sheaf on X (see [8], I, §2.3) and let (ω(f), 0) be its section over X\overline{F}_{p} whose restrictions to I\overline{F}_{p} and E\overline{F}_{p} are ω(f) and 0, respectively. Since J^{o} = Pic_{X/Z_{p}[ζ_{p}]}^{o} we have canonical isomorphisms:

Cot(J^{o}) = H^{0}(X, ω_{X/Z_{p}[ζ_{p}]}), Cot(\overline{J} o ; \overline{F}_{p}) = H^{0}(X\overline{F}_{p}, ω_{X\overline{F} ; p/\overline{F}_{p}}),  (6.7.2)

hence we can view (ω(f), 0) as an element of Cot(\overline{J} o ; \overline{F}_{p}). One can check that as such, (ω(f), 0) is an eigenvector for the subring H of End(J^{o}) defined in 6.4:

\begin{matrix}∀l : T_{l}^{∗}((ω(f), 0)) = a_{l}(ω(f), 0)), \\ ∀a ∈ F_{p}^{∗} : ⟨a⟩_{p}^{∗}(ω(f), 0)) = a^{k−2}(ω(f), 0)), \\ ∀a ∈ (Z/NZ)^{∗} : ⟨a⟩_{N}^{∗}(ω(f), 0)) = ε(a)(ω(f), 0)). \end{matrix}  (6.7.3)

To check this one needs to work out the relation between the isomorphism in 6.7.1 and the T_{l}, ⟨a⟩_{N} and ⟨a⟩_{p}, and one uses the formulas in 6.6.

6.8 The proof of Theorem 2.6. Let f be as in 6.7. The action of H on (ω(f), 0) gives us a character λ: H → \overline{F}_{p} such that for all h in H we have h^{∗}(ω(f), 0) = λ(h)(ω(f), 0). Let F = im(λ) and let m = ker(λ); hence m is a maximal ideal of H. Let H_{p} be the p-adic completion of H and let H_{m} be the m-adic completion. Since H, being a subring of End(J_{Q}), is a free Z-module of finite rank, we have

H_{p} = H ⊗_{Z} Z_{p} = H_{m} × H′_{m},  (6.8.1)

where H′_{m} is the product of the completions of H at the maximal ideals over p other than m. Let ϵ and ϵ′ be the corresponding idempotents in H_{p}: 1 = ϵ + ϵ′, ϵ^{2} = ϵ, ϵ^{′2} = ϵ′, ϵϵ′ = 0, ϵH_{p} = H_{m}, ϵ′H_{p} = H′_{m}. Note that for any H_{p}-module M (for example J^{o}[p^{n}](T) for some Z_{p}[ζ_{p}]-scheme T) we have M = ϵM ⊕ ϵ′M = M[ϵ′] ⊕ M[ϵ].

Let η = \frac{1}{p−1} ∑_{a∈F_{p}^{∗}}⟨a⟩_{p} ∈ H_{p}; let η′ = 1−η. Then η and η′ are orthogonal idempotents; for any H_{p}-module M we have M = ηM ⊕ η′M and ηM is the submodule of ⟨a⟩_{p}-invariants of M. Since λ(⟨a⟩_{p}) = a^{k−2} for all a ∈ F_{p}^{∗}, and 3 ≤ k ≤ p, we have ϵη = ϵ, ϵη′ = 0.

Let J^{o}[p^{∞}] be the “p-divisible group” (J^{o}[p^{n}])_{n≥0} over Z_{p}[ζ_{p}] (for the definition of a pdivisible group see for example [29], [6], III or [28], VI). Note that the J^{o}[p^{n}] are quasi-finite and flat but not necessarily finite over Z[ζ_{p}]. Since H_{p} acts on J^{o}[p^{∞}], it splits as a direct sum:

J^{o}[p^{∞}] = G ⊕ G′,  (6.8.2)

with G_{n} = J^{o}[p^{n}][ϵ′] and G′_{n} = J^{o}[p^{n}][ϵ]. We also have the direct sum decomposition J^{o}[p^{∞}] = J^{o}[p^{∞}][η′] ⊕ J^{o}[p^{∞}][η] and G = J^{o}[p^{∞}][η′, ϵ′]. Let J^{o}[p^{∞}]^{f} = (J^{o}[p^{n}]^{f})_{n} be the finite part (i.e., “la partie fixe” in [12], 7, IX, §2.2.3) of J^{o}[p^{∞}]. Since the ⟨a⟩_{p} act trivially on the torus T in \overline{J}^{o} (see 6.6), they act trivially on J^{o}[p^{∞}]/J^{o}[p^{∞}]^{f} by [12], 7, IX, 5.5.8. It follows that all the G_{n} are finite over Z_{p}[ζ_{p}].

Let \overline{G} = G ×_{Z_{p}[ζ_{p}]} F_{p} = \overline{J}^{o}[p^{∞}][ϵ′]. The exact sequence 6.3.1 induces an exact sequence of p-divisible groups over F_{p}:

0 ⟶ T [p^{∞}] ⟶ \overline{J}^{o}[p^{∞}] ⟶ J_{I}[p^{∞}] ⊕ J_{E}[p^{∞}] ⟶ 0.  (6.8.3)

Again, since the ⟨a⟩_{p} act trivially on T, we get an isomorphism

\overline{J}^{o}[p^{∞}][η′] ˜ ⟶ (J_{I}[p^{∞}] ⊕ J_{E}[p^{∞}])[η′] = J_{I}[p^{∞}][η′] ⊕ J_{E}[p^{∞}][η′],  (6.8.4)

where the last equality follows from the fact that the ⟨a⟩_{p} preserve J_{I} and J_{E} (see 6.6.1). It also follows that H_{p} preserves the direct sum decomposition in 6.8.4. Therefore we finally get:

\overline{G} = J_{I}[p^{∞}][η′, ϵ′] ⊕ J_{E}[p^{∞}][η′, ϵ′] = J_{I}[p^{∞}][ϵ′] ⊕ J_{E}[p^{∞}][ϵ′] = \overline{G}_{I} ⊕ \overline{G}_{E},  (6.8.5)

where the last equality is a definition. Since the \overline{G}_{I}_{,n} and the \overline{G}_{E}_{,n} are finite group schemes over the perfect field F_{p}, we get direct sum decompositions into étale, multiplicative and local-local parts:

\overline{G}_{I} = \overline{G}_{I}^{e} ⊕ \overline{G}_{I}^{m} ⊕ \overline{G}_{I}^{l}, \overline{G}_{E} = \overline{G}_{E}^{e} ⊕ \overline{G}_{E}^{m} ⊕ \overline{G}_{E}^{l}.  (6.8.6)

From the fact that T_{p} acts on J_{I} as ⟨p⟩_{N}V, on J_{E} as F (see 6.6.1) and that λ(T_{p}) = a_{p} = 0, we conclude that \overline{G}_{I}^{m} = 0 = \overline{G}_{E}^{e} since V (resp. F) acts both as a unit and a non-unit on \overline{G}_{I}^{m} (resp. \overline{G}_{E}^{e}).

Let us now argue that \overline{G}_{I}^{l} ≠ 0. Since \overline{G}_{I}^{m} = 0, it suffices to show that Cot(\overline{G}_{I}), the cotangent space of \overline{G}_{I} at 0, is not zero (here Cot(\overline{G}_{I}) = lim_{⟵} Cot(\overline{G}_{I,n}) which equals Cot(\overline{G}_{I,1}) since the \overline{G}_{I} is a p-divisible group over F_{p}). But we have:

0 ≠ ω(f) ∈ Cot(J_{I,\overline{F}_{p}})[ϵ^{′∗}] = Cot(J_{I}[p^{∞}]\overline{F}_{p})[ϵ^{′∗}] = Cot(J_{I}[p^{∞}][ϵ′]\overline{F}_{p}) = Cot(\overline{G}_{I}) ⊗_{F_{p}} \overline{F}_{p}, (6.8.7)

where we used that “Cot” is an additive functor and that ϵ′ is an idempotent.

The next thing we want to show is that \overline{G}_{I} and \overline{G}_{E} become Cartier dual over F_{p}(ζ_{N}). Let (·, ·)_{n} denote the Weil pairing on J[p^{n}]_{Q}, and let ⟨·, ·⟩_{n} be the pairing on J[p^{n}]_{Q(ζ_{pN})} given by ⟨x, y⟩_{n} = (x, w(y))_{n}, where w is the automorphism defined in 6.4. It follows from 6.4.4 that the h ∈ H_{p} are self-adjoint with respect to the ⟨·, ·⟩_{n}: ⟨h(x), y⟩_{n} = ⟨x, h(y)⟩_{n} for all x, y and h. In particular this is true for ϵ and ϵ′, which shows that the decomposition 6.8.2, pulled back to Q(ζ_{pN}), is orthogonal with respect to the ⟨·, ·⟩_{n}.

According to [12], 7, IX, Thm. 5.4, the (·, ·)_{n} extend to J^{o}[p^{∞}] and their restrictions to \overline{J}^{o}[p^{∞}] are induced from a principal polarization on J_{I} ⊕ J_{E} which one can check to be equal to the product of the canonical principal polarizations on J_{I} and J_{E}. Since w interchanges I and E it follows that the ⟨·, ·⟩_{n} induce a perfect duality between \overline{G}_{I,F_{p}(ζ_{N})} and \overline{G}_{E,F_{p}(ζ_{N})}. We conclude that \overline{G}_{E} = \overline{G}_{E}^{l} and that Cot(\overline{G}_{E}) ≠ 0. From 6.6.1 and 6.7.3 it follows that g(σ) acts on Cot(\overline{G}_{I}) and Cot(\overline{G}_{E}) by χ(σ)^{2−k} and 1, respectively.

Let ρ_{f}: G_{Q} → GL_{2}(F) be the continuous semi-simple representation corresponding to f as in 2.3. Let W be the 2-dimensional F-vector space scheme over Q corresponding to ρ_{f}. Let K, K′, D and D′ be as in 5.4. By construction, each G[p^{n}](\overline{Q}) has all Jordan-Hölder quotients isomorphic to irreducible subrepresentations of ρ_{f}. Now ρ_{f,p} has to be either in case 2.4.1 or 2.4.2. Suppose that ρ_{f,p} is in case 2.4.1. Then there is an exact sequence

0 ⟶ W′_{K} ⟶ W_{K} ⟶ W′′_{K} ⟶ 0,

where W′_{K} and W′′_{K} are 1-dimensional F-vector space schemes over K, corresponding to characters Gal(\overline{K}/K) → F^{∗} of level 1. But then W′_{K′} and W′′_{K′} are constant, hence all G[p^{n}]_{K′} have all Jordan-Hölder quotients isomorphic to F_{K′}. It follows from [21], §3.3.5 that then \overline{G}[p^{n}] = \overline{G}[p^{n}]^{e} ⊕ \overline{G}[p^{n}]^{m}, but this is nonsense since we know that \overline{G}[p^{n}] = \overline{G}[p^{n}]^{l}. We conclude that ρ_{f,p} is in case 2.4.2, say with characters ϕ and ϕ′ of level 2. In that case ρ_{f,p} is absolutely irreducible hence certainly ρ_{f} is absolutely irreducible. A recent result by Boston, Lenstra and Ribet ([2], Theorems 1 and 2) states that G[m](\overline{Q}) is isomorphic to a direct sum of d copies of ρ_{f} for some d > 0. By [21], §3.3.5 both W_{K′} and G[m]_{K′} have unique extensions to W_{D′} and G[m] over D′. It follows that G[m] ≅ ⊕_{i=1}^{d}W_{D′} and that this isomorphism is compatible with the action of Gal(K′/K).

Let F′ be a finite extension of F containing F_{p^{2}}. Then W_{K} ⊗_{F} F′ splits as F′-vector space scheme over K into a direct sum V ⊕ V′ where V and V′ correspond to the characters ϕ and ϕ′: Gal(\overline{K}/K) → F_{p^{2}}^{∗} ⊂ F^{′∗}, respectively. Let V (ϕ) and V (ϕ′) be the F_{p^{2}}-vector space schemes over K corresponding to ϕ and ϕ′: Gal(\overline{K}/K) → F_{p^{2}}^{∗}. Let r′ = [F′ : F_{p^{2}}]. Then we have V ≅ ⊕_{1}^{r′}V (ϕ) and V′ ≅ ⊕_{1}^{r′}V (ϕ′) as F_{p^{2}}-vector space schemes over K. Again, since local-local group schemes extend uniquely over D′, we get an isomorphism, compatible with the action of Gal(K′/K), of F_{p^{2}}-vector space schemes over D′:

G[m] ⊗_{F} F′ ≅ ( ⊕_{1}^{d} ⊕_{1}^{r′}V (ϕ)_{D′}) ⨁ ( ⊕_{1}^{d} ⊕_{1}^{r′}V (ϕ′)_{D′}).  (6.8.8)

Since \overline{G}[m] = \overline{G}_{I}[m]⊕\overline{G}_{E}[m], the action of Gal(K′/K) on Cot(\overline{G}[m]) consists of the characters χ^{2−k} and 1, both with positive multiplicities. Since V (ϕ) and V (ϕ′) are isomorphic as group schemes over K, the Gal(K′/K)-representations Cot(V (ϕ)\overline{F}_{p}) and Cot(V (ϕ′)\overline{F}_{p}) are isomorphic. Now write ϕ = ψ^{a_{0}}ψ^{′a_{1}} with ψ and ψ′ the fundamental characters of level 2 and a_{0} > a_{1}. It follows that Gal(K′/K) must act on Cot(V (ϕ)\overline{F}_{p}) and Cot(V (ϕ′)\overline{F}_{p}) each via the characters χ^{2−k} and 1, both with positive multiplicities. By 5.5.2 and 5.5.1 this means that we must have a_{0} = k − 1 and a_{1} = 0, i.e., ϕ = ψ^{k−1}, ϕ′ = ψ^{′k−1}. This completes the proof of Theorem 2.6 for k > 2.

The proof for k = 2 is similar but much easier, so we only give a sketch. Instead of X_{1}(pN) we use the curve X := X_{1}(N) and its jacobian J over Z_{p}; X and J have good reduction over Z_{p}. The Kodaira-Spencer isomorphism then gives us a differential form ω(f) on X\overline{F}_{p}. Let H be the subring of End(J) generated by all T_{l} and ⟨a⟩_{N}. We have T_{l}^{∗}ω(f) = a_{l}ω(f) and ⟨a⟩_{N}^{∗}ω(f) = ε(a)ω(f) for all l and a. We define λ, F, m, H_{m}, ϵ, ϵ′, G and \overline{G} as above. Since T_{p} acts on \overline{J} as F + ⟨p⟩_{N}V and a_{p} = 0, we have \overline{G} = \overline{G}^{l} and we get two characters ϕ = ψ^{a_{0}}ψ^{′a_{1}} and ϕ′ = ϕ^{p} as above. The fact that G has good reduction over Z_{p} then implies that a_{0} = 1 and a_{1} = 0. □

The proof above actually also proves the following result, that we will use in §9.

6.9 Proposition. With the notation as above in the case k > 2, let \overline{W} = W_{Z_{p}[ζ_{p}]} ⊗_{Z_{p}[ζ_{p}]} F_{p}. Then Gal(Q_{p}(ζ_{p})/Q_{p}) acts on \overline{W} via the characters χ^{2−k} and χ^{0} = 1. Let \overline{W} = \overline{W} (2−k)⊕\overline{W} (0) be the decomposition into eigenspaces. Then \overline{W} (2−k) and \overline{W} (0) are “1-dimensional” F-vector space schemes (i.e., they satisfy condition (∗∗) in §5.1) over F_{p} that are annihilated by F and by V. Let d be the multiplicity of ρ_{f} in J_{Q}[m](\overline{Q}) as above. Then

J_{I}[m] ≅ ⊕_{1}^{d} \overline{W} (2 − k) and J_{E}[m] ≅ ⊕_{1}^{d} \overline{W} (0),

where the action of H on J_{I} and J_{E} is described by the diagonal entries of the matrices in 6.6.1.

With the notation as above in the case k = 2, let \overline{W} = W_{Z_{p}} ⊗_{Z_{p}} F_{p}. Then \overline{W} [F] = \overline{W} [V] is a “1-dimensional” F-vector space scheme over F_{p} and

\overline{J}[m] ≅ ⊕_{1}^{d} \overline{W} .

7 Reduction to weight at most p+1.

7.1. In this section we will give a proof of Thm. 3.4, using certain spaces S(N, k) that were introduced in [27]. Let p be any prime number and let N ≥ 5 be prime to p; let X := X_{1}(N)\overline{F}_{p}. For any integer k, multiplication by the Hasse-invariant gives a short exact sequence of sheaves on X:

0 ⟶ \underline{ω}^{⊗k−p+1} ⟶^{·A} \underline{ω}^{⊗k} ⟶ ⊕_{x s.s.}ω_{E_{x}}^{⊗k} ⟶ 0.  (7.1.1)

Let S(N, k) = H^{0}(X, ⊕_{x s.s.}ω_{E_{x}}^{⊗k}); this is the space of modular forms of type (N, k) on supersingular elliptic curves. We let the T_{l}, l ≠ p, act on the S(N, k) by the usual formulas (see 2.2). The long exact sequences arising from 7.1.1, together with the following proposition, show that ⊕_{k}M(N, k) and ⊕_{k}S(N, k) have the same systems of eigenvalues for the T_{l}^{∗}, l ≠ p. To see this, consider the S(N, k) and M(N, k) as Artinian modules over the polynomial ring \overline{F}_{p}[T_{l} | l ≠ p]; systems of eigenvalues correspond to maximal ideals, etc.

7.2 Proposition. There exists an element B in S(N, p+1) inducing isomorphisms

·B: S(N, k) ˜ ⟶ S(N, k + p + 1),

for all k ∈ Z, such that T_{l}^{∗}(Bf) = lBT_{l}^{∗}(f) for all f ∈ S(N, k) and l ≠ p. For all k ∈ Z there exists an isomorphism, compatible with the T_{l}^{∗}, l ≠ p, between S(N, k+p^{2}−1) and S(N, k).

Proof. We will give three constructions of such a B.

1. According to Robert, [23], Thm. B, one may take B to be the reduction mod p of the Eisenstein series E_{p+1}, if p ≥ 5.

2. Let x ∈ s.s.X(\overline{F}_{p}). Since A has a simple zero at x, it gives a non-zero element in ω_{E_{x}}^{⊗p−1} ⊗ m_{x}/m_{x}^{2} (here m_{x} is the maximal ideal in O_{X,x}). The Kodaira-Spencer isomorphism then gives a non-zero element in ω_{E_{x}}^{⊗p+1}.

3. (See [15].) Let E be a supersingular elliptic curve over \overline{F}_{p}. There is an exact sequence 0 → H^{0}(E, Ω) → H_{DR}^{1}(E) → H^{1}(E, O) → 0. Since F: E → E^{(p)} induces 0: H^{0}(E^{(p)}, Ω) → H^{0}(E, Ω) and 0: H^{1}(E^{(p)}, O) → H^{1}(E, O), it induces a map H^{1}(E^{(p)}, O) → H^{0}(E, Ω). This map is an isomorphism; via Serre duality we can interpret it as an element of ω_{E_{x}}^{⊗p+1}.

It follows easily from lemma 7.4 that the elements B as described in (2) and (3) have the desired property. Multiplication by B^{p−1} is the desired isomorphism S(N, k) → S(N, k+p^{2}−1). Another way to get such an isomorphism is as follows (see [27]). Every supersingular elliptic curve over \overline{F}_{p} has a unique F_{p^{2}}-structure for which the Frobenius endomorphism is −p. Every morphism between supersingular elliptic curves over \overline{F}_{p} descends to F_{p^{2}}. It follows that each ω_{E_{x}}^{⊗p^{2}−1} has a canonical \overline{F}_{p}-basis; these bases are respected by all isogenies of degree prime to p. □

Proposition 7.2 shows that all systems of eigenvalues occurring in ⊕_{k}M(N, k) occur, up to twist, in ⊕_{1≤k≤p+1}S(N, k); we want to prove the same for ⊕_{1≤k≤p+1}M(N, k). The relevant part of the long exact sequence arising from 7.1.1 is:

M(N, k) ⟶ S(N, k) ⟶ H^{1}(X, \underline{ω}^{⊗k−p+1}).  (7.2.1)

Let k′ = p+1−k, then Serre duality and the Kodaira-Spencer map give an isomorphism:

H^{1}(X, \underline{ω}^{⊗k−p+1}) ⟶^{SD} H^{0}(X, Ω ⊗ \underline{ω}^{⊗p−1−k})^{∨} ⟶^{KS^{∨}} H^{0}(X, \underline{ω}^{⊗k′}(−cusps))^{∨} = M^{0}(N, k′)^{∨}.

Let ζ ∈ \overline{F}_{p}^{∗} have order N. Let w be the automorphism of [Γ_{1}(N)]\overline{F}_{p} sending (E/S, α) to (wE/S, wα), where β: E → wE is the N-isogeny with ker(β) = im(α), im(wα) = ker(β^{t}) and ⟨α(1), (wα)(1)⟩_{β} = ζ (compare 6.4.3). We let w act on modular forms by: (w^{∗}f)(E, α) = β^{∗}f(wE, wα). 7.3 Proposition. Let Φ be the composition of the map S(N, k) → H^{1}(X, \underline{ω}^{⊗k−p+1}) as in 7.2.1 and w^{∗∨}^{∘}(KS)^{∨}^{∘}SD. For all k and all l ≠ p the following diagram is commutative:

M(N, k) ⟶ S(N, k) ⟶^{Φ} M^{0}(N, k′)^{∨}
↓ T_{l}^{∗}  ↓ T_{l}^{∗}  ↓ l^{k−1}T_{l}^{∗∨}
M(N, k) ⟶ S(N, k) ⟶^{Φ} M^{0}(N, k′)^{∨}.

Proof. Let l ≠ p. Let Y be the smooth proper connected curve over \overline{F}_{p} parametrizing pairs (ϕ: E → E′, α), with ϕ an l-isogeny of elliptic curves over an \overline{F}_{p}-scheme S and α ∈ [Γ_{1}(N)](E/S), such that ϕ^{∘}α is injective. There are two maps, S and T (for “source” and “target”) from Y to X corresponding to (ϕ: E → E′, α) ↦ (E, α) and (ϕ: E → E′, α) ↦ (E′, ϕ^{∘}α), respectively. Let E/X be the universal curve over X and let ϕ: S^{∗}E → T^{∗}E be the universal isogeny over Y. Then ϕ induces an isomorphism ϕ^{∗}: T^{∗}\underline{ω}^{⊗k} ˜→ S^{∗}\underline{ω}^{⊗k}. For i ≥ 0 we define a Hecke operator T_{l}^{∗} on H^{i}(X, \underline{ω}^{⊗k}) by:

lT_{l}^{∗}: H^{i}(X, \underline{ω}^{⊗k}) ⟶^{T^{∗}} H^{i}(Y, T^{∗}\underline{ω}^{⊗k}) ⟶^{ϕ^{∗}} H^{i}(Y, S^{∗}\underline{ω}^{⊗k}) ⟶^{S_{∗}} H^{i}(X, \underline{ω}^{⊗k}),

where S_{∗} is induced by the trace map S_{∗}S^{∗}\underline{ω}^{⊗k} = \underline{ω}^{⊗k} ⊗ S_{∗}O_{Y} → \underline{ω}^{⊗k}. For i = 0 these are the T_{l}^{∗} defined as before. The exact sequence 7.2.1 is compatible with the T_{l}^{∗}. Now consider the following diagram:

H^{1}(X, \underline{ω}^{⊗k−p+1}) ⟶^{SD} H^{0}(X, Ω_{X} ⊗ \underline{ω}^{⊗p−1−k})^{∨}  ⟶^{KS^{∨}} H^{0}(X, \underline{ω}^{⊗k′}(−cusps))^{∨}
↓ T^{∗}  ↓ T_{∗}^{∨}  ↓ T_{∗}^{∨}
H^{1}(Y, T^{∗}\underline{ω}^{⊗k−p+1}) ⟶^{SD} H^{0}(Y, Ω_{Y} ⊗ T^{∗}\underline{ω}^{⊗p−1−k})^{∨} ⟶^{KS_{T^{∗}E}^{∨}} H^{0}(Y, T^{∗}\underline{ω}^{⊗k′}(−cusps))^{∨}
↓ ϕ^{∗}  ↓ l^{k}(id⊗ϕ^{t∗})^{∨}  ↓ l^{k−1}ϕ^{t∗∨}
H^{1}(Y, S^{∗}\underline{ω}^{⊗k−p+1}) ⟶^{SD} H^{0}(Y, Ω_{Y} ⊗ S^{∗}\underline{ω}^{⊗p−1−k})^{∨} ⟶^{KS_{S^{∗}E}^{∨}} H^{0}(Y, S^{∗}\underline{ω}^{⊗k′}(−cusps))^{∨}
↓ S_{∗}  ↓ S^{∗∨}  ↓ S^{∗^{∨}}
H^{1}(X, \underline{ω}^{⊗k−p+1}) ⟶^{SD} H^{0}(X, Ω_{X} ⊗ \underline{ω}^{⊗p−1−k})^{∨}  ⟶^{KS^{∨}} H^{0}(X, \underline{ω}^{⊗k′}(−cusps))^{∨}.

(7.3.1)

We claim that this diagram is commutative. For the upper left square, this follows from tr_{Y} = tr_{X}^{∘}T_{∗}, where tr_{X}: H^{1}(X, Ω_{X}) ˜→\overline{F}_{p} and tr_{Y}: H^{1}(Y, Ω_{Y}) ˜→\overline{F}_{p} are the trace maps used in the construction of Serre duality. For the lower left square one uses tr_{Y} = tr_{X}^{∘}S_{∗}. Consider the middle left square. Let ⟨·, ·⟩_{T} be the natural pairing

T^{∗}\underline{ω}^{⊗k−p+1} × Ω_{Y} ⊗ T^{∗}\underline{ω}^{⊗p−1−k} ⟶ Ω_{Y}

and let ⟨·, ·⟩_{S} be the analog of this for S. Then one has:

⟨ϕ^{∗}·, ·⟩_{S} = ⟨ϕ^{t∗}ϕ^{∗}·, (id ⊗ ϕ^{t∗})·⟩_{T} = l^{k−p+1}⟨·, (id ⊗ ϕ^{t∗})·⟩_{T} .

The upper and lower right squares are commutative just because the Kodaira-Spencer morphism is compatible with base change. The commutativity of the middle right square follows from the following lemma.

7.4 Lemma. Let S → T be any morphism of schemes; let ϕ: E → E′ be a morphism of elliptic curves over S. Then the following diagram is commutative:

ω_{E/S}^{⊗2}  ⟵^{ϕ^{∗}}  ω_{E′/S}^{⊗2}
KS_{E/S} ↘  ↙ (deg ϕ)KS_{E′/S}
Ω_{S/T}

Proof. One uses the formula for the Kodaira-Spencer map in terms of the Gauss-Manin connection and the cup product on H_{DR}^{∗}, compare [10], Prop. 3.15. □

Let T_{l}^{t∗} be the operator l^{−1}T_{∗}ϕ^{t∗}S^{∗} on M^{0}(N, k′). The commutativity of 7.3.1 implies that KS^{∨}^{∘}SD^{∘}T_{l}^{∗} = l^{k−1}T_{l}^{t∗∨}^{∘}KS^{∨}^{∘}SD. The formula for T_{l}^{t∗} in the style of 2.2.1 is:

(T_{l}^{t∗}f)(E/\overline{F}_{p}, α) = \frac{1}{l} ∑_{ϕ} ϕ^{t∗}(f(E′/\overline{F}_{p}, α′)),  (7.4.1)

where the sum is over the l-isogenies ϕ: E′ → E of elliptic curves with Γ_{1}(N)-structures. We leave it to the reader to check that w^{∗}T_{l}^{t∗} = T_{l}^{∗}w^{∗}. This finishes the proof of Prop. 7.3. □

7.5 Proof of Theorem 3.4. Let us first note that the functor Hom\overline{F}_{p}(·, \overline{F}_{p}), on the category of modules of finite \overline{F}_{p}-dimension over the polynomial ring \overline{F}_{p}[T_{l} | l ≠ p], preserves supports. Using this, Theorem 3.4 is a consequence of the existence of the long exact cohomology sequence of 7.1.1 combined with Propositions 7.2 and 7.3, if N ≥ 5. We are now left with the cases N = 1, 2, 3 or 4. If N = 4 or N = 2 (hence p ≠ 2) we have M(N, k) = H^{0}(X, \underline{ω}^{⊗k})^{G}, where X = X(4)^{ζ−can} (ζ some 4^{th} root of 1 in \overline{F}_{p}) and G the subgroup \{({1 \atop 0} {∗ \atop 1})\} or \{({1+2∗ \atop 0} {∗ \atop 1+2∗})\} of SL_{2}(Z/4Z). Note that p |̸#G. If N = 3 we are in the same situation by taking X = X(3)^{ζ−can} and G = \{({1 \atop 0} {∗ \atop 1})\}. Finally, if N = 1 and p > 3 we take X = X(3)^{ζ−can} and G = SL_{2}(Z/3Z) or X = X(4)^{ζ−can} and G = SL_{2}(Z/4Z). In all these cases the functor “G-invariants” from \overline{F}_{p}[G]- modules to \overline{F}_{p}-vector spaces is exact and for any \overline{F}_{p}[G]-module V, the canonical map V^{G} → V_{G} from invariants to co-invariants is an isomorphism. We have an exact sequence as in Prop. 7.3:

H^{0}(X, \underline{ω}^{⊗k}) ⟶ ⊕_{xs.s.}ω_{E_{x}}^{⊗k} ⟶^{Φ} H^{0}(X, \underline{ω}^{⊗k′}(−cusps))^{∨}  (7.5.1)

where Φ is the composition of ⊕_{xs.s.}ω_{E_{x}}^{⊗k} → H^{1}(X, \underline{ω}^{⊗k−p+1}) and KS^{∨}^{∘}SD. We let the T_{l}, l ≠ p (and l ≠ 2 or 3 if N = 1), act on the last term in 7.5.1 by l^{k−1}T_{l}^{∗∨}. We let g ∈ G act on the three terms in 7.5.1 by g^{∗}, g^{∗} and g^{−1∗∨} respectively. The sequence 7.5.1 is then compatible with all these actions. Theorem 3.4 now follows by taking G-invariants, except for N = 1 and p = 2 or 3. For these remaining two cases we refer to [25], Théorème 3. Let us remark that it is not hard to give a proof in these last two cases that is similar to the proof above (just compute the systems of eigenvalues in the sum of all S(N, k) and show directly that they occur in ⊕_{k=2}^{p+1}M(N, k)). □

8 Finiteness at p.

8.1. Let p be a prime, let F be a finite extension of F_{p} and let r := [F : F_{p}]. Let ρ_{p}: G_{p} → GL_{2}(F) be a continuous representation of a decomposition group G_{p} in G_{Q} (see §2.4). We suppose that ρ_{p} is of the form:

ρ_{p} = ( \begin{matrix}χε_{2} & ∗ \\ 0 & ε_{1} \end{matrix} ) ,  (8.1.1)

with ε_{1} and ε_{2} unramified F^{∗}-valued characters of G_{p}. Let V_{Q_{p}} be the F-vector space scheme over Q_{p} defined by ρ_{p} (see §5.1). Let D be the integral closure of Z_{p} in the maximal unramified extension K of Q_{p} in \overline{Q}_{p}.

8.2 Proposition. The following conditions are equivalent:

1. ρ_{p} is “peu ramifié” (see [26], 2.4.7),

2. V_{Q_{p}} can be extended to a finite flat F-vector space scheme V_{D} over D,

3. V_{Q_{p}} can be extended to a finite flat F-vector space scheme V_{Z_{p}} over Z_{p},

4. V_{Q_{p}} can be extended to a finite flat group scheme V_{D} over D,

5. V_{Q_{p}} can be extended to a finite flat group scheme V_{Z_{p}} over Z_{p}.

If these conditions are satisfied we say that ρ_{p} is finite.

Proof. The implications (3) ⇒ (2) ⇒ (4) and (3) ⇒ (5) ⇒ (4) are obvious. Now suppose that (4) holds. Let V_{D} be the maximal (see [21], 2.2.3) extension of V_{K} to a finite flat group scheme over D. Then the actions of F and Gal(K/Q_{p}) on V_{K} extend uniquely to V_{D}. This shows that V_{D} extends V_{K} as F-vector space scheme and that the descent data from K to Q_{p} for V_{K} extend to descent data for V_{D}. We have now shown that (2), (3), (4) and (5) are all equivalent.

8.3 Lemma. Let W be an F-vector space scheme over a scheme S. For any extension 0 → W → V → F_{S} → 0 of F-vector space schemes (i.e., the sequence is an exact sequence of f.p.p.f.-sheaves on S) and an element a in F let V_{a} be the inverse image in V of the section a of F_{S}. Then the assignment V ↦ V_{1} induces an equivalence between the category of extensions of F_{S} by W and the category of torsors, on S, for the underlying additive group scheme of W.

Proof. For V such an extension and a ∈ F^{∗}, multiplication by a on V induces an isomorphism V_{1}[a] → V_{a}, where V_{1}[a] is the torsor obtained from V_{1} by twisting the action of W by multiplication by a on W. This shows how to reconstruct the extension from V_{1}. □

Because of 8.1.1, V_{K} sits in an exact sequence:

0 ⟶ F_{K}^{D} ⟶ V_{K} ⟶ F_{K} ⟶ 0,  (8.3.1)

where F_{K}^{D} is the Cartier dual of F_{K}. Let us choose an F_{p}-basis of F, i.e., an isomorphism ⊕_{i=1}^{r}F_{p} → F of additive groups. By Lemma 8.3, extensions as 8.3.1 are classified up to isomorphism by H_{f.p.p.f.}^{1}(K, F^{D}) ≅ ⊕_{i=1}^{r}H_{f.p.p.f.}^{1}(K, μ_{p}). The long exact cohomology sequence arising from 0 → μ_{p} → G_{m} → G_{m} → 0, plus the fact that Pic(K) = {0}, show that H_{f.p.p.f.}^{1}(K, μ_{p}) = K^{∗}/K^{∗p}. Under this identification, [x] ∈ K^{∗}/K^{∗p} corresponds to the μ_{p}-torsor of p^{th} roots of x.

In the same way, extensions of F_{D} by F_{D}^{D} are classified by ⊕_{i=1}^{r}D^{∗}/D^{∗p}. If V_{K} corresponds to the element ([x_{1}], …, [x_{r}]), then V_{K} can be extended to an extension of F_{D} by F_{D}^{D} if and only if ρ_{p} is “peu ramifié”, i.e., v(x_{i}) ≡ 0 (p) for all i, where v is the valuation on K. For p > 2 any extension of V_{K} to V_{D} as in (2) is automatically an extension of F_{D} by F_{D}^{D} (see [21], 3.3.3). If p = 2 and an extension V_{D} as in (2) exists, the functoriality of Ext and a non-zero morphism F_{D} → F_{D}^{D} give an extension V_{D} of F_{D} by F_{D}^{D}. This completes the proof of Proposition 8.2. □

8.4. Let ρ_{p} and V_{Q_{p}} be as in §8.1. The fact that the extension 8.3.1 comes from an extension

0 ⟶ V′′_{Q_{p}} ⟶ V_{Q_{p}} ⟶ V′_{Q_{p}} ⟶ 0,

with V′_{Q_{p}} and V′′_{Q_{p}} twists of F_{Q_{p}}^{D} and F_{Q_{p}}, gives restrictions on the torsor V_{K,1} corresponding to 8.3.1. To be precise, one finds that

σ^{∗}V_{K,1} ≅ V_{K,1}[λ_{1}λ_{2}^{−1}],  (8.4.1)

where σ is the automorphism of Spec(D) with σ^{#}(x) ≡ x^{p} mod p and λ_{i} = ε_{i}(σ^{#}), i ∈ {1, 2}. Let λ = λ_{1}λ_{2}^{−1} ∈ F^{∗} and let X^{s} + a_{n−1}X^{s−1} + ⋯ + a_{0} be its minimal polynomial over F_{p}. Let e_{j}, 1 ≤ j ≤ r/s, be an F_{p}(λ)-basis of F. Then F has an F_{p}-basis λ^{i}e_{j}, with 0 ≤ i < s and 1 ≤ j ≤ r/s. Now F = ⊕_{j=1}^{r/s}F_{p}(λ)e_{j} is a decomposition of F into λ-stable F_{p}-subspaces of F. Let V_{K,1} correspond to ([x_{i,j}])_{i,j} ∈ ⊕_{i,j}K^{∗}/K^{∗p}. Then V_{K,1}[λ] and σ^{∗}V_{K,1} correspond to ([y_{i,j}])_{i,j} and ([σ^{#}x_{i,j}])_{i,j} respectively, where y_{i,j} = x_{i+1,j} if i < s − 1 and y_{s−1,j} = x_{0,j}^{−a_{0}}x_{1,j}^{−a_{1}} ⋯ x_{s−1,j}^{−a_{s−1}}. Hence we have the equations:

∀j : [σ^{#}x_{i,j}] = [x_{i+1,j}] if i < s − 1 and [σ^{∗}x_{s−1,j}] = [x_{0,j}^{−a_{0}}x_{1,j}^{−a_{1}} ⋯ x_{s−1,j}^{−a_{s−1}}].  (8.4.2)

Now let α_{i,j} = v([x_{i,j}]) ∈ F_{p}. Then equations 8.4.2 give:

∀j : α_{0,j} = α_{1,j} = ⋯ = α_{s−1,j} and α_{0,j}(1 + a_{s−1} + ⋯ + a_{0}) = 0.  (8.4.3)

From this we conclude that either α_{i,j} = 0 for all (i, j) (i.e., ρ_{p} is “peu ramifié”) or 1 + a_{s−1} + ⋯ + a_{0} = 0 (i.e., λ = 1 and ε is trivial).

8.5 Proposition. (Compare [26], §2.4, Remarques.) If ρ_{p} is “très ramifié” then ε_{1} = ε_{2}, all x_{i} in [26], (2.4.6) can be chosen in Q_{p}^{∗} and the integer m in [26], (2.4.6), equals 1 or 2 if p > 2 and 1 or 2 or 3 if p = 2. □

Suppose now that ρ_{p} is “peu ramifié”, i.e., α_{i,j} = 0 for all i and j. Let U := {x ∈ D^{∗} | x ≡ 1 (p)} and let τ: \overline{F}_{p} → D be the Teichmüller lift. Then we have an isomorphism Z × U × \overline{F}_{p}^{∗} → K^{∗}, sending (n, u, a) to p^{n}uτ(a). Moreover, if p > 2, we can view the [x_{i,j}] as elements of \overline{F}_{p} via the isomorphism

D^{∗}/D^{∗p} = U/U^{p} ˜ ⟶ \overline{F}_{p}, [1 + pa] ↦ a mod p.  (8.5.1)

The equations 8.4.2 then give the following proposition.

8.6 Proposition. Suppose that p > 2 and that ρ_{p} is “peu ramifié”. Let λ = (ε_{1}ε_{2}^{−1})(Frob_{p}), let s = [F_{p}(λ) : F_{p}] and let n be the order of λ in F^{∗}. Let f ∈ F_{p}[X] be the minimal polynomial of λ. Then the [x_{i,j}] ∈ D^{∗}/D^{∗p}, defined in §8.4, satisfy [x_{i,j}] = (σ^{#})^{i}[x_{0,j}] for all j, and the image y of any of them under the isomorphism 8.5.1 satisfies (f(Frob_{p}))(y) = 0. This last equation has all p^{s} solutions in F_{p^{n}} since f divides X^{n} − 1 in F_{p}[X]. Conversely, any set of [x_{i,j}] satisfying the equations just mentioned comes from a ρ_{p} that is “peu ramifié”. □

9 Multiplicity one results.

9.1. In this section we will extend some results of Mazur, Ribet and Gross that state that in certain cases the Galois representation ρ_{f} occurs with multiplicity one in a certain subspace of the p-torsion of the jacobian of the modular curve that is used to construct ρ_{f}. These results are not used in this article.

9.2 Theorem. Let f be a cuspidal eigenform of type (N, k, ε), defined over \overline{F}_{p}, with p |̸N and 2 ≤ k ≤ p+1. Let J_{Q} be the jacobian of the curve X_{1}(pN)_{Q} if k > 2 and let J_{Q} be the jacobian of X_{1}(N)_{Q} if k = 2. Let H be the subring of End(J_{Q}) generated by all T_{l} and ⟨a⟩_{N}, and ⟨b⟩_{p} if k > 2. Let m be the maximal ideal of H corresponding to f, and let F = H/m ⊂ \overline{F}_{p}. Suppose that the representation ρ_{f}: G_{Q} → GL_{2}(\overline{F}_{p}) is irreducible. Then J_{Q}(\overline{Q})[m] is an F-vector space of dimension two in each of the following cases:

1. 2 ≤ k < p,

2. k = p and a_{p}^{2} ≠ ε(p), where T_{p}^{∗}f = a_{p}f,

3. k = p and ρ_{f} is ramified at p,

4. k = p+1 and there is no form g of type (N, 2, ε) with ρ_{g} ≅ ρ_{f}.

Proof. Case (4) is dealt with in [18], Main Theorem, for ε = 1, but their proof works in general. Cases (1) and (2) are dealt with in [10], Thm. 12.10(1), assuming that a_{p} ≠ 0 and that f is a newform. The assumption that f is a newform is not used in the proof of [10], Thm. 12.10(1).

It remains to prove (1) and (2) in the case a_{p} = 0, and (3). To do this, let us first take care of all cases with a_{p} = 0. Suppose that a_{p} = 0 and that 2 < k ≤ p. Let G, \overline{G}, \overline{G}_{I} etc. be as in §6.8 and Prop. 6.9. The arguments that follow are motivated by pages 116–118 in [17]. Let M(−) denote the contravariant Dieudonné module functor of [19]. Note that since we work with group schemes over F_{p}, the Dieudonné ring Z_{p}[F, V] is commutative. Let M = M(J_{E}[m]); the reason to work with J_{E} and not with J_{I} will be explained below. By Prop. 6.9 we have M ≅ ⊕_{1}^{d}M(\overline{W} (0)), where M(\overline{W} (0)) is a one-dimensional F-vector space on which F and V act by zero. Since M(−) is contravariant and exact we have M = M(J_{E}[p])/mM(J_{E}[p]). According to Cor. 5.11 in [19], there is a functorial isomorphism

M(J_{E}[p]) ˜ ⟶ H_{DR}^{1}(J_{E}).

Under this isomorphism, the Hodge filtration

0 ⟶ H^{0}(J_{E}, Ω^{1}) ⟶ H_{DR}^{1}(J_{E}) ⟶ H^{1}(J_{E}, O) ⟶ 0

corresponds to the filtration

0 ⟶ M(J_{E}[F]) ⟶ M(J_{E}[p]) ⟶ M(J_{E}[V]) ⟶ 0.

Applying − ⊗_{H} F to the isomorphism H^{1}(J_{E}, O) ˜→ M(J_{E}[V]) = M(J_{E}[p])/V M(J_{E}[p]) gives an isomorphism

H^{1}(J_{E}, O) ⊗_{H} F ˜ ⟶ M/V M ≅ ⊕_{1}^{d}(M(\overline{W} (0))/V M(\overline{W} (0))) = ⊕_{1}^{d}M(\overline{W} (0)).

This isomorphism shows that d=1 if and only if H^{1}(J_{E}, O) ⊗_{H} F is a one-dimensional Fvector space. Serre’s duality on E gives an isomorphism of H-modules between the F_{p}-dual of H^{1}(J_{E}, O) ⊗_{H} F and H^{0}(E, Ω_{E}^{1})[m], where h ∈ H acts on H^{0}(E, Ω_{E}^{1}) = H^{0}(J_{E}, Ω_{J_{E}}^{1}) by h^{t∗}. In particular, T_{p} acts on H^{0}(E, Ω_{E}^{1}) by F^{t∗} = V^{∗}. This explains why we work with J_{E}; working with J_{I} results in T_{p} acting on H^{0}(I, Ω_{I}^{1}) by V^{t∗} = F^{∗} = 0. Lemma 9.3 finishes the proof for a_{p}=0 and 2 < k ≤ p.

Suppose now that a_{p}=0 and k=2. Let X := X_{1}(N)_{F_{p}} and let J be the jacobian of X. Let M = M(\overline{G}[m]) = M(J[m]). By Prop. 6.9 we have M ≅ ⊕_{1}^{d}M(\overline{W}), where M(\overline{W}) is a twodimensional F-vector space on which F and V have rank one. As above we get an isomorphism

H^{1}(J, O) ⊗_{H} F ˜ ⟶ M/V M ≅ ⊕_{1}^{d}M(\overline{W})/V M(\overline{W}).

We conclude that d=1 if and only if H^{0}(X, Ω)[m] is a one-dimensional F-vector space, where h ∈ H acts on H^{0}(X, Ω)[m] by h^{t∗}. Let ζ ∈ \overline{F}_{p} be a primitive N^{th} root of unity and let w be the automorphism of X\overline{F}_{p} as defined in §6.4. Then we have h^{t} = whw^{−1} for all h in H. Hence w^{∗}: H^{0}(X, Ω) → H^{0}(X, Ω) is an isomorphism of H-modules, with h ∈ H acting on source and target by h^{t∗} and h^{∗}, respectively. The result now follows from the “standard” formulas for the action of the T_{l}^{∗} on q-expansions.

To prove that d=1 in case (3), we may assume that a_{p}≠0. Let W be an F-vector space scheme over Q such that W (\overline{Q}) gives ρ_{f}. Let J_{Q} be the jacobian over Q used in the construction of ρ_{f}. Let H and m be as before. By [2], Thm’s. 1 and 2, we have J_{Q}[m] ≅ ⊕_{1}^{d}W. From the construction of ρ_{f} as in [10], §11–12 (compare our §6) it follows that for any embedding W ↪ J_{Q}[m], the Zariski closure W_{Z_{p}[ζ_{p}]} of W in J_{Z_{p}[ζ_{p}]} is finite and flat over Z_{p}[ζ_{p}]. By Thm. 2.5 such a W_{Z_{p}[ζ_{p}]} sits in a short exact sequence

0 ⟶ W′ ⟶ W_{Z_{p}[ζ_{p}]} ⟶ W′′ ⟶ 0,  (9.2.1)

where W′ and W′′ are “one-dimensional” F-vector space schemes over Z_{p}[ζ_{p}]. By Raynaud’s result [21], §3.3.5, W′ and W′′ can only be etale or multiplicative. The assumption that ρ_{f} is ramified at p means that the exact sequence 9.2.1, when restricted to Q_{p}(ζ_{p}), does not split. Now one remarks that this means that W′′ is etale and W′ is multiplicative. Indeed, an extension of etale by etale is etale, hence unramified; an extension of multiplicative by multiplicative splits (consider the Cartier duals); an extension of multiplicative by etale splits (use the connected component of identity), compare [26], page 190. It follows that J_{Z_{p}[ζ_{p}]}[m]^{e} is an F-vector space scheme of dimension d. But according to [10], Thm.’s 12.8 and 12.9 the dimension of the etale part is one, hence d = 1. □

9.3 Lemma. Let H act on H^{0}(E, Ω_{E}) = H^{0}(J_{E}, Ω_{J_{E}}) as above: h ∈ H acts by h^{t∗}. Let m ⊂ H be a maximal ideal with F = H/m of characteristic p. Then H^{0}(E, Ω_{E})[m] is a F-vector space of dimension at most one.

Proof. (Compare [17], Chapter II, Prop. 9.3.) Let λ: F → \overline{F}_{p}. For l prime let a_{l} = λ(T_{l}). Let ε: (Z/NZ)^{∗} → \overline{F}_{p}^{∗} be given by ε(a) = λ(⟨a⟩_{N}). Let d ∈ Z/(p−1)Z be determined by λ(⟨b⟩_{p}) = b^{d} for all b ∈ F_{p}^{∗}. What we must show is that the intersection, in H^{0}(E\overline{F}_{p}, Ω_{E\overline{F} ; p}), of the kernels of T_{l}^{t∗} − a_{l}, ⟨a⟩_{N}^{t∗} − ε(a) and ⟨b⟩_{p}^{t∗} − b^{d} for all l prime, a ∈ (Z/NZ)^{∗} and b ∈ F_{p}^{∗} has dimension at most one over \overline{F}_{p}. To do this, we consider q-expansions at the cusp (Tate(q^{pN}), q^{N}, q^{p}) of E. Let ω be a differential form on E\overline{F}_{p} and let ω(q) = (∑ b_{n}q^{n})dq/q be its q-expansion. Suppose that ⟨a⟩_{N}^{t∗}ω = ε(a)ω and ⟨b⟩_{p}^{t∗}ω = b^{d}ω for all a and b. Then one finds the following formulas (compare [14], §1.11):

(T_{l}^{t∗}ω)(q) = \{ \begin{matrix}(∑_{n≥1}(b_{nl} + ε(l)^{−1}l^{1−d}b_{n/l})q^{n})dq/q & if l |̸pN \\ (∑_{n≥1} b_{nl}q^{n})dq/q & if l|pN \end{matrix}  (9.3.1)

These formulas clearly show what we need.

Another way to prove this lemma is to use an automorphism w of E\overline{F}_{p} with the property that T_{l}^{t} = wT_{l}w^{−1}⟨l^{−1}⟩_{p} for all l ≠ p (compare §6.4). Then one can use the “standard” formulas for the action of the T_{l} on q-expansions. □

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