MODULARITY OF CERTAIN POTENTIALLY CRYSTALLINE GALOIS REPRESENTATIONS

BRIAN CONRAD, FRED DIAMOND, AND RICHARD TAYLOR

Introduction

Conjectures of Langlands, Fontaine and Mazur [16] predict that certain Galois representations

ρ : Gal(\overline{Q}/Q) → GL_{2}(\overline{Q}_{ℓ})

(where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(\overline{Q}/Q) on the ℓ-adic Tate module of an elliptic curve defined over Q, and so implies the Shimura-Taniyama-Weil conjecture.

Wiles’ breakthrough in [34], completed by [33] and extended in [10], provided results of the form

\overline{ρ} modular ⇒ ρ modular

where \overline{ρ} is the reduction of ρ. These results were subject to hypotheses on the local behavior of ρ at ℓ, i.e., the restriction of ρ to a decomposition group at ℓ, and to irreducibility hypotheses on \overline{ρ}. In this paper, we build on the methods of [34], [33] and [10] and relax the hypotheses on local behavior. In particular, we treat certain ℓ-adic representations which are not semistable at ℓ, but potentially semistable.

We do this using results of [5] generalizing a theorem of Ramakrishna [24] (for crystalline representations) along lines suggested by the work of Fontaine and Mazur [16]. These results show that certain potentially crystalline deformation problems are smooth, allowing us to define certain universal deformations for \overline{ρ} with the necessary Galois-theoretic properties to apply Wiles’ method. To carry out the proof that these deformations are indeed realized in the cohomology of modular curves (i.e., that the universal deformation rings are Hecke algebras), we need to identify the corresponding cohomology groups and prove they have the modular-theoretic properties needed to apply Wiles’ method. As in [14] and [10], the identification is made by matching local behavior of automorphic representations and Galois representations via the local Langlands correspondence (together with Fontaine’s theory at the prime ℓ). We work directly with cohomology of modular curves instead of using the Jacquet-Langlands correspondence, and we use the simplification of [34] provided by [11] and Fujiwara [17] independently.

The main technical restriction in this paper is that we only treat representations which arise from ℓ-divisible groups over tamely ramified extensions of Q_{ℓ}. There are further hypotheses which might be removed with suitable generalizations of Fontaine’s “slope 1/2” theory [16]. We do obtain sufficiently strong results here to give the following application:

Date: June 20, 1997 (Preliminary version).

One of the authors (B.C.) was supported by an N.S.F. Postdoctoral Fellowship. Another (F.D.)

was supported by the C.N.R.S. during some of the research and is grateful to Université de Paris-Sud for its hospitality. All of the authors are grateful to Centre Émile Borel at the Institut Henri Poincaré for its hospitality.

Theorem 0.1. If E is an elliptic curve over Q with conductor not divisible by 27, then E is modular.

Note that the hypothesis on the conductor of E is satisfied if and only if E acquires semistable reduction over a tamely ramified extension of Q_{3}.

1. Deformation Algebras

1.1. Notation. We fix an odd prime ℓ and algebraic closures \overline{Q}, \overline{R} = C, and \overline{Q}_{p} for all p. Choose embeddings of \overline{Q} into C and \overline{Q}_{p} for all p, so we realize the Galois group G_{p} = Gal(\overline{Q}_{p}/Q_{p}) as a decomposition group inside of G_{Q} = Gal(\overline{Q}/Q). Let I_{p} denote the inertia subgroup, Frob_{p} ∈ G_{p}/I_{p} the arithmetic Frobenius element, and W_{p} the Weil subgroup of G_{p} (i.e., the preimage of the subgroup of G_{p}/I_{p} ≃ Ẑ generated by Frob_{p}.) We define \overline{F}_{p} to be the residue field of the valuation ring of \overline{Q}_{p}. The order p^{n} subfield of \overline{F}_{p} is denoted F_{p^{n}}.

For any field F of characteristic distinct from ℓ and having a fixed choice of separable closure F_{s}, with G_{F} = Gal(F_{s}/F) the resulting Galois group, define ϵ : G_{F} → Z_{ℓ}^{×} to be the ℓ-adic cyclotomic character. Define ω : G_{F} → Z_{ℓ}^{×} to be the Teichmüller lift of ϵ mod ℓ. For any Z_{ℓ}[G_{F}]-module V, define V (n) = V ⊗_{Z_{ℓ}} ϵ^{n} for all n ∈ Z. For a representation ρ of G_{F} and L/F a subextension of F_{s}/F, let ρ|_{L} denote ρ|_{G_{L}}.

We will let η_{p,n} denote the character I_{p} → W (F_{p^{n}})^{×} obtained from the reciprocity map F^{×} → G_{F}^{ab} of local class field theory, where F is the field of fractions of W (F_{p^{n}}). We write η_{n} for η_{ℓ,n}, ϵ_{n} for η_{n}^{−1}, ω_{n} : I_{ℓ} → F_{ℓ^{n}}^{×} for the reduction mod ℓ of ϵ_{n} and ω̃_{n} for the Teichmüller lift of ω_{n}. Thus ϵ_{1} = ϵ and ω̃_{n} satisfies

σ(ℓ^{1/(ℓ^{n}−1)}) = ω̃_{n}(σ)ℓ^{1/(ℓ^{n}−1)}.

1.2. Potentially Barsotti-Tate representations. Fix a finite extension K of Q_{ℓ} in \overline{Q}_{ℓ} with valuation ring 𝒪 and residue field k. Consider a continuous representation

ρ : G_{ℓ} → GL(M)

where M is a vector space of finite dimension d over K. Fontaine [15] has defined what it means for ρ to be potentially semistable. If ρ is potentially semi-stable, then one can associate to it a continuous representation

W D(ρ) : W_{ℓ} → GL(D)

where D is a vector space over \overline{Q}_{ℓ} of dimension d. This representation is defined by choosing a sufficiently large finite extension F of Q_{ℓ} so that

D_{st,F} (M) = (B_{st} ⊗_{Q_{ℓ}} M)^{G_{F}}

is free of rank d over F_{0} ⊗_{Q_{ℓ}} K, where F_{0} denotes the maximal unramified subextension of F. (See [15] and [16, §10] for definitions and further details.) The module D_{F} = D_{st,F} (M) ⊗_{K} \overline{Q}_{ℓ} is equipped with a Frob_{ℓ} ⊗ 1-linear bijective endomorphism ϕ, a semi-linear action of Gal(F/Q_{ℓ}) which commutes with ϕ (and a filtration after extending scalars to F ⊗_{Q_{ℓ}} \overline{Q}_{ℓ} which will play no role here). We define a representation of W_{ℓ} on D_{F} by letting σ act as the product of its image in Gal(F/Q_{ℓ}) and ϕ^{−n} if σ maps to Frob_{ℓ}^{n} in W_{ℓ}/I_{ℓ}. As this representation commutes with the action of ϕ we see that D_{F} is isomorphic to F ⊗_{Q_{ℓ}} D for a well-defined representation D of W_{ℓ} with coefficients in \overline{Q}_{ℓ}. Note that I_{F} acts trivially so W D(ρ) is continuous, even with \overline{Q}_{ℓ} given the discrete topology. One also finds that W D(ρ) is independent of the choice of F as above, and that if K′ is a finite extension of K, then W D(ρ ⊗_{K} K′) isomorphic to W D(ρ).

Suppose that F is a finite extension of Q_{ℓ} with ring of integers 𝒪_{F} and L is a free 𝒪-module of finite rank with a continuous action of G_{F}. The representation

ρ′ : G_{F} → GL(L)

is called Barsotti-Tate if there is an ℓ-divisible group Γ over 𝒪_{F} such that

L ≅ proj lim_{n} Γ[ℓ^{n}](\overline{F})

as a Z_{ℓ}[G_{F}]-module. The notion depends only on the representation (ρ′ ⊗_{𝒪} \overline{Q}_{ℓ})|_{I_{F}}, and we refer to ρ′ ⊗_{𝒪} K as being Barsotti-Tate whenever ρ′ is. We say also that a representation ρ : G_{ℓ} → GL(M) is Barsotti-Tate over F if ρ|_{F} is Barsotti-Tate. If ρ is Barsotti-Tate over F, then it is also potentially semistable and W D(ρ) is trivial on I_{F}.

1.3. Types of local deformations. Fix a continuous two-dimensional representation

\overline{ρ} : G_{ℓ} → GL(V)

over k such that End_{k[G_{ℓ}]} V = k. One then has a universal deformation ring R_{V,𝒪} for \overline{ρ} (see appendix A).

An ℓ-type is an equivalence class of two-dimensional representations

τ : I_{ℓ} → GL(D)

over \overline{Q}_{ℓ} with open kernel. For each ℓ-type τ, we shall define a certain quotient R_{V,𝒪}^{D} of the complete local Noetherian 𝒪-algebra R_{V,𝒪} .

A deformation ρ of V to the the ring of integers 𝒪′ of a finite extension of K in \overline{Q}_{ℓ}, is of type τ if

1. ρ is Barsotti-Tate over F for any finite extension F of Q_{ℓ} such that τ|_{I_{F}} is trivial;

2. the restriction of W D(ρ) to I_{ℓ} is equivalent to τ;

3. the character ϵ^{−1} det ρ is the Teichmüller lift of the prime-to-ℓ order character ω^{−1} det \overline{ρ} : G_{ℓ} → \overline{F}_{ℓ}^{×}.

We say that a prime ideal 𝔭 of R_{V,𝒪} is of type τ if there exist a K′ and an 𝒪-algebra homomorphism R_{V,𝒪} → 𝒪′ with kernel 𝔭 such that the corresponding deformation is of type τ. If 𝔭 is of type τ, then so is the deformation corresponding to any 𝒪-algebra homomorphism with kernel 𝔭 and values in some 𝒪′.

If there do not exist any prime ideals 𝔭 of type τ, we define R_{V,𝒪}^{D} = 0. Otherwise, define R_{V,𝒪}^{D} to be the quotient of R_{V,𝒪} by the intersection of all 𝔭 of type τ. We say that a deformation of \overline{ρ} is weakly of type τ if the associated local 𝒪-algebra map R_{V,𝒪} → R factors through the quotient R_{V,𝒪}^{D} . We say that τ is acceptable for \overline{ρ} if R_{V,𝒪}^{D} ≠ 0 and if there is a surjective local 𝒪-algebra map 𝒪[[X]] ↠ R_{V,𝒪}^{D} .

It is straightforward to check that the above notions are well-behaved with respect to extension of the field K. In particular, if K′ is a finite extension of K with valuation ring 𝒪′ and residue field k′, then 𝒪′ ⊗_{𝒪} R_{V,𝒪}^{D} is naturally isomorphic to R_{V ⊗_{k}k′,𝒪′}^{D} and τ is acceptable for \overline{ρ} if and only if τ is acceptable for \overline{ρ} ⊗_{k} k′.

We make the following conjectures, although they are considerably stronger than the results that will actually be important in the sequel. What is important for the sequel is only the question of which ℓ-types are acceptable for a given \overline{ρ}|_{G_{ℓ}}.

Conjecture 1.4. A deformation ρ : G_{ℓ} → GL(M) of \overline{ρ} to 𝒪′ is weakly of type τ if and only if it is of type τ.

Conjecture 1.5. Suppose that τ = ω̃^{i} ⊕ ω̃^{j}. Then R_{V,𝒪}^{D} ≠ (0) if and only if \overline{ρ}|_{I_{ℓ}} ⊗_{k} k̄ is of one of the following three forms:

• ( \begin{matrix}ω^{1+i} & ∗ \\ 0 & ω^{j} \end{matrix} ) and in the case j ≡ i mod ℓ − 1, ∗ is peu-ramifié (in the sense

of Serre [30]),

• ( \begin{matrix}ω^{1+j} & ∗ \\ 0 & ω^{i} \end{matrix} ) and in the case j ≡ i mod ℓ − 1, ∗ is peu-ramifié,

• ω_{2}^{1+{j−i}+(ℓ+1)i} ⊕ ω_{2}^{ℓ−{j−i}+(ℓ+1)j}, where {a} denotes the unique integer in

the range from 0 to ℓ − 2 congruent to a modulo ℓ − 1. In the first two of these three cases R_{V,𝒪}^{D} ≅ 𝒪[[X]] and so τ is acceptable for \overline{ρ}. In the last case, if j ≡ i mod ℓ − 1, then R_{V,𝒪}^{D} ≅ 𝒪[[X]] and so τ is acceptable for \overline{ρ}.

Conjecture 1.6. Suppose that τ = ω̃_{2}^{m} ⊕ ω̃_{2}^{ℓm} where m ∈ Z/(ℓ^{2} − 1)Z and m = i + (ℓ + 1)j with i = 1, …, ℓ and j ∈ Z/(ℓ − 1)Z. Then R_{V,𝒪}^{D} ≠ (0) if and only if \overline{ρ}|_{I_{ℓ}} ⊗_{k} k̄ is of one of the following four forms:

• ( \begin{matrix}ω^{i+j} & ∗ \\ 0 & ω^{1+j} \end{matrix} ) and in the case i = 2, ∗ is peu-ramifié,

• ( \begin{matrix}ω^{1+j} & ∗ \\ 0 & ω^{i+j} \end{matrix} ) and in the case i = ℓ − 1, ∗ is peu-ramifié,

• ω_{2}^{1+m} ⊕ ω_{2}^{ℓ(1+m)},

• ω_{2}^{ℓ+m} ⊕ ω_{2}^{1+ℓm}, In all these cases R_{V,𝒪}^{D} ≅ 𝒪[[X]] and so τ is acceptable for \overline{ρ}.

It will be convenient to say that a type τ is strongly acceptable for \overline{ρ}|_{G_{ℓ}} if it is acceptable and if one of the following is true:

• \overline{ρ}|_{I_{ℓ}} ≅ ( \begin{matrix}ω^{m} & ∗ \\ 0 & ω^{n} \end{matrix} ) for some m, n ∈ Z/(ℓ − 1)Z (and in the case m −

1 ≡ n mod ℓ − 1, ∗ is peu-ramifié), and τ is equivalent to ω̃^{m−1} ⊕ ω̃^{n} or ω̃_{2}^{{m−n}+1+(n−1)(ℓ+1)} ⊕ ω̃_{2}^{ℓ({m−n}+1)+(n−1)(ℓ+1)}.

• \overline{ρ}|_{I_{ℓ}} ⊗_{k} k̄ ≅ ω_{2}^{1+m} ⊕ ω_{2}^{ℓ(1+m)} for some m ∈ Z/(ℓ^{2} − 1)Z, and τ ≅ ω̃_{2}^{m} ⊕ ω̃_{2}^{ℓm} Note that in particular we are conjecturing that we can suppress the assumption of acceptability in the definition of strong acceptability. In section 2 we shall explain what we know about these conjectures.

1.7. Global Galois representations. Fix a finite extension K of Q_{ℓ} in \overline{Q}_{ℓ} with valuation ring 𝒪, uniformizer λ and residue field k, and a continuous global twodimensional Galois representation

\overline{ρ} : G_{Q} → GL(V).

We shall suppose \overline{ρ} satisfies the following hypotheses.

• The determinant of \overline{ρ}(c) is −1, where c denotes complex conjugation.

• The restriction of \overline{ρ} to Q(\sqrt{(−1)^{(ℓ−1)/2}ℓ}) is absolutely irreducible.

• The conductor of \overline{ρ} (see [30]) divides the conductor of all of its twists by k̄^{×}-valued characters.

• The centraliser of \overline{ρ}(G_{ℓ}) consists only of scalar matrices.

We remark that the second condition is simply for convenience and will be removed as an assumption from all our main theorems by an easy twisting argument. Let S be a finite set of rational primes which does not contain ℓ, and let τ : I_{ℓ} → GL(D) be an ℓ-type which is strongly acceptable for \overline{ρ}|_{G_{ℓ}}.

We will say that a deformation ρ : G_{Q} → GL(M) of \overline{ρ} to an object R of 𝒞_{𝒪}^{∗} (see appendix A) is of type (S, τ) if the following hold:

• ρ|_{G_{ℓ}} is weakly of type τ.

• If p ∉ S ∪ \{ℓ\} and the order of \overline{ρ}(I_{p}) is not ℓ, then ρ(I_{p}) →^{∼} \overline{ρ}(I_{p}).

• If p ∉ S ∪ \{ℓ\} and the order of \overline{ρ}(I_{p}) is ℓ, then M/M^{I_{p}} is free of rank one over R.

• ϵ^{−1} det ρ has finite order prime to ℓ.

One checks that the subsets of deformations of type (S, τ) satisfies the representability criterion in appendix A, and we let R_{V,𝒪}^{S,D} denote the universal deformation ring. We write simply R_{V,𝒪} and R_{V,𝒪}^{D} for the deformation rings associated to \overline{ρ}|_{G_{ℓ}}. Then R_{V,𝒪} , R_{V,𝒪}^{D} and R_{V,𝒪}^{S,D} are complete local Noetherian 𝒪-algebras with residue field k, well-behaved with respect to extension of K.

1.8. Galois cohomology. We will let ad^{0} V denote the representation of G_{Q} on the trace zero endomorphisms of V. The trace pairing gives rise to a Galois equivariant perfect pairing

ad^{0} V × ad^{0} V (1) → k(1).

If 𝔪 denotes the maximal ideal of R_{V,𝒪}^{D} , then there is a natural map from the dual of 𝔪/(λ, 𝔪^{2}) to H^{1}(Q_{ℓ}, ad^{0} V). Using the trace pairing and local Poitou-Tate duality we get a map from H^{1}(Q_{ℓ}, ad^{0} V (1)) → 𝔪/(λ, 𝔪^{2}). We will denote the kernel of this map H_{D}^{1}(Q_{ℓ}, ad^{0} V (1)). We will let H_{S,D}^{1}(Q, ad^{0} V (1)) denote those classes in H^{1}(Q, ad^{0} V (1)) which localise trivially at all primes in S, to an element of H_{D}^{1}(Q_{ℓ}, ad^{0} V (1)) at ℓ and to an element of H^{1}(F_{p}, (ad^{0} V (1))^{I_{p}}) at all primes p ∉ S∪{ℓ}. The usual calculations give rise to the following lemma (see for example section 2 of [7], especially corollary 2.43).

Lemma 1.9. Suppose that τ is acceptable for \overline{ρ}|_{G_{ℓ}}. Then R_{V,𝒪}^{S,D} can be topologically generated as an 𝒪-algebra by

dim_{k} H_{S,D}^{1}(Q, ad^{0} V (1)) + ∑_{p∈S} dim_{k} H^{0}(Q_{p}, ad^{0} V (1))

elements.

The proof of this lemma makes essential use of the assumption that τ is acceptable for \overline{ρ}|_{G_{ℓ}}.

Now suppose that we are given a deformation ρ : G_{Q} → GL(M) of \overline{ρ} of type (S, τ) to 𝒪. Let θ^{S,D} : R_{V,𝒪}^{S,D} → 𝒪, θ^{D} : R_{V,𝒪}^{D} → 𝒪 and θ : R_{V,𝒪} → 𝒪 denote the corresponding homomorphisms. We let ad M and ad^{0} M denote the representations of G_{Q} on the endomorphisms and trace-zero endomorphisms of M respectively. Let H_{D}^{1}(Q_{ℓ}, ad^{0} M ⊗_{𝒪} K/𝒪) denote the image of

Hom_{𝒪} (ker θ^{D}/(ker θ^{D})^{2}, K/𝒪) ⟶ Hom_{𝒪} (ker θ/(ker θ)^{2}, K/𝒪)
≅ H^{1}(Q_{ℓ}, ad M ⊗_{𝒪} K/𝒪).

It is easy to see that the image is in fact contained in

H^{1}(Q_{ℓ}, ad^{0} M ⊗_{𝒪} K/𝒪) ⊂ H^{1}(Q_{ℓ}, ad M ⊗_{𝒪} K/𝒪).

We let H_{S,D}^{1}(Q, ad^{0} M ⊗_{𝒪} K/𝒪) denote the set of elements of H^{1}(Q, ad^{0} M ⊗_{𝒪} K/𝒪) which localise at ℓ to an element of H_{D}^{1}(Q_{ℓ}, ad^{0} M ⊗_{𝒪} K/𝒪) and at each prime p ∉ S ∪ {ℓ} to an element of H^{1}(F_{p}, (ad^{0} M ⊗_{𝒪} K/𝒪)^{I_{p}}). Again the usual calculations give the following lemma (see for example section 2 of [7]).

Lemma 1.10. In the above situation we have an isomorphism

Hom_{𝒪} (ker θ^{S,D}/(ker θ^{S,D})^{2}, K/𝒪) ≅ H_{S,D}^{1}(Q, ad^{0} M ⊗_{𝒪} K/𝒪).

Corollary 1.11. Suppose that p ∉ S ∪ {ℓ}, let S′ = S ∪ {p} and let θ^{S′,D} denote the composite R_{V,𝒪}^{S′,D} → R_{V,𝒪}^{S,D} → 𝒪. Then

length_{𝒪} (ker θ^{S′,D})/(ker θ^{S′,D})^{2} ≤ length_{𝒪} (ker θ^{S,D})/(ker θ^{S,D})^{2} + length_{𝒪} ad^{0} M(1)^{I_{p}}/(Frob_{p} − 1) ad^{0} M(1)^{I_{p}}.

2. Local Calculations

In this section, we study local deformation problems. Consider a continuous two-dimensional representation \overline{ρ} : G_{ℓ} → GL(V) over k. Choose a subfield F ⊆ \overline{Q}_{ℓ} with finite degree over Q_{ℓ} and with absolute ramification index e = e(F) ≤ ℓ − 1. We write A for the valuation ring of F, and I_{F} for the inertia subgroup of G_{F}.

A finite discrete G_{ℓ}-module ρ : G_{ℓ} → Aut(M) is called A-flat if there exists a finite flat commutative group scheme 𝒢 over A and a G_{F} -module isomorphism ρ|_{G_{F}} ≃ 𝒢(\overline{Q}_{ℓ}).

2.1. A-ordinary deformations. Assume now that \overline{ρ} is reducible and End_{k[G_{ℓ}]} V = k. Then \overline{ρ} is isomorphic to a non-semisimple representation

\overline{ρ} ≃ (\begin{matrix}η_{a}ω^{m} & ∗ \\ 0 & η_{b}ω^{n} \end{matrix})

for some a, b ∈ k^{×}, m, n ∈ Z/(ℓ−1) with a ≠ b or m ≠ n (where η_{x} : G_{ℓ} → k^{×} is the unique continuous unramified character sending Frob_{ℓ} to x ∈ k^{×}). If m ≠ n + 1 or a ≠ b, then dim_{k} H^{1}(G_{ℓ}, (η_{b}ω^{n})^{−1}η_{a}ω^{m}) = 1, so the non-split condition implies that \overline{ρ} is determined up to k[G_{ℓ}]-module isomorphism by the specification (and ordering) of its diagonal characters. See the discussion above [5, Thm 2.4.4] for more details on this. In the case m = n + 1, we assume ∗ is peu ramifié. This condition is automatically satisfied if a ≠ b, and it is satisfied for a unique isomorphism class of representations in the case a = b.

Let F denote the unramified extension of Q_{ℓ}(μ_{ℓ}) of degree |k^{×}|. We say that a deformation M of \overline{ρ} to an object R of 𝒞_{𝒪}^{∗} is A-ordinary if its determinant is ϵη, with η the Teichmüller lift of (det \overline{ρ})ω^{−1} : G_{ℓ} → k^{×}, and there is an exact sequence of R[G_{ℓ}]-modules

0 → M^{(−1)} → M → M^{(0)} → 0

with M^{(−1)} and M^{(0)} free of rank one over R, and I_{F} acting via ϵ on M^{(−1)} and trivially on M^{(0)}. (Note that the G_{ℓ} actions on M^{(−1)}/𝔪_{R} and M^{(0)}/𝔪_{R} are necessarily via η_{a}ω^{m} and η_{b}ω^{n} respectively.) Using the criterion in Appendix A, one checks that there is a universal A-ordinary deformation of \overline{ρ}, where in the case m = n + 1, we impose a further restriction gotten from the property of finite G_{ℓ}- modules M, that M ⊗_{Z_{ℓ}} Z_{ℓ}(ω̃^{−n}) be Z_{ℓ}-flat. We let R_{V,𝒪}^{A−ord} denote the universal A-ordinary deformation ring, canonically a quotient of R_{V,𝒪} .

We shall now define a subspace of H^{1}(G_{ℓ}, ad^{0} V) denoted H_{A−ord}^{1}(G_{ℓ}, ad^{0} V). Let W^{(0)} denote the subspace of ad^{0} V consisting of those trace-zero endomorphisms preserving the filtration

0 → V^{(−1)} → V → V^{(0)} → 0,

and let W^{(−1)} = Hom_{k}(M^{(0)}, M^{(−1)}) ⊂ W^{(0)}. There are thus canonical exact sequences

0 ⟶ W^{(0)} ⟶ ad^{0} V ⟶ Hom_{k}(M^{(−1)}, M^{(0)}) ⟶ 0, and
0 ⟶ W^{(−1)} ⟶ W^{(0)} ⟶ Hom_{k}(M^{(−1)}, M^{(−1)}) ⟶ 0.

We define a subspace C_{1} of H^{1}(I_{ℓ}, W^{(−1)}) as follows: If m ≠ n + 1, we let C_{1} = H^{1}(I_{ℓ}, W^{(−1)}). If m = n + 1, then W^{(−1)} ≅ k(1) as k[I_{ℓ}]-modules, and we define C_{1} as the kernel of the map

H^{1}(I_{ℓ}, W^{(−1)}) ≅ H^{1}(I_{ℓ}, k(1)) ≅ (Q_{ℓ}^{un})^{×} ⊗ k → k,

where the second isomorphism is the Kummer map and the last homomorphism is induced by the valuation on Q_{ℓ}^{un}. Let C_{2} denote the image of C_{1} in H^{1}(I_{ℓ}, W^{(0)}), let C_{3} denote the preimage of C_{2} in H^{1}(G_{ℓ}, W^{(0)}), and finally, let H_{A−ord}^{1}(G_{ℓ}, ad^{0} V) be the image of C_{2} under the natural inclusion.

Theorem 2.2. There is a surjective homomorphism of 𝒪-algebras

𝒪[[T]] → R_{V,𝒪}^{A−ord}.

Proof. Letting 𝔪 denote the maximal ideal R_{V,𝒪}^{A−ord}, it suffices to prove that 𝔪/(λ, 𝔪^{2}) is at most one-dimensional over k. This is done by checking that the image of Hom_{k}(𝔪/(λ, 𝔪^{2}), k) in H^{1}(G_{ℓ}, ad V) is contained in the subspace H_{A−ord}^{1}(G_{ℓ}, ad^{0} V), and then computing the dimension of this subspace. Both steps are now standard; see [34, I.2] for the case m ≠ n + 1 and [7, 2.4] for the case m = n + 1. ■

Now suppose that ρ : G_{ℓ} → GL_{2}(𝒪′) is a deformation of \overline{ρ} of type τ, with τ = ω̃^{i} ⊕ ω̃^{j} for some i and j. In this case there is an ℓ-divisible group Γ over A with an action of 𝒪′ such that ρ|_{G_{F}} is isomorphic to the representation defined by the action of 𝒪′[G_{F}] on the Tate module of Γ:

M = proj lim Γ[ℓ^{n}](\overline{Q}_{ℓ}).

The canonical connected-étale sequence for Γ gives rise to an exact sequence

0 → M^{0} → M → M^{ét} → 0

of free 𝒪′-modules with an action of G_{F}.

Lemma 2.3. The 𝒪′-modules M^{0} and M^{ét} are each free of rank one, and I_{F} acts via ϵ on M^{0}.

Proof. First observe that it suffices to prove that M^{ét} ≠ 0, for I_{F} acts trivially on M^{ét} and det ρ|_{I_{F}} = ϵ.

Suppose now that M^{ét} = 0, so Γ[ℓ] is connected. Since V ≅ M/λM has a nonzero element fixed by G_{F}, the same is true of

(M/ℓM)[λ] ⊂ Γ[ℓ](\overline{Q}_{ℓ}).

Therefore there is a nontrivial map

μ_{ℓ,F} → Γ[ℓ]_{F} → Γ[ℓ].

Since Γ[ℓ] is connected and e = ℓ−1, the schematic closure of the image is isomorphic to μ_{ℓ,A}. It follows that the Cartier dual of Γ[ℓ] is not connected.

Now let Γ^{D} denote the dual ℓ-divisible group, and M^{D} its Tate module. As 𝒪′[G_{F}]-modules, we have

M^{D} ≅ Hom_{Z_{ℓ}}(M, Z_{ℓ}(1)) ≅ Hom_{𝒪′} (M, 𝒪′(1))

(the first isomorphism is canonical, the second depends on a choice of generator for the different of 𝒪′). We have just proved that Γ^{D}[ℓ] is not connected, so M^{D,ét} ≠ 0. The lemma follows on observing that the first paragraph of the proof now applies to M^{D}, showing that I_{F} acts via ϵ on M^{D,0} ≠ 0. ■

We can now deduce the following from Theorem 2.2.

Corollary 2.4. Suppose that \overline{ρ} is reducible with End_{k[G_{ℓ}]} V = k. If \overline{ρ} admits a lifting of type τ : I_{ℓ} → GL(D) (i.e., if R_{τ} (\overline{ρ}) ≠ 0) with τ ≅ ω̃^{i} ⊕ ω̃^{j}, then τ is acceptable for \overline{ρ}.

Proof. Let 𝔭 be a prime ideal of R_{V,𝒪} of type τ, so 𝔭 is the kernel of a homomorphism R_{V,𝒪} → 𝒪′ such that the associated deformation to 𝒪′ is of type τ. The lemma shows this deformation is A-ordinary, so the map R_{V,𝒪} → 𝒪′ factors through and R_{V,𝒪}^{A−ord}. Therefore the kernel of the map

R_{V,𝒪} → R_{V,𝒪}^{A−ord}

is contained in the intersection of all prime ideals 𝔭 of type τ. It follows that R_{V,𝒪}^{D} is a quotient of R_{V,𝒪}^{A−ord}, hence a quotient of 𝒪[[T]]. ■

2.5. A-flat deformations. Now we return to the case of arbitrary F with e ≤ ℓ−1. We no longer assume that \overline{ρ} is reducible. We assume that V is A-flat with 𝒢 connected and having connected dual. Under these conditions 𝒢 is the unique finite flat A-group scheme with generic fiber G_{F} -module isomorphic to \overline{ρ}_{0}|_{G_{F}}, with 𝒢 determined up to unique isomorphism (see [5, §2.1] for details; the connectedness conditions ensure uniqueness when e = ℓ − 1).

Let M denote the (contravariant) Dieudonne module of the closed fiber of 𝒢. We assume that the canonical sequence of groups

0 → M/V M →^{F} M = M/ℓM → M/F M → 0

is exact. This exactness condition is automatically satisfied when 𝒢 ≃ Γ[ℓ] for Γ_{/A} an ℓ-divisible group, and so is an extremely natural condition; also, it is needed in the results we will require from [5] below.

We now give the complete list of isomorphism classes of representations \overline{ρ} satisfying the above conditions. (See Corollary 2.2.3, Lemma 2.4.2 and Theorem 2.4.4 of [5].) The irreducible \overline{ρ} which arise are exactly those continuous \overline{ρ} : G_{ℓ} → GL_{2}(k) satisfying \overline{ρ}|_{I_{ℓ}} ≃ ω_{ℓ,2}^{m} ⊕ ω_{ℓ,2}^{ℓm}, with em ≡ e mod ℓ^{2} − 1, The reducible \overline{ρ} which arise can be written as non-semisimple representations

\overline{ρ} ≃ (\begin{matrix}η_{a}ω^{m} & ∗ \\ 0 & η_{b}ω^{n} \end{matrix})

for a, b ∈ k^{×}, m, n ∈ Z/(ℓ − 1) . The precise possibilities are as follows. There must exist an integer j satisfying 1 ≤ j ≤ e − 1, en ≡ j mod ℓ − 1, e|j(ℓ + 1) and m = n + 1 − j(ℓ + 1)/e mod ℓ − 1. These conditions imply e |̸(ℓ − 1), m ≠ n, and for ℓ ≡ −1 mod 4 they also imply m ≠ n + 1. For ℓ ≡ −1 mod 4, the unique (up to isomorphism) non-split extensions of η_{b}ω^{n} by η_{a}ω^{m} for a, b ∈ k^{×} and n and m as above are exactly the \overline{ρ} which arise. For ℓ ≡ 1 mod 4, we get the same list, except that the cases with m = n + 1 (which occur precisely when e = (ℓ + 1)/2, n = (ℓ − 1)/2) and a = b are given by the unramified k-twists the (unique up to isomorphism) non-split F_{ℓ}-representation of the form

ω^{(ℓ−1)/2} ⊗ (\begin{matrix}ω & ∗ \\ 0 & 1 \end{matrix})

for which ∗ is peu-ramifié.

The property of being A-flat is closed under taking submodules, quotients and finite products, and since End_{k[G_{ℓ}]} V = k for the representations above, there is an associated universal A-flat deformation ring R_{V,𝒪}^{A−flat} which is a quotient of R_{V,𝒪} . Consider also the fixed determinant character χ : G_{ℓ} → 𝒪^{×} of the form ϵη, with η the Teichmüller lift of (det \overline{ρ})ω^{−1} : G_{ℓ} → k^{×}. There is a universal deformation of \overline{ρ} which is A-flat with determinant χ, and the universal deformation ring is naturally a quotient of R_{V,𝒪}^{A−flat} which we denote R_{V,𝒪}^{A−flat,χ}. The main deformation-theoretic result we need in the local theory is provided by Theorems 4.1.1 and 4.1.2 of [5].

Theorem 2.6. There are 𝒪-algebra isomorphisms

R_{V,𝒪}^{A−flat} ≃ 𝒪[[T_{1}, T_{2}]] and R_{V,𝒪}^{A−flat,χ} ≃ 𝒪[[T]].

Corollary 2.7. If \overline{ρ} admits a lifting of type τ : I_{ℓ} → GL(D) (i.e., if R_{τ} (\overline{ρ}) ≠ 0) and τ is trivial on the inertia group of F, then τ is acceptable for \overline{ρ}.

Proof. Let 𝔭 be a prime ideal of R_{V,𝒪} of type τ, so 𝔭 is the kernel of a homomorphism R_{V,𝒪} → 𝒪′ such that the associated deformation to 𝒪′ is of type τ. Since this deformation is A-flat and has determinant χ, the map R_{V,𝒪} → 𝒪′ factors through R_{V,𝒪}^{A−flat,χ}. Therefore the kernel of the map

R_{V,𝒪} → R_{V,𝒪}^{A−flat,χ}

is contained in the intersection of all prime ideals 𝔭 of type τ. It follows that R_{V,𝒪}^{D} is a quotient of R_{V,𝒪}^{A−flat,χ}, which, by the preceding theorem, is isomorphic to 𝒪[[T]]. ■

2.8. Twisted A-flat deformations. We now consider a variant of the A-flat deformation problem. We still fix a finite extension F/Q_{ℓ} with e(F) ≤ ℓ−1 and study certain deformations of a given \overline{ρ} : G_{ℓ} → GL_{2}(k). However, instead of requiring \overline{ρ}|_{G_{F}} to arise as the generic fiber of a finite flat A-group scheme, we fix a (ramified) quadratic character ψ on G_{F} and require that \overline{ρ}⊗ψ is the generic fiber of a finite flat A-group scheme. Moreover, we impose the same connectedness/unipotence conditions and the same exactness hypothesis on Dieudonné modules as above, and we study those deformations of \overline{ρ} whose ψ-twist is A-flat (in the same sense as above).

In contrast to the (untwisted) A-flat setting, the case e|(ℓ − 1) now occurs, and we shall actually restrict our attention to this case. In §4.2 of [5], it is explained how the methods then carry over to this setting. In particular, under the above hypotheses, End_{k[G_{ℓ}]} V = k, so there is a universal deformation ring R_{V,𝒪} for \overline{ρ}. The property of the ψ-twist being A-flat is preserved under taking submodules, quotients and finite products, so there is an associated quotient of R_{V,𝒪} , which we denote R_{V,𝒪}^{A−ψ⊗flat}. We can also consider the quotient R_{V,𝒪}^{A−ψ⊗flat,χ} representing such deformations with fixed determinant χ = ϵη. Theorem 4.2.1 of [5] then gives:

Theorem 2.9. Under the above hypotheses, there are 𝒪-algebra isomorphisms

R_{V,𝒪}^{A−ψ⊗flat} ≃ 𝒪[[T_{1}, T_{2}]] and R_{V,𝒪}^{A−ψ⊗flat,χ} ≃ 𝒪[[T]].

Corollary 2.10. Suppose that e(F)|(ℓ − 1), ψ is a quadratic character of G_{F} and τ ⊗ψ is trivial on the inertia group of F. If \overline{ρ} admits a lifting of type τ : I_{ℓ} → GL(D) (i.e., if R_{τ} (\overline{ρ}) ≠ 0) and \overline{ρ} ⊗ ψ is A-flat and satisfies the above connectedness, unipotence and Dieudonné module hypotheses, then τ is acceptable for \overline{ρ}.

3. Some Representations of Finite Groups

3.1. Representations of GL_{2}(F_{p}). Let us recall the classification of irreducible representations of GL_{2}(F_{p}) over an algebraically closed field F of characteristic zero. Any such representation is isomorphic to one of the following, where we have fixed an embedding i : F_{p^{2}} → M_{2}(F_{p}).

• For any character χ : F_{p}^{×} → F^{×} the representation χ ∘ det.

• For any character χ : F_{p}^{×} → F^{×} the representation sp_{χ} = sp⊗(χ∘det), where sp is the representation of GL_{2}(F_{p}) on the space of functions P^{1}(F_{p}) → F with average value zero.

• For any pair of characters χ_{1} ≠ χ_{2} : F_{p}^{×} → F^{×} the representation I(χ_{1}, χ_{2}) on the space of functions f : GL_{2}(F_{p}) → F which satisfy

f (( \begin{matrix}a_{1} & b \\ 0 & a_{2} \end{matrix} ) g) = χ_{1}(a_{1})χ_{2}(a_{2})f(g).

(This is isomorphic to the representation induced from the following character on the subgroup of upper-triangular matrices in GL_{2}(F_{p}):

( \begin{matrix}a_{1} & b \\ 0 & a_{2} \end{matrix} ) ↦ χ_{1}(a_{1})χ_{2}(a_{2}).)

• For any character χ : F_{p^{2}}^{×} → F^{×} with χ ≠ χ^{p} a representation Θ(χ) of dimension p − 1 which is characterised by

Θ(χ) ⊗ sp ≃ Ind_{F_{p^{2}}^{×}}^{GL_{2}(F_{p})} χ.

The only isomorphisms between these representations are I(χ_{1}, χ_{2}) ≅ I(χ_{2}, χ_{1}) and Θ(χ) ≅ Θ(χ^{p}). For convenience, we include the character table of GL_{2}(F_{p}):

Representation

Conjugacy

class of:  χ ∘ det  sp_{χ}  I(χ_{1}, χ_{2})  Θ(χ)

( \begin{matrix}a & 0 \\ 0 & a \end{matrix} )  χ(a)^{2}  pχ(a)^{2}  (p + 1)χ_{1}(a)χ_{2}(a)  (p − 1)χ(a)
( \begin{matrix}a & 1 \\ 0 & a \end{matrix} )  χ(a)^{2}  0  χ_{1}(a)χ_{2}(a)  −χ(a)
( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ∉ F_{p}^{×} χ(ab)  χ(ab) χ_{1}(a)χ_{2}(b) + χ_{1}(b)χ_{2}(a)  0
i(c) ∉ F_{p}^{×}  χ(c^{p+1}) −χ(c^{p+1})  0  −χ(c) − χ(c^{p})

We recall also the classification of irreducible representations of GL_{2}(F_{ℓ}) in characteristic ℓ. We will let σ_{n} denote the natural representation of GL_{2}(F_{ℓ}) on Symm^{n}(F_{ℓ}^{2}) for each n ∈ Z_{≥0}. For m ∈ (Z/(ℓ − 1)Z) and 0 ≤ n ≤ ℓ − 1, we will let σ_{n,m} denote σ_{n} ⊗ det^{m}. These representations \overline{σ}_{n,m} exhaust the isomorphism classes of irreducible representations of GL_{2}(F_{ℓ}) in characteristic ℓ. The Brauer character of σ_{n,m} is given by:

\begin{matrix}( \begin{matrix}a & 0 \\ 0 & a \end{matrix} ) & ↦ & (n + 1)ã^{2m+n}, \\ ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ∉ F_{p}^{×} & ↦ & ã^{m}b̃^{m}(ã^{n+1} − b̃^{n+1})/(ã − b̃), \\ i(c) ∉ F_{p}^{×} & ↦ & c̃^{m(ℓ+1)}(c̃^{(n+1)ℓ} − c̃^{n+1})/(c̃^{ℓ} − c̃), \end{matrix}

where ˜ indicates Teichmüller lift. Using Brauer characters one finds:

Lemma 3.2. Let L be a finitely-generated free 𝒪-module with an action of GL_{2}(F_{ℓ}) such that V = L ⊗_{𝒪} \overline{Q}_{ℓ} is irreducible.

1. If V ≅ χ ∘ det with χ(a) = ã^{m}, then L ⊗_{𝒪} k ≅ σ_{0,m}.

2. If V ≅ sp_{χ} with χ(a) = ã^{m}, then L ⊗_{𝒪} k ≅ σ_{ℓ−1,m}.

3. If V ≅ I(χ_{1}, χ_{2}) with χ_{i}(a) = ã^{m_{i}}, then L ⊗_{𝒪} k has two irreducible constituents: σ_{{m_{1}−m_{2}},m_{2}} and σ_{{m_{2}−m_{1}},m_{1}}; where 0 < {m} < ℓ − 1 and {m} ≡ m mod ℓ − 1.

4. If V ≅ Θ(χ) with χ(a) = a^{i+(ℓ+1)j} where 1 ≤ i ≤ ℓ and j ∈ Z/(ℓ − 1)Z, then L⊗_{𝒪} k has one or two irreducible constituents: σ_{i−2,1+j} and σ_{ℓ−1−i,i+j}. Both occur unless i = 1 (when only the second one occurs) or i = ℓ (when only the first occurs).

3.3. Representations of GL_{2}(Z/p^{n}Z). We shall also need to consider certain representations of GL_{2}(Z/p^{n}Z) for n > 1 which generalize the representations Θ(χ) for n = 1. Let σ denote Frob_{p} on A = W (F_{p^{2}}), choose an isomorphism

M_{2}(Z_{p}) ≅ End_{Z_{p}} A = A ⊕ Aσ,

and let ϖ_{n} denote the projection GL(A) → G = GL_{2}(Z/p^{n}Z). Let m = [n/2] and define subgroups N ⊂ H of G as follows:

N = ϖ_{n} ({ x + yσ | x ∈ 1 + p^{n}A, y ∈ p^{n−m}A })
H = ϖ_{n} ({ x + yσ | x ∈ A^{×}, y ∈ p^{m}A }) .

Thus [N : 1] = p^{2m} and [G : H] = ϕ(p^{2m}).

Suppose that χ : A^{×} → F^{×} is a character of conductor p^{n}. We assume also that χ/(χ ∘ σ) also has conductor p^{n} (or equivalently, no twist of χ by a character factoring through the norm A^{×} → Z_{p}^{×} has conductor less than p^{n}; see [18, §3.2]). If n is even, we define a character β_{χ} of H/N by β_{χ}(x + yσ) = χ(x). If n is odd, we let β_{χ} denote the p-dimensional representation of H/N such that

tr β_{χ}(x + yσ) = \{ \begin{matrix}pχ(x), & if y ≡ 0 mod p^{m+1}A and x ≡ x^{σ} mod pA; \\ −χ(x), & if y ≡ 0 mod p^{m+1}A and x ≢ x^{σ} mod pA; \\ 0, & if x + yσ is not conjugate to an element as above. \end{matrix}

(The existence of such a representation can be proved using Brauer’s criterion.) Now let Θ(χ) denote the ϕ(p^{n})-dimensional representation Ind_{H}^{G} β_{χ}. The isomorphism class of Θ(χ) is independent of the choice of isomorphism M_{2}(Z_{p}) ≅ End_{Z_{p}} A, since any two such isomorphisms differ by conjugation by an element of GL_{2}(Z_{p}).

Lemma 3.4. Suppose that n ≥ 1 and χ_{1} and χ_{2} are a pair of characters of A^{×} as above. Suppose that L_{i} for i = 1, 2, is a free 𝒪-module with an action of G = GL_{2}(Z/p^{n}Z) such that L_{i} ⊗_{𝒪} \overline{Q}_{ℓ} is isomorphic to Θ(χ_{i}). Let B denote the subgroup of G consisting of matrices which mod p are upper triangular, and let C = A^{×} ∩ B.

1. The restrictions Θ(χ_{i})|_{B} are irreducible,

2. Θ(χ_{1}) ≃ Θ(χ_{2}) ⇔ χ_{2} ∈ { χ_{1}, χ_{1} ∘ σ },

3. Θ(χ_{1})|_{B} ≃ Θ(χ_{2})|_{B} ⇔ χ_{2}|_{C} ∈ { χ_{1}|_{C}, χ_{1} ∘ σ|_{C} },

4. if p(p − 1) is not divisible by ℓ, then L_{i} ⊗_{𝒪} k is absolutely irreducible,

5. if χ_{1} ≡ χ_{2} mod λ, then L_{1} ⊗_{𝒪} k ≅ L_{2} ⊗_{𝒪} k.

Proof. The first three assertions are proved by computing ⟨Θ(χ_{1}), Θ(χ_{2})⟩ on B and G. If #B is not divisible by ℓ, then the reduction L ⊗_{𝒪} k is also irreducible by [28, §15.5]. The last assertion follows since the two representations have the same Brauer character. ■

3.5. Duality. Let K, 𝒪, λ and k be as above, and let 𝒪′ denote the valuation ring of K′ = K(λ′), where λ′ = λ^{1/2}. Suppose that V is a K-vector space with an action of GL_{2}(Z/p^{n}Z) such that V ⊗_{K} \overline{Q}_{ℓ} is one of the representations considered above, i.e., that n = 1 and V is absolutely irreducible, or that n > 1 and V ⊗_{K} \overline{Q}_{ℓ} is isomorphic to Θ(χ) for some character χ : F_{p^{2}}^{×} → \overline{Q}_{ℓ}^{×}. In each case, there is a nondegenerate pairing ( , ) on V such that

(gu, gv) = ψ(det g)(u, v),

where ψ : (Z/p^{n}Z)^{×} → K^{×} is the central character of the representation. For n = 1, this is clear from the character table, and for Θ(χ) with n > 1, this follows from the analogous assertion for β_{χ}.

We will need the following lemma.

Lemma 3.6. Suppose G is a finite group which acts absolutely irreducibly on a finite dimensional K-vector space V. Let ( , ) be a non-degenerate pairing on V such that (gu, gv) = ψ(g)(u, v) for some character ψ : G → K^{×}. Let V′ = V ⊗_{K} K(λ^{1/2}). Then there is a G-invariant 𝒪′-lattice L′ ⊂ V′ which is self-dual for ( , ).

Proof. First note that by Schur’s lemma, the pairing is symmetric or alternating. Choose a G-invariant lattice L_{1} in V which contains its dual lattice L_{1}^{⊥}. Then ( , ) induces a perfect pairing L_{1}/L_{1}^{⊥} × L_{1}/L_{1}^{⊥} → K/𝒪. Let X be a maximal isotropic G-submodule of L_{1}/L_{1}^{⊥} and replace L_{1} by L_{2} the preimage of X^{⊥}. Then one sees that L_{2} ⊃ L_{2}^{⊥} ⊃ λL_{2}. It is then easy to check that L′ = L_{2}^{⊥} ⊕ λ′L_{2} will suffice. ■

4. Galois representations for modular forms.

4.1. ℓ-adic representations associated to modular forms. Now let us recall some facts about the ℓ-adic representations attached to certain automorphic representations of GL_{2}(A). Recall that we have fixed embeddings \overline{Q} into C and into \overline{Q}_{p} for all p. Suppose that π ≅ ⊗′_{v}π_{v} is a cuspidal automorphic representation of GL_{2}(A) such that π_{∞} is a lowest discrete series representation of GL_{2}(C) with trivial central character. Recall that the set of such representations π is in one-to-one correspondence with the set of weight two newforms. The theory of Eichler and Shimura associates a continuous irreducible two-dimensional representation

ρ_{π} : G_{Q} → GL(V_{π})

over \overline{Q}_{ℓ} to π which is characterised as follows: For any prime p ≠ ℓ such that π_{p} is unramified, ρ_{π} is unramified at p and ρ_{π}(Frob_{p}) has characteristic polynomial

X^{2} − t_{p}X + ps_{p},

where t_{p} denotes the eigenvalue of the Hecke operator

T_{p} = [GL_{2}(Z_{p}) ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) GL_{2}(Z_{p})]

on π_{p}^{GL_{2}(Z_{p})}, and s_{p} denotes that of

S_{p} = [GL_{2}(Z_{p}) ( \begin{matrix}p & 0 \\ 0 & p \end{matrix} ) GL_{2}(Z_{p})] .

Let S(π) denote the set of primes p such that π_{p} is ramified, and let S be any finite set of primes. Then the set of t_{p} for p ∉ S ∪ S(π) generate a number field over which π is defined, and ρ_{π} is defined over the extension of K these generate. If π and π′ are such that the corresponding eigenvalues t_{p} and t′_{p} coincide for all p ∉ S ∪ S(π) ∪ S(π′), then in fact π = π′. For any automorphism σ in G_{K}, there is an automorphic representation denoted π^{σ} such that ρ_{π}^{σ} = ρ_{π^{σ}}.

For any prime p, the local Langlands correspondence associates to π_{p} a certain continuous semi-simple two-dimensional representation W D(π_{p}) : W_{p} → GL(D_{π_{p}}) over \overline{Q} with discrete topology. Our convention here is that W D(π_{p}) is the restriction to W_{p} of σ(π_{p}) ⊗ | |^{−1}, where σ(π_{p}) is as in [2] and we have identified Q_{p}^{×} with W_{p}^{ab} by the Artin map (which, with our conventions, sends p to a preimage of Frob_{p} ∈ W_{p}/I_{p}). Thus Π ↦ W D(Π) establishes a bijection between (a) isomorphism classes of irreducible admissible infinite-dimensional representations of GL_{2}(Q_{p}) defined over \overline{Q}, and (b) isomorphism classes of continuous semi-simple representations W_{p} → GL_{2}(\overline{Q}). The bijection has the following properties:

• If χ is a continuous character Q_{p}^{×} → \overline{Q}^{×}, then W D(Π⊗χ∘det) ≅ W D(Π)⊗χ.

• The determinant of W D(Π) is ϵχ_{Π}, where χ_{Π} is the central character of Π.

A theorem of Carayol ([2, Thm. A], generalizing results of Langlands and Deligne) shows that

\overline{Q}_{ℓ} ⊗\overline{Q} W D(π_{p}) ≅ (ρ_{π}|_{W_{p}})^{ss}

for p ≠ ℓ. The restriction ρ_{π}|_{G_{ℓ}} is potentially semi-stable, and if π_{ℓ} is not special then ρ_{π}|_{G_{ℓ}} is Barsotti-Tate over any extension F of Q_{ℓ} such that W D(π_{ℓ})|_{I_{F}} is trivial. Moreover, the main theorem of [26] shows that

\overline{Q}_{ℓ} ⊗\overline{Q} W D(π_{ℓ}) ≅ W D(ρ_{π}|_{G_{ℓ}}).

4.2. The local Langlands correspondence. We shall need to recall some properties of the correspondence Π ↔ W D(Π). Before doing so, we define, for each n ≥ 0, open subgroups

U_{0}(p^{n}) ⊃ U_{ℓ}(p^{n}) ⊃ U_{1}(p^{n}) ⊃ U(p^{n})

of GL_{2}(Z_{p}) as follows. We set

• U_{0}(p^{n}) = ϖ_{n}^{−1} \{( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )\};

• U_{ℓ}(p^{n}) = ϖ_{n}^{−1} \{( \begin{matrix}∗ & ∗ \\ 0 & a \end{matrix} ) | a ∈ (Z/p^{n}Z)^{×} has ℓ-power order\};

• U_{1}(p^{n}) = ϖ_{n}^{−1} \{( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} )\};

• U(p^{n}) = ker ϖ_{n}; where ϖ_{n} denote the natural projection GL_{2}(Z_{p}) → GL_{2}(Z/p^{n}Z). For V = U_{0}(p^{n}),

U_{ℓ}(p^{n}) or U_{1}(p^{n}) with n ≥ 1, we let U_{p} denote the Hecke operator V ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) V

on Π^{V}. If V and V′ are two such subgroups with V′ ⊂ V, then the operators denoted U_{p} are compatible with the natural inclusion Π^{V} → Π^{V′}.

Lemma 4.3. There is an integer c = c(Π) ≥ 0 such that

dim Π^{U_{1}(p^{m})} = max{0, m − c + 1}

for all m ≥ 0.

1. If c = 0 and n > 0, then the characteristic polynomial of U_{p} on Π^{U_{0}(p^{n})} = Π^{U_{1}(p^{n})} is X^{n−1}(X^{2} −t_{p}X +ps_{p}), where t_{p} (respectively, s_{p}) is the eigenvalue of T_{p} (respectively, S_{p}) on π_{p}^{GL_{2}(Z_{p})}.

2. If c > 0 and n > 0, then the characteristic polynomial of U_{p} on Π^{U_{1}(p^{c+n})} is X^{n}(X − u_{p}), where u_{p} is the eigenvalue of U_{p} on Π^{U_{1}(p^{c})}.

With the above notation, we have the following well-known properties of the correspondence Π ↔ W D(Π).

Lemma 4.4. 1. Suppose that c = 0. Then W D(Π) is unramified and the char-

acteristic polynomial of Frob_{p} on W D(Π) is X^{2} − t_{p}X + ps_{p}.

2. Suppose that c = 1. Let χ denote the character of Z_{p}^{×} defined by the action of F_{p}^{×} ≅ U_{0}(p)/U_{1}(p) on Π^{U_{1}(p)}. Then W D(Π)|_{I_{p}} ≅ 1 ⊕ χ ∘ η_{p,1}. If χ is trivial, then W D(Π) ≅ ξ ⊕ ξ| |^{−1} where ξ is unramified and ξ(Frob_{p}) = u_{p}. If χ is not trivial, then Frob_{p} acts via u_{p} on W D(Π)^{I_{p}}.

3. Suppose that c > 1. Then c is the conductor of W D(Π). Moreover u_{p} ≠ 0 if and only if W D(Π)^{I_{p}} ≠ 0, in which case Frob_{p} acts via u_{p} on W D(Π)^{I_{p}}.

Note that the only case where c(Π) is not the conductor of W D(Π) is when c(Π) = 1 and the central character of Π is unramified. We refer to such Π as unramified special representations. We let e(Π) = 1 if W D(Π) is unramified special, and e(Π) = dim W D(Π)^{I_{p}} otherwise. In general, if π is an automorphic representation as above and p ≠ ℓ, then dim ρ_{π}^{I_{p}} = e(π_{p}).

Corollary 4.5. The characteristic polynomial of U_{p} on π_{p}^{U_{1}(p^{c(πp)+e(πp)})} is of the form xf(x) where f(x) has degree e(π) and roots which are ℓ-adic units.

We also have the following relationship between W D(Π) and the action of GL_{2}(Z/p^{n}Z) ≅ GL_{2}(Z_{p})/U(p^{n}) on Π^{U(p^{n})}.

Lemma 4.6. 1. Suppose that χ : F_{p}^{×} → \overline{Q}^{×}. If ⟨sp_{χ}⊕χ∘det, Π^{U(p)}⟩_{GL_{2}(F_{p})} ≠ 0, then W D(Π)|_{I_{p}} ≅ χ ∘ η_{p,1} ⊕ χ ∘ η_{p,1}. Conversely if W D(Π)|_{I_{p}} ≅ χ ∘ η_{p,1} ⊕ χ ∘ η_{p,1}, then either Π^{U(p)} ≅ sp_{χ} ⊕ χ ∘ det and Π is the twist of an unramified representation, or Π^{U(p)} ≅ sp_{χ} and Π is special.

2. Suppose that χ_{1} ≠ χ_{2} : F_{p}^{×} → \overline{Q}^{×}. If ⟨I(χ_{1}, χ_{2}), Π^{U(p)}⟩_{GL_{2}(F_{p})} ≠ 0, then W D(Π)|_{I_{p}} ≅ (χ_{1} ∘ η_{p,1}) ⊕ (χ_{2} ∘ η_{p,1}). Conversely if W D(Π)|_{I_{p}} ≅ (χ_{1} ∘ η_{p,1}) ⊕ (χ_{2} ∘ η_{p,1}), then Π^{U(p)} ≅ I(χ_{1}, χ_{2}).

3. Suppose that χ : A^{×} = W (F_{p^{2}})^{×} → \overline{Q}^{×} is as in §3.3 with conductor p^{n}A. If ⟨Θ(χ), Π^{U(p^{n})}⟩_{GL_{2}(Z/p^{n}Z)} ≠ 0, then W D(Π)|_{I_{p}} ≅ (χ∘η_{p,2})⊕(χ∘Frob_{p} ∘η_{p,2}). Conversely if W D(Π)|_{I_{p}} ≅ (χ ∘ η_{p,2}) ⊕ (χ ∘ Frob_{p} ∘ η_{p,2}), then Π^{U(p^{n})} ≅ Θ(χ).

These assertions follow from explicit descriptions of the local Langlands correspondence. The first two parts can already be deduced from the properties listed above, together with the classification of representations of GL_{2}(F_{p}). For the third part, see [18, §3].

5. Hecke Algebras and Modules

5.1. Definition of Hecke algebras. We use the notation of §1.7. In particular,

\overline{ρ} : G_{Q} → GL(V)

is absolutely irreducible, τ is an ℓ-type strongly acceptable for \overline{ρ}|_{G_{ℓ}} and S is a finite set of primes not containing ℓ. We suppose from now on that \overline{ρ} is modular, meaning there exist an automorphic representation as in §4 and a finite extension K′ of K such that ρ_{π} ≅ \overline{Q}_{ℓ} ⊗_{𝒪′} ρ for some deformation ρ of \overline{ρ} to 𝒪′. We let 𝒩_{S} denote the set of π such that this holds for some ρ of type (S, τ). We shall write R_{S} for the universal deformation ring R_{S}, and ρ_{S} for the universal deformation.

Let S(\overline{ρ}) denote the set of primes p such that p = ℓ or \overline{ρ} is ramified at p. Let T (\overline{ρ}) denote the set of primes p such that p ≡ −1 mod ℓ, \overline{ρ}|_{G_{p}} is irreducible and \overline{ρ}|_{I_{p}} is reducible. We let T̃_{S} denote the polynomial algebra over 𝒪 generated by the variables T_{p} and S_{p} for p ∉ S ∪ S(\overline{ρ}). We define the Hecke algebra T_{S} as the image of the 𝒪-algebra homomorphism

T̃_{S} → ∏_{π∈𝒩 _{S}} \overline{Q}_{ℓ},

sending T_{p} ↦ t_{p} and S_{p} ↦ s_{p} in each component. We let I_{S} denote the kernel of T̃_{S} → T_{S}. We also let T̃′_{S} denote the polynomial algebra over T_{S} generated by the variables U_{p} for p ∈ S, and let I′_{S} denote the ideal generated by I_{S} and the set of U_{p} for p ∈ S.

For each prime p we will define open subgroups V_{S,p}, normal in U_{S,p} ⊂ GL_{2}(Z_{p}), an element w_{S,p} ∈ GL_{2}(Q_{p}) and a finite-dimensional irreducible representation σ_{S,p} of U_{S,p}/V_{S,p} over \overline{Q}. For p ≠ ℓ, let c_{p} denote the conductor of \overline{ρ}|_{G_{p}} and e_{p} = dim_{k} \overline{ρ}^{I_{p}}. If p ∈ T (\overline{ρ}), then c_{p} is even and we can choose a character χ_{p} : W (F_{p^{2}})^{×} → \overline{Q}^{×} of order prime to ℓ whose reduction \overline{χ}_{p} satisfies

\overline{ρ}|_{I_{p}} ⊗_{k} \overline{F}_{ℓ} ≅ \overline{χ}_{p} ∘ η_{p,2} ⊕ \overline{χ}_{p} ∘ Frob_{p} ∘ η_{p,2}.

• If p ∉ S ∪ T (\overline{ρ}) ∪ \{ℓ\} then U_{S,p} = V_{S,p} = U_{ℓ}(p^{c_{p}}), w_{S,p} = ( \begin{matrix}0 & −1 \\ p^{c_{p}} & 0 \end{matrix} ) and

σ_{S,p} = 1.

• If p ∈ S − T (\overline{ρ}) then U_{S,p} = V_{S,p} = U_{ℓ}(p^{c_{p}+e_{p}}), w_{S,p} = ( \begin{matrix}0 & −1 \\ p^{c_{p}+e_{p}} & 0 \end{matrix} ) and

σ_{S,p} = 1.

• If p ∈ T (\overline{ρ}) − S then U_{S,p} = GL_{2}(Z_{p}), V_{S,p} = U(p^{c_{p}/2}), w_{S,p} = 1 and σ_{S,p} = Θ(χ_{p}).

• If p ∈ T (\overline{ρ}) ∩ S then U_{S,p} = U_{0}(p), V_{S,p} = U(p^{c_{p}/2}), w_{S,p} = 1 and σ_{S,p} = Θ(χ_{p})|_{U_{S,p}/V_{S,p}}.

• If p = ℓ, then U_{S,p} = GL_{2}(Z_{ℓ}), V_{S,p} = U(ℓ), w_{S,p} = 1 and σ_{S,p} is

χ ∘ det, if τ ≅ χ ∘ ω_{1} ⊕ χ ∘ ω_{1} for some χ : F_{ℓ}^{×} → \overline{Q}^{×};
I(χ_{1}, χ_{2}), if τ ≅ χ_{1} ∘ ω_{1} ⊕ χ_{2} ∘ ω_{1} with χ_{1} ≠ χ_{2} : F_{ℓ}^{×} → \overline{Q}^{×};
Θ(χ),  if τ ≅ χ ∘ ω_{2} ⊕ χ^{ℓ} ∘ ω_{2} with χ ≠ χ^{ℓ} : F_{ℓ^{2}}^{×} → \overline{Q}^{×}.

We will set U_{S} = ∏_{p} U_{S,p}, and V_{S} = ∏_{p} V_{S,p}, w_{S} = ∏_{p} w_{S,p} and σ_{S} = ⊗_{p}σ_{S,p}. We have defined w_{S} for later use. The point of the definitions of U_{S} and σ_{S} is the following lemma.

Lemma 5.2. Suppose that ρ_{π} ≅ \overline{Q}_{ℓ} ⊗_{𝒪′} ρ for some deformation ρ of \overline{ρ} to the ring of integers 𝒪′ of a finite extension K′ of K. Then π ∈ 𝒩_{S} if and only if

Hom_{U_{S}}(σ_{S}, π^{∞}) ≠ (0).

In that case, the eigenvalues of U_{p} on Hom_{U_{S}}(σ_{S}, π^{∞}) for each p ∈ S are either 0 or ℓ-adic units, and the subspace on which U_{p} = 0 for all p ∈ S is 1-dimensional.

Proof. This follows from the results recalled in the preceding section together with the analysis of possible lifts of \overline{ρ}|_{G_{p}} for p ≠ ℓ (see [3] and [14]). In particular that analysis shows that c(π_{p}) + e(π_{p}) = c_{p} + e_{p}. Moreover if p ∉ T (\overline{ρ}), c_{p} = c(π_{p}) and det ρ|_{I_{p}} has order prime to ℓ, then ρ|_{I_{p}} satisfies the local condition at p in the

definition of type (S, τ). On the other hand, if p ∈ T (\overline{ρ}), then π_{p}^{U(p^{cp/2})} as a module for GL_{2}(Z_{p}) is isomorphic to Θ(χ_{p}ψ_{p}) for some character ψ_{p} of ℓ-power order. Note that in that case Θ(χ_{p}ψ_{p})|_{U_{0}(p)} ≅ Θ(χ_{p})|_{U_{0}(p)}. ■

Corollary 5.3. The set 𝒩_{S} is finite, T_{S} is finitely generated as an 𝒪-module and the natural map

T_{S} ⊗_{𝒪} \overline{Q}_{ℓ} → ∏_{π∈𝒩 _{S}} \overline{Q}_{ℓ}

is an isomorphism.

5.4. The universal modular deformation. For each π in 𝒩_{S}, the universal property of the deformation ring R_{S} provides an 𝒪-algebra homomorphism R_{S} → \overline{Q}_{ℓ} so that ρ_{π} is the extension of scalars of the universal deformation. The map

R_{S} → ∏_{π∈𝒩 _{S}} \overline{Q}_{ℓ}

has image T_{S} since R_{S} is topologically generated by traces. We let ϕ_{S} denote the resulting surjective 𝒪-algebra homomorphism

R_{S} → T_{S}.

Note that whenever S ⊂ S′, we have a natural commutative diagram of 𝒪-algebra homomorphisms

T̃_{S′} → T_{S′} ← R_{S′}
↓ ↓ ↓
T̃_{S} → T_{S} ← R_{S},

with all maps surjective except T̃_{S′} → T̃_{S} which is injective.

5.5. Definition of Hecke modules. It is convenient to fix an auxiliary prime r ∉ S(\overline{ρ}) such that no lift of \overline{ρ} can be ramified at r (see lemma 2 of [14]). Thus we have 𝒩_{S} = 𝒩_{S∪{r}}, so T_{S∪{r}} ≅ T_{S}. We also assume the field K is sufficiently large that it contains all quadratic extensions of some field K_{0} such that

• \overline{ρ} is defined over k_{0};

• σ_{∅,p} ⊗\overline{Q} \overline{Q}_{ℓ} is defined over K_{0} for each p ∈ T (\overline{ρ}) ∪ \{ℓ\}. For each p ∈ T (\overline{ρ}) ∪ \{ℓ\}, we fix a lattice M_{p} as in lemma 3.6 for σ_{∅,p} and a pairing ( , )_{p} inducing an isomorphism

M_{p} → Hom_{𝒪} (M_{p}, 𝒪(χ_{p}))

of 𝒪U_{∅,p}-modules, where χ_{p} = ψ_{p} ∘ det, ψ_{p} being the central character of σ_{∅,p}. Letting M_{S} denote the model ⊗_{p∉S}M_{p} over 𝒪 for σ_{S}, we have a pairing ( , )_{S} on M_{S} which induces an isomorphism

M_{S} → Hom_{𝒪} (M_{S}, 𝒪(χ_{S}))

of 𝒪U_{S}-modules, where χ_{S} = ψ_{S} ∘ det, where ψ_{S} = ⊗p ∈ Sψ_{p}.

If U is an open compact subgroup of GL_{2}(A^{∞}) then we will let Y_{U} denote the modular curve

GL_{2}(Q)\((GL_{2}(A^{∞})/U) × (C − R)) ≅ GL_{2}(Q)\GL_{2}(A)/UU_{∞},

where U_{∞} = O_{2}(R)R^{×}. We let X_{U} denote its compactification obtained by adjoining cusps. Recall that X_{U} is not necessarily connected (its connected components are in bijection with Ẑ^{×}/ det U), and that X_{U} has a model over Q. Our convention for the definition of this model is that Y_{U}(\overline{Q}) is in canonical bijection with the set of equivalence classes of pairs (E, α), where E is an elliptic curve over \overline{Q} and α : A × A ⟶^{∼} (proj lim_{n} E[n]) ⊗Ẑ A. We consider (E_{1}, α_{1}) ∼ (E_{2}, α_{2}) if there is an isogeny ϕ : E_{1} → E_{2} such that ϕ ∘ α_{1} = α_{2} ∘ u for some u ∈ U. The point GL_{2}(Q) · (xU, τ) corresponds to the elliptic curve C/Λ_{τ} where Λ_{τ} = Zτ ⊕ Z and α is defined by composing x with the isomorphism obtained using (τ, 1) as a basis over Ẑ for proj lim_{n} E[n].

We obtain an admissible GL_{2}(A^{∞})-module

ℋ = inj lim_{U} H^{1}(X_{U}, \overline{Q})

where the limit is with respect to the natural maps on cohomology induced by X_{V} → X_{U} whenever V ⊂ U. Then ℋ decomposes as ⊕_{π}(B_{π}^{+} ⊕ B_{π}^{−}) where each B_{π}^{±} is a model for π. We recover each H^{1}(X_{U}, \overline{Q}) from ℋ as the subspace ℋ^{U} of U-invariants.

We let X_{S} = X_{V_{S}} and consider H^{1}(X_{S}, 𝒪). We have natural compatible actions on it of T̃′_{S}, and G_{S} = U_{S}/V_{S}. If r is in S, then we define L_{S} to be the T_{S}-module

Hom_{𝒪[G_{S}]} (M_{S}, H^{1}(X_{S}, 𝒪))[I′_{S}].

(If I is an ideal in a ring R and M is an R-module, we write M[I] for the largest submodule of M whose annihilator contains I.) If r is not in S, then we set L_{S} = L_{S∪{r}}.

Lemma 5.6. The (T_{S} ⊗_{𝒪} K)-module L_{S} ⊗_{𝒪} K is free of rank two.

Proof. We may assume r is in S and replace K by \overline{Q}_{ℓ}. Writing H^{1}(X_{S}, \overline{Q}_{ℓ}) as ℋ ⊗\overline{Q} \overline{Q}_{ℓ} and decomposing ℋ, we obtain an isomorphism of T_{S} ⊗_{𝒪} \overline{Q}_{ℓ}-modules

L_{S} ⊗_{𝒪} \overline{Q}_{ℓ} ≅ ⨁(B′_{π} ⊗\overline{Q} \overline{Q}_{ℓ})[I_{S}]

where B′_{π} is the subspace of Hom_{U_{S}}(σ_{S}, B_{π}^{+} ⊕ B_{π}^{−}) killed by U_{p} for all p in S. By lemma 5.2, this space is two-dimensional whenever π is in 𝒩_{S}. The lemma follows, since for p not in S ∪ S(\overline{ρ}), the operators T_{p} and S_{p} act on B′_{π} as t_{p} and s_{p}, respectively. ■

There is also a natural action of G_{Q} on L_{S} compatible with that of T_{S}. This can be defined, for example, via the isomorphism of Hom_{Z_{ℓ}}(H^{1}(X_{S}, Z_{ℓ}), Z_{ℓ}) with the ℓadic Tate module of the Jacobian of X_{S}. Each π ∈ 𝒩_{S} gives rise to a homomorphism θ_{π,S} : T_{S} → \overline{Q}_{ℓ} sending T_{p} ↦ t_{p} and S_{p} ↦ s_{p}. The resulting representation of G_{Q} on Hom_{T_{S}}(L_{S}, \overline{Q}_{ℓ}) is isomorphic to ρ_{π}, and we therefore have the following lemma:

Lemma 5.7. There is an isomorphism

Hom_{\overline{Q}_{ℓ}}(L_{S}, \overline{Q}_{ℓ}) ≅ ⨁_{π∈𝒩 _{S}} V_{π}

of representations of G_{Q} over T_{S} ⊗_{𝒪} \overline{Q}_{ℓ} ≅ ⨁_{π∈𝒩 _{S}} \overline{Q}_{ℓ}.

5.8. The main results. Recall that we are assuming \overline{ρ} is as in §1.7 and that it is modular. We defined 𝒩_{S} and T_{S} in §5.1, ϕ_{S} in §5.4 and L_{S} in §5.5. We have not yet shown that these objects are non-trivial. We shall deduce the following lemma from related results in the literature in section 6.

Proposition 5.9. With the above notation and hypotheses, 𝒩_{S} ≠ ∅.

Our main result is the following theorem; we give the proof in the rest of this section, subject to some propositions which are proved in §6.

Theorem 5.10. With the above hypotheses and notation, the following hold:

1. ϕ_{S} is an isomorphism;

2. T_{S} is a complete intersection;

3. L_{S} is free over T_{S}.

5.11. Reduction to the case S = ∅. Following Wiles [34], we shall deduce the result for arbitrary S from the result for S = ∅.

For each p ∉ S ∪ {ℓ}, we define an element μ_{p} ∈ T_{S} as follows:

• If e_{p} = 2, then μ_{p} = (p − 1)(T_{p}^{2} − S_{p}(1 + p)^{2}).

• If e_{p} = 1 and det \overline{ρ} is unramified at p, then μ_{p} = p^{2} − 1.

• If e_{p} = 1 and det \overline{ρ} is ramified at p, then μ_{p} = p − 1.

• If e_{p} = 0 and p ∈ T (\overline{ρ}) then μ_{p} = p + 1.

• If e_{p} = 0 and p ∉ T (\overline{ρ}) then μ_{p} = 1. If S′ is a finite set of primes disjoint from S ∪ {ℓ}, we let μ_{S′} = ∏_{p∈S′} μ_{p}.

The key proposition we prove in §6 for the induction step is the following:

Proposition 5.12. 1. There exists a pairing ⟨ , ⟩_{S} on L_{S} which induces an

isomorphism

L_{S} → Hom_{𝒪} (L_{S}, 𝒪)

of T_{S}-modules.

2. If S ⊂ S′ and ℓ ∉ S′, then there exists a T_{S′}-module homomorphism

i_{S}^{S′} : L_{S} → L_{S′}

such that i_{S,S′} ⊗_{𝒪} k is injective and

j_{S′}^{S}i_{S}^{S′}L_{S} = μ_{S′−S}L_{S}

where j_{S′}^{S} is the adjoint of of i_{S}^{S′} with respect to the pairings ⟨ , ⟩_{S} and ⟨ , ⟩_{S′}.

Fix for the moment an element π of 𝒩_{∅}. Note that if part 1 or 2 of theorem 5.10 holds for some K such that T_{S} is defined, then it holds for all such K, and similarly for part 3 provided L_{S} is defined. We may therefore assume that K contains the eigenvalues t_{p} and s_{p} for all p ∉ S(\overline{ρ}). Let 𝔭_{S} denote the kernel of θ_{π,S} and J_{S} the annihilator of 𝔭_{S} in T_{S}. Define

C_{S,θ} = L_{S}/(L_{S}[𝔭_{S}] + L_{S}[J_{S}]).

This is isomorphic to the cokernel of the map

L[𝔭_{S}] → Hom_{𝒪} (L[𝔭_{S}], 𝒪)

defined by the restriction of ⟨ , ⟩_{S}. Thus C_{S,θ} has finite length over 𝒪, and by Proposition 5.12 we have

length_{𝒪} C_{S∪{p},θ} = length_{𝒪} C_{S,θ} + 2 · length_{𝒪} (𝒪/θ_{π,S}(μ_{p})).

Since θ_{π,S}(μ_{p}) is a unit times the determinant of Frob_{p} − 1 on ad^{0} V_{π}(1)^{I_{p}}, we can combine this with Theorem 2.4 of [11] and Corollary 1.11 to conclude that theorem 5.10 for all S follows from the special case S = ∅.

5.13. The case of S = ∅. We now turn to the proof of Theorem 5.10 in the case S = ∅. We will use the improvement on the method of Taylor and Wiles ([33]) found by Diamond ([11]) and Fujiwara ([17]). We keep the above hypotheses and notation, but only consider finite sets S of primes with the following properties. If p ∈ S then

• p ∉ S(\overline{ρ}),

• p ≡ 1 mod ℓ,

• \overline{ρ}(Frob_{p}) has distinct eigenvalues α_{1,p} and α_{2,p}.

One checks as in lemma 2.44 of [7], that for each p ∈ S, the restriction to G_{p} of the universal deformation ρ_{S} is equivalent to χ_{1,p}^{S} ⊕ χ_{2,p}^{S} for some characters

χ_{i,p}^{S} : G_{p} → R_{S}^{×},

where the reduction of χ_{i,p}^{S} mod the maximal ideal of R_{S} is unramified and sends Frob_{p} to α_{i,p} for i = 1, 2. The restrictions χ_{i,p}^{S}|_{I_{p}} are of the form ξ_{i,p}^{S} ∘ ω_{p} where ω_{p} denotes the mod p cyclotomic character I_{p} → (Z/pZ)^{×}. We let Δ_{S} = ∏_{p∈S} Δ_{p} where Δ_{p} denotes the ℓ-Sylow subgroup of (Z/pZ)^{×}. We regard R_{S} as an 𝒪[Δ_{S}]- algebra by mapping

ξ_{1,p}^{S} : Δ_{p} → R_{S}^{×}

for each p ∈ S. This makes L_{S} an 𝒪[Δ_{S}]-module. We let 𝔞_{S} denote the augmentation ideal of 𝒪[Δ_{S}]. The last key result whose proof we postpone until §6 is the following proposition:

Proposition 5.14. The 𝒪[Δ_{S}]-module L_{S} is free. The map π_{∅,S} : L_{S} → L_{∅} induces an isomorphism

L_{S}/𝔞_{S}L_{S} ⟶^{∼} L_{∅}.

We also need the following lemma, which is proved exactly as theorem 2.49 of [7] using lemma 1.9.

Lemma 5.15. There exists an integer r ≥ 0 such that for any integer n > 0, there exists a finite set of primes S_{n} disjoint from S(\overline{ρ}) such that

1. if p ∈ S_{n} then p ≡ 1 mod ℓ^{n};

2. if p ∈ S_{n} then \overline{ρ}(Frob_{p}) has distinct eigenvalues;

3. #S_{n} = r;

4. R_{S_{n}} can be topologically generated by r elements as an 𝒪-algebra.

We can then apply theorem 2.1 of [11] to complete the proof of theorem 5.10 in the case S = ∅. (See the proof of theorem 3.1 of [11].)

6. Cohomology of modular curves

In this section we will give the proofs of Propositions 5.9, 5.12 and 5.14. We maintain the notation of the preceding section. In particular, we consider a representation \overline{ρ} : G_{Q} → GL_{2}(k) which is irreducible and modular. Recall that T̃_{S} is the polynomial algebra generated by the operators T_{p} and S_{p} for p not in S ∪ S(\overline{ρ}). We define 𝔪_{S} to be the kernel of the map T̃_{S} → k defined by T_{p} ↦ tr(\overline{ρ}(Frob_{p})), S_{p} ↦ p^{−1} det(\overline{ρ}(Frob_{p})). We consider also the polynomial algebra T̃′_{S} over T̃_{S} generated by the operators U_{p} for p ∈ S, and the maximal ideal 𝔪′_{S} generated by 𝔪_{S} and the operators U_{p} for p ∈ S. Recall that since \overline{ρ} is irreducible, the maximal ideals 𝔪_{S} and 𝔪′_{S} are not Eisenstein. (We say a maximal ideal 𝔪 of T̃_{S} or T̃′_{S} is Eisenstein if there exists an integer N > 0 such that T_{p} − 2 ∈ 𝔪 and S_{p} − 1 ∈ 𝔪 for all p ∈ S ∪ S(\overline{ρ}) with p ≡ 1 mod N; see [13, Proposition 2], for example.)

For n ≥ 0, we let L_{n} denote the SL_{2}(Z)-module Symm^{n} Z^{2}. Recall that if Γ_{1}(N) ⊂ Γ ⊂ Γ_{0}(N) for some N ≥ 1 and S ∪S(\overline{ρ}) contains the set of prime divisors of N, then there is a natural action of T̃_{S} on H^{1}(Γ, L_{n} ⊗ M) for any 𝒪-module M. (See for example, [31, Chapter 8].) The following is a consequence of the results of Ribet and others (see [9, Corollary 1.2]).

Theorem 6.1. Suppose that 0 ≤ n ≤ ℓ − 1 and \overline{ρ}|_{I_{ℓ}} is of the form • ( \begin{matrix}ω^{n+1} & ∗ \\ 0 & 1 \end{matrix} ) with ∗ peu ramifié if n = 0, or

• ( \begin{matrix}ω_{2}^{n+1} & 0 \\ 0 & ω_{2}^{(n+1)ℓ} \end{matrix} ).

Let N be any integer divisible by the conductor of ρ, let Γ denote the group of

matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) such that d mod N has ℓ-power order and let S be a

set of primes such that S ∪ S(\overline{ρ}) contains the set of prime divisors of N. Then 𝔪_{S} contains the annihilator of H^{1}(Γ, L_{n} ⊗K), hence is in the support of H^{1}(Γ, L_{n} ⊗𝒪) and H^{1}(Γ, L_{n} ⊗ k).

Suppose that V is an open compact subgroup of GL_{2}(A^{∞}). We assume that V is of the form ∏_{p} V_{p} with V_{p} ⊂ GL_{2}(Z_{p}) and that V_{r} ⊂ U_{1}(r^{2}) for some fixed prime r as in §5.5. Suppose that Σ is a finite set of primes, and that for each p in Σ we are given a finitely generated 𝒪-module M_{p} with a left action of V_{p} which is continuous for the discrete topology on V_{p}. We can then associate to the V -module M = ⊗_{p}M_{p} a locally constant sheaf

ℱ_{M} = GL_{2}(Q)\(GL_{2}(A) × M^{op})/V U_{∞}

on Y_{V}. If r is in S, then we let Y_{S} denote Y_{U_{S}} and ℱ_{S} denote the sheaf on Y_{S} associated to the U_{S}-module Hom_{𝒪} (M_{S}, 𝒪). (See §5.1 and §5.5 for the definitions of U_{S} and M_{S}.) If r is not in S, then we let Y_{S} = Y_{S∪{r}} and ℱ_{S} = ℱ_{S∪{r}}.

If for all p ∉ S ∪ S(\overline{ρ}) we have V_{p} = GL_{2}(Z_{p}) and V_{p} acts trivially on M, then there is a natural action of T̃_{S} on the cohomology groups

H_{c}^{1}(Y_{V} , ℱ_{M}) and H^{1}(Y_{V} , ℱ_{M}).

Standard arguments yield the following useful technical result.

Lemma 6.2. Suppose that V, M and S are as above and let 𝔪 be a non-Eisenstein maximal ideal of T̃_{S} with finite residue field.

1. The map H_{c}^{1}(Y_{V} , ℱ_{M})_{𝔪} → H^{1}(Y_{V} , ℱ_{M})_{𝔪} is an isomorphism.

2. If M is free over 𝒪, then the natural map

H^{1}(Y_{V} , ℱ_{M})_{𝔪} ⊗_{𝒪} k → H^{1}(Y_{V} , ℱ_{M⊗_{𝒪} k})_{𝔪}

is an isomorphism.

3. If 0 → M′ → M → M′′ → 0 is an exact sequence of 𝒪[V]-modules, then the sequence

0 → H^{1}(Y_{V} , ℱ_{M′})_{𝔪} → H^{1}(Y_{V} , ℱ_{M})_{𝔪} → H^{1}(Y_{V} , ℱ_{M′′})_{𝔪} → 0

is exact.

4. If V′ ⊂ V and satisfies the hypotheses above for S, then

H^{1}(Y_{V} , ℱ_{M})_{𝔪} → H^{1}(Y_{V′}, ℱ_{M})_{𝔪}^{V}

is an isomorphism.

5. If V acts trivially on M, then

H^{1}(X_{V} , 𝒪)_{𝔪} ⊗_{𝒪} M → H^{1}(Y_{V} , ℱ_{M})_{𝔪}

is an isomorphism.

If in addition we have U_{1}(p^{n}) ⊂ V_{p} ⊂ U_{0}(p^{n}) for some n > 0 and V_{p} acts trivially on M for all p ∈ S, then this action extends naturally to an action of T̃′_{S}. Furthermore, the lemma holds for 𝔪 = 𝔪′_{S}. In particular, T̃′_{S∪{r}} acts on

H_{c}^{1}(Y_{S}, ℱ_{S}) and H^{1}(Y_{S}, ℱ_{S})

and the lemma yields natural isomorphisms

H_{c}^{1}(Y_{S}, ℱ_{S})_{𝔪} → H^{1}((Y_{S}, ℱ_{S})_{𝔪} → Hom_{G_{S}}(M_{S}, H^{1}(X_{S}, 𝒪))_{𝔪}

where 𝔪 = 𝔪′_{S∪{r}}. One can also check that the natural map

L_{S} = Hom_{G_{S}}(M_{S}, H^{1}(X_{S}, 𝒪))[I′_{S∪{r}}] → Hom_{G_{S}}(M_{S}, H^{1}(X_{S}, 𝒪))_{𝔪}

is an isomorphism, so we conclude:

Lemma 6.3. There are natural isomorphisms of T̃′_{S∪{r}}-modules

H_{c}^{1}(Y_{S}, ℱ_{S})_{𝔪} ≅ H^{1}(Y_{S}, ℱ_{S})_{𝔪} ≅ L_{S}

identifying T_{S} = T_{S∪{r}} with the localization at 𝔪 of the image of T̃′_{S∪{r}} in End_{𝒪} H^{1}(Y_{S}, ℱ_{S}) (or End_{𝒪} H_{c}^{1}(Y_{S}, ℱ_{S})).

6.4. Proof of proposition 5.9. Suppose for the moment that 0 ≤ n ≤ ℓ − 1 and \overline{ρ}|_{I_{ℓ}} is of the form

• ( \begin{matrix}ω^{n+1} & ∗ \\ 0 & 1 \end{matrix} ) with ∗ peu ramifié if n = 0, or

• ( \begin{matrix}ω_{2}^{n+1} & 0 \\ 0 & ω_{2}^{(n+1)ℓ} \end{matrix} ).

Setting S = T (\overline{ρ}) ∪ \{r\}, we find that the group Γ = SL_{2}(Z) ∩ U_{S} satisfies the hypotheses of theorem 6.1. Furthermore

H^{1}(Y_{S}, ℱ_{M}) ≅ H^{1}(Γ, L_{n} ⊗ k)

as a T̃′_{S}-module where M is the module for U_{S,ℓ} = GL_{2}(Z_{ℓ}) defined by the action of GL_{2}(F_{ℓ}) on Hom(L_{n}, k). Therefore 𝔪_{S} is in the support of H^{1}(Y_{S}, ℱ_{M}). On examining the list of possible τ which can be considered strongly acceptable for \overline{ρ} (see the definition after Conjecture 1.6), we see from lemma 3.2 that M is a constituent of Hom_{𝒪} (M_{S}, k). It follows from lemma 6.2 that 𝔪_{S} is in the support of H^{1}(Y_{S}, ℱ_{S}), so L_{S} is non-zero and 𝒩_{S} is non-empty. Moreover by twisting by a power of the Teichmüller character, we see that this holds without the assumption on \overline{ρ}|_{I_{ℓ}} imposed in Theorem 6.1.

Now choose an automorphic representation π in 𝒩_{S}. Then for each p ∈ T (\overline{ρ}), ρ_{π}|_{I_{p}} is necessarily of the form

χ′_{p} ∘ η_{p,2} ⊕ χ′_{p} ∘ Frob_{p} ∘η_{p,2}

for some character χ′_{p} of W (F_{p^{2}}) such that \overline{χ}′_{p} = \overline{χ}_{p}. Suppose now that 𝒪 is sufficiently large that each representation Θ(χ′_{p}) has a model M′_{p} over 𝒪, and set

M′_{\{r\}} = M_{ℓ} ⊗_{𝒪} ⨂_{p∈T (\overline{ρ})} M′_{p}.

It follows from lemma 6.2 and the last part of lemma 4.6 that

Hom_{𝒪[G_{{r}}]} (M′_{{r}}, H^{1}(X_{{r}}, 𝒪))_{𝔪_{S}} ≅ H^{1}(Y_{∅}, ℱ_{M′_{{r}}})_{𝔪_{S}} ≠ 0.

Finally, since M′_{{r}} ⊗_{𝒪} k ≅ M_{{r}} ⊗ k by lemma 3.4, we conclude from lemma 6.2 that 𝔪_{S} is in the support of

H^{1}(Y_{{r}}, ℱ_{{r}}) = H^{1}(Y_{∅}, ℱ_{∅}),

hence 𝒩_{∅} is not empty.

6.5. Proof of proposition 5.12. We first define the pairings ⟨ , ⟩_{S} on L_{S} inducing isomorphisms

L_{S} → Hom_{𝒪} (L_{S}, 𝒪)

of T_{S}-modules. We may assume r ∈ S. We let W_{S} denote the involution of Y_{S} defined by g ↦ (det g)^{−1}gw_{S}. There is a natural isomorphism of sheaves

W_{S}^{∗}ℱ_{S} → ℱ_{S}(χ_{S}) = ℱ_{S} ⊗_{𝒪} ℱ_{χ_{S}∘det},

where χ_{S} = σ_{S}|_{A^{×}∩U_{S}}. (Recall that ℱ_{S} is the sheaf associated to Hom_{𝒪} (M_{S}, 𝒪).) The U_{S}-equivariant perfect pairing chosen in §5.5 gives rise to one on Hom_{𝒪} (M_{S}, 𝒪) which defines an isomorphism of sheaves ℱ_{S}(χ_{S}) → ℱ_{M_{S}}. We thus obtain an isomorphism

H_{c}^{1}(Y_{S}, ℱ_{S}) ⟶^{∼} H_{c}^{1}(Y_{S}, W_{S}^{∗}ℱ_{S}) ⟶^{∼} H_{c}^{1}(Y_{S}, ℱ_{M_{S}}).

The cup product gives rise to a pairing

H_{c}^{1}(Y_{S}, ℱ_{M_{S}}) × H^{1}(Y_{S}, ℱ_{S}) → H_{c}^{2}(X_{S}, 𝒪) ≅ 𝒪,

which defines an isomorphism

H_{c}^{1}(Y_{S}, ℱ_{M_{S}}) → Hom_{𝒪} (H^{1}(Y_{S}, ℱ_{S}), 𝒪).

We thus obtain an isomorphism

H_{c}^{1}(Y_{S}, ℱ_{S}) → Hom_{𝒪} (H^{1}(Y_{S}, ℱ_{S}), 𝒪),

which one can check is T̃′_{S}-linear. Localizing at 𝔪 and applying Lemma 6.3, we obtain the desired isomorphism

L_{S} → Hom_{𝒪} (L_{S}, 𝒪)

of T̃_{S}-modules, arising from an alternating pairing ⟨ , ⟩_{S} on L_{S}.

Now we suppose p is a prime not in S ∪ {ℓ} and we define a homomorphism

i_{S,S′} : L_{S} → L_{S′}

of T_{S′}-modules where S′ = S ∪ {p}. We use the identity map if p = r, and then

we can can assume r ∈ S. We let η_{p^{n}} = ( \begin{matrix}p^{−n} & 0 \\ 0 & 1 \end{matrix} )_{p}. Let T̃_{S}^{(p)} denote the

polynomial algebra over 𝒪 generated by the variables T_{q} and S_{q} for q ∉ S′ ∪ S(\overline{ρ})

and U_{q} for q ∈ S. We regard T̃_{S}^{(p)} as a subalgebra of both T̃′_{S} and T̃′_{S′}, and let

𝔪_{S}^{(p)} = 𝔪′_{S} ∩ T̃_{S}^{(p)} = 𝔪′_{S′} ∩ T̃_{S}^{(p)}.

If 0 ≤ n ≤ e_{p}, then g ↦ gη_{p^{n}} defines a map Y_{S′} → Y_{S} and η_{p^{n}}^{∗}ℱ_{S} is canonically

isomorphic to ℱ_{S′}. We also use η_{p^{n}} to denote the induced T̃_{S}^{(p)}-linear map on cohomology

H^{1}(Y_{S}, ℱ_{S}) → H^{1}(Y_{S′}, ℱ_{S′}).

Now consider the map

ι_{S}^{S′} : H^{1}(Y_{S}, ℱ_{S})_{𝔪_{S}^{(p)}} → H^{1}(Y_{S′}, ℱ_{S′})_{𝔪_{S}^{(p)}}

defined by • x ↦ pη_{1}x − η_{p}T_{p}x + η_{p^{2}}S_{p}x, if e_{p} = 2;

• x ↦ pη_{1}x − η_{p}U_{p}x, if e_{p} = 1;

• x ↦ η_{1}x, if e_{p} = 0. Using lemma 6.2 we can identify H^{1}(Y_{S}, ℱ_{S})_{𝔪_{S}^{(p)}} with the T_{S′}-module L_{S}. One then checks that the image of ι_{p} is in H^{1}(Y_{S′}, ℱ_{S′})_{𝔪_{S}^{(p)}}[U_{p}], which can be identified

with the T_{S′}-module L_{S′}. We thus obtain the desired map i_{S}^{S′} : L_{S} → L_{S′}. These are compatible for varying p in the sense that i_{S∪{p}}^{S′} ∘ i_{S}^{S∪{p}} = i_{S∪{q}}^{S′} ∘

i_{S}^{S∪{q}} if S′ = S ∪ {p, q} for distinct primes p, q ∉ S ∪ {ℓ}. We can therefore inductively define i_{S}^{S′} : L_{S} → L_{S′} if S ⊂ S′ and ℓ ∉ S′.

To complete the proof of the proposition, we can assume S′ = S ∪ {p}. First consider the calculation of j_{S′}^{S}i_{S}^{S′}. The assertion holds for p = r since μ_{r} is a unit in T_{S}. We can then assume r ∈ S.

In the case e_{p} = 2, one finds that the adjoint of η_{p^{n}} is the trace map

H_{c}^{1}(Y_{S′}, ℱ_{S′}) → H_{c}^{1}(Y_{S}, ℱ_{S})

with respect to η_{p^{2−n}}, which we denote η_{p^{2−n}}^{t}. After localizing at 𝔪_{S}^{(p)}, we can compute the composites η_{p^{m}}^{t}η_{p^{n}} on H^{1}(X_{S}, 𝒪), for example. Finally, we are reduced to the calculation of

d(p, −T_{p}, S_{p}) ( \begin{matrix}T_{p}^{2} − (p + 1)S_{p} & pT_{p} & p(p + 1) \\ pT_{p} & p(p + 1) & pS_{p}^{−1}T_{p} \\ p(p + 1) & pS_{p}^{−1}T_{p} & S_{p}^{−2}(T_{p}^{2} − (p + 1)S_{p}) \end{matrix} ) ( \begin{matrix}p \\ −T_{p} \\ S_{p} \end{matrix} )

where d is the prime-to-ℓ part of ϕ(p^{2}). The result is −dμ_{p}.

The case of e_{p} = 1 is similar, except that one gets −pU_{p}(p^{2} − U_{p}^{∗}U_{p}) where

U_{p}^{∗} = V_{S,p} ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} )_{p} V_{S,p}. One then uses that

U_{p}^{∗}U_{p} = \{ \begin{matrix}1, & if π_{p} is unramified special, \\ p, & if π_{p} is principal series with e_{p} = 1. \end{matrix}

Note that this also shows that U_{p} is an automorphism of L_{S}.

In the case e_{p} = 0, one gets j_{S′}^{S}i_{S}^{S′} = p + 1 if p ∈ T (\overline{ρ}) and j_{S′}^{S}i_{S}^{S′} = 1 if p ∉ T (\overline{ρ}). We now turn to the proof that L_{S} ⊗_{𝒪} k → L_{S′} ⊗_{𝒪} k is injective. Again we may

assume r ∈ S and S′ = S ∪ {p} for some p ∉ S ∪ {ℓ}.

First we treat the case e_{p} = 0. There is nothing to prove if p ∉ T (\overline{ρ}), and if p is in T (\overline{ρ}) then the assertion is immediate from lemma 6.2.

The remaining cases of proposition 5.12 follow from lemma 6.2 and the lemma below, the first part of which is essentially due to Ihara [21] and the second to Wiles [34]. For the following lemma, we let V_{1}(N) = ∏_{q} U_{1}(q^{v_{q}(N)}), V_{0}(N) = ∏_{q} U_{0}(q^{v_{q}(N)}) and V (N) = ∏_{q} U(q^{v_{q}(N)}). We also let V_{1,0}(N, N′) denote V_{1}(N) ∩

V_{0}(N′) and V_{1}(N, N′) denote V_{1}(N) ∩ V (N′). We set η = η_{p} = ( \begin{matrix}p^{−1} & 0 \\ 0 & 1 \end{matrix} )_{p}; we

also use η to denote the map H^{1}(X_{V} , ℱ_{M}) → H^{1}(X_{V′}, ℱ_{M}) induced on cohomology whenever ηV′η^{−1} ⊂ V and V_{p} acts trivially on M.

Lemma 6.6. Suppose that D and N are relatively prime positive integers with N > 3 and p not dividing NDℓ. Let 𝔪 = 𝔪_{Σ} where Σ is a finite set of primes contining those dividing NDpℓ. Suppose that M is a k[GL_{2}(Z/DZ)]-module finitedimensional over k.

1. The map

H^{1}(X_{V_{1}(N)}, ℱ_{M})_{𝔪}^{2} ⟶^{1⊕η} H^{1}(X_{V_{1}(Np)}, ℱ_{M})_{𝔪}

is injective.

2. If s ≥ 1 then

(0) → H^{1}(X_{V_{1}(Np^{s−1})}, ℱ_{M})_{𝔪} ⟶^{(−η,1)} H^{1}(X_{V_{1}(Np^{s})}, ℱ_{M})_{𝔪}^{2} ⟶^{1⊕η} H^{1}(X_{V_{1}(Np^{s+1})}, ℱ_{M})_{𝔪}

is exact.

Proof. We first explain the proof of the second part. Let V = η^{−1}V_{1}(Np^{s})η and L = H^{1}(X_{V_{1}(N,p^{s})}, ℱ_{M})_{𝔪}. Then using lemma 6.2, we see that we are required to check the exactness of

(0) ⟶ L^{V_{1}(Np^{s−1})} ⟶ L^{V} ⊕ L^{V_{1}(Np^{s})} ⟶ L^{V ∩V_{1}(Np^{s})},

where the nontrivial maps are given by x ↦ (−x, x) and (x, y) ↦ x + y. Thus it suffices to check that V_{1}(Np^{s−1}) is generated by V_{1}(Np^{s}) and V, which is a straightforward calculation.

We now turn to the proof of the first part of the lemma. By lemma 6.2, it suffices to show that if (x, y) is in the kernel of

H^{1}(X_{V_{1}(N)}, ℱ_{M})^{2} ⟶^{1⊕η} H^{1}(X_{V_{1,0}(N,p)}, ℱ_{M})

then x restricts to zero in H^{1}(X_{V} , ℱ_{M}) for some open subgroup V ⊂ V_{1}(N). However we can rewrite this map as

H^{1}(Γ_{1}(N), M)^{2} ⟶ H^{1}(Γ_{1}(N) ∩ Γ_{0}(p), M)

with the map induced by 1 ⊕ μ where μ = ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) ∈ GL_{2}(Z[1/p]). Thus it

suffices to show that if (x, y) is in the kernel, then there is a congruence subgroup Γ ⊂ Γ_{1}(N) with Res x = 0 in H^{1}(Γ, M). Let Δ ⊂ SL_{2}(Z[1/p]) denote the subgroup

of elements congruent to ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) modulo N. Then Δ is the amalgam of Γ_{1}(N)

and μ^{−1}Γ_{1}(N)μ over Γ_{1}(N) ∩ Γ_{0}(p) (see Serre [29, II.1.4]). Thus we have an exact sequence

H^{1}(Δ, M) → H^{1}(Γ_{1}(N), M) ⊕ H^{1}(μ^{−1}Γ_{1}(N)μ, M) → H^{1}(Γ_{1}(N) ∩ Γ_{0}(p), M).

(See [29, II.2.8].) Thus it suffices to show that if x ∈ H^{1}(Δ, \overline{σ}) there is a congruence subgroup Δ′ ⊂ Δ with 0 = Res x ∈ H^{1}(Δ′, M). We may choose a subgroup Δ′ of finite index in Δ such that Res x = 0, and since Δ has the congruence subgroup property (see [27]), Δ′ will be a congruence subgroup. ■

6.7. Proof of proposition 5.14. For the rest of the section, S will denote a fixed set of primes as in §5.13. Recall that r is a fixed prime such that no lift of \overline{ρ} can be ramified at r. It is convenient to choose another such prime r′ and a character ψ : (Z/r′Z)^{×} → 𝒪^{×} of order greater than 2. For each prime p, we define open subgroups V_{1,p} ⊂ V_{0,p} ⊂ GL_{2}(Z_{p}) as follows:

• V_{1,p} = V_{0,p} = U_{{r},p} if p ∉ S ∪ {r′}, where U_{{r,r′},p} was defined in §5.1;

• V_{1,p} = U_{1}(p) and V_{0,p} = U_{ℓ}(p) if p ∈ S;

• V_{1,p} = V_{0,p} = U_{0}(r′^{2}) if p = r′.

We then set V_{1} = ∏_{p} V_{1,p} and V_{0} = ∏_{p} V_{0,p}, so V_{1} ⊂ V_{0} ⊂ U_{{}∅}. We identify U′_{S}/U′_{0,S} with Δ_{S} = ∏_{p∈S} Δ_{p}, where Δ_{p} denotes the ℓ-Sylow subgroup of (Z/pZ)^{×}. Recall that we defined a representation σ_{∅} of U_{∅} in §5.1. Now we let σ = σ_{∅} ⊗ ψ_{r′}^{−2} where ψ_{r′} is the character of U_{0}(r′^{2}) → (Z/r′Z)^{×} gotten by composing with ψ. If χ is a Dirichlet character and π is an automorphic representation, we write simply π · χ for π · χ_{A} ∘ det where χ_{A} is the character of A^{×}/Q^{×}R^{×} corresponding to χ.

The analogue of lemma 5.2 for V_{1} and V_{0} is the following:

Lemma 6.8. Suppose that ρ_{π⊗ψ} ≅ \overline{Q}_{ℓ} ⊗_{𝒪′} ρ for some deformation ρ of \overline{ρ} to the ring of integers 𝒪′ of a finite extension K′ of K.

1. The space

Hom_{V_{0}}(σ, π^{∞}) ≠ (0)

is non-zero if and only if π · ψ ∈ 𝒩_{∅}. In that case the dimension is 3 · 2^{s}.

2. The space

Hom_{V_{1}}(σ, π^{∞}) ≠ (0)

is non-zero if and only if the following hold:

• π · ψχ is in 𝒩_{S} for some character χ unramified outside S;

• c(π_{p}) ≤ 1 for all p ∈ S. In that case the character χ is unique, and the space has dimension 3 · 2^{s−d} where d is the number of primes dividing the conductor of χ.

Proof. For part 1, we note that if p ∈ S, c(π_{p}) ≤ 1 and π_{p} has unramified central character, then ρ_{π⊗ψ^{−1}} is unramified at p (using the above description of the universal deformation, or the analysis of local lifts in [14]). So if Hom_{V_{0}}(σ, π^{∞}) is not zero, then c(π_{p}) = 0 for each p ∈ S and it follows that Hom_{U_{{r,r′}}}(σ_{{r,r′}}, (π·ψ)^{∞}) is not zero. Therefore π ⊗ ψ ∈ 𝒩_{{r,r′}} = 𝒩_{∅}. The dimension computation for π ∈ 𝒩_{∅} is straightforward.

For part 2, we note that if p ∈ S, then the central character χ_{π_{p}} of π_{p} has ℓ-power order. We may therefore choose a finite order character χ of A^{×} so that χ_{p}^{2} = χ_{π_{p}}^{−1}. If Hom_{V_{1}}(σ, π^{∞}) is not zero, then neither is Hom_{U_{S∪{r,r′}}}(σ_{S∪{r,r′}}, (π · ψχ)^{∞}), so π · ψχ is in 𝒩_{S∪{r,r′}} = 𝒩_{S}. The uniqueness of χ is clear and the dimension calculation is straightforward. ■

Choose a model M for σ over 𝒪. For i = 0, 1, we let L_{i} = H^{1}(Y_{V_{i}}, ℱM̌)_{𝔪} where M̌ = Hom_{𝒪} (M, 𝒪) and 𝔪 is the kernel of the map

T̃_{S∪{r,r′}} → k
T_{p}  ↦ ψ(p)^{−1} tr(\overline{ρ}(Frob_{p}))
S_{p}  ↦ ψ(p)^{−2}p^{−1} det(\overline{ρ}(Frob_{p})).

Lemma 6.2 lets us identify L_{i} with Hom_{𝒪[V_{i}]} (M, H^{1}(X_{V} , 𝒪))_{𝔪} for V ⊂ V_{1} ∩ ker σ.

Lemma 6.9. The 𝒪-rank of L_{0} is 3 · 2^{s}#𝒩_{∅}. The 𝒪-rank of L_{1} is 3 · 2^{s}#𝒩_{S}.

Proof. The first assertion follows from lemma 6.8 and the argument of 5.6. Simliarly one finds

rank_{𝒪} L_{1} = ∑_{π′∈𝒩 ′_{S}} dim(Hom_{V_{1}}(σ, π′^{∞})),

where 𝒩′_{S} is the set of automorphic representations as in part 2 of lemma 6.8. Note that if π ∈ 𝒩_{S}, then for each p in S, ρ_{π}|_{I_{p}} is equivalent to a representation of the form ξ_{p} ⊕ ξ_{p}^{−1} where ξ has ℓ-power order. It follows that there are 2^{d} twists of π in 𝒩′_{S}, where d is the number of primes in S such that ξ_{p} is ramified. ■

There is also a natural action Δ_{S} on L_{1} compatible with that of T_{S∪{r,r′}}; in fact, the action Δ_{p} factors through the homomorphism

∏_{p∈S}(Z/pZ)^{×} → T_{S∪{r,r′}}^{×}

sending q^{−1} to the image of S_{q} for each prime q ∉ S such that q ≡ 1 mod N(\overline{ρ})r^{2}ℓ. The key lemma for the proof of proposition 5.14 is the following:

Lemma 6.10. The 𝒪[Δ_{S}]-module L_{1} is free of rank equal to the 𝒪-rank of L_{0}.

Proof. Since L_{0} is isomorphic to L_{1}^{Δ_{S}}, it suffices to prove L_{1} is free over 𝒪[Δ_{S}]. Since L_{1} is an 𝒪[Δ_{S}]-module summand of H^{1}(Y_{V_{1}}, ℱM̌), it suffices to prove that H^{1}(Y_{V_{1}}, ℱM̌) is free. Letting Γ_{i} = GL_{2}(Q) ∩ V_{i}, we have

H^{1}(Y_{V_{1}}, ℱM̌) ≅ H^{1}(Γ_{1}, M̌)

as a module for Δ_{S} ≅ Γ_{0}/Γ_{1}. Since Γ_{1} and Γ_{2} are fundamental groups of connected non-compact Riemann surfaces, they are free groups, so H^{i}(Γ_{1}, A) = H^{i}(Γ_{1}, A) = 0 for j > 1 and any Γ_{0}-module A. Note also that this holds for i = 0 and A = M̌ ⊗_{𝒪} k since \overline{ψ}^{2} is non-trivial. Therefore H^{1}(Γ_{1}, M̌) is torsion-free over 𝒪, and it suffices to prove that H^{i}(Δ_{S}, H^{1}(Γ_{1}, M̌)) = 0 for i > 0 (see [1, VI.8.10], for example), and this is immediate from the Hochschild-Serre spectral sequence. ■

We now complete the proof of proposition 5.14. First note that the image of 𝔞_{S} in R_{V,𝒪}^{∅,D} is trivial, so we have a surjective map T_{S}/𝔞_{S}T_{S} → T_{∅}. By lemma 5.3 this map is an isomorphism after tensoring over 𝒪 with \overline{Q}_{ℓ}, hence after tensoring with K. It follows that the rank of T_{S}[𝔞_{S}] is the same as that of T_{∅}. Setting 𝔟_{S} = Ann_{𝒪[Δ_{S}]} 𝔞_{S}, we also see that the rank of T_{S}[𝔟_{S}] is the same as that of the kernel of T_{S} → T_{∅}. Since

i_{∅}^{S}(L_{∅}) ⊂ L_{S}[𝔞_{S}]

and i_{∅}^{S} is injective with torsion-free cokernel (by lemma 5.12), we conclude that equality holds. Similarly we find that L_{S}[𝔟_{S}] = ker j_{S}^{∅}. Furthermore, using the surjectivity of j_{∅}^{S} and the formula for j_{S}^{∅}i_{∅}^{S}, we conclude that

L_{S}/(L_{S}[𝔟_{S}] + L_{S}[𝔞_{S}]) ⟶^{∼} L_{∅}/#Δ_{S}(L_{∅}).

The first assertion of proposition 5.14 now follows from theorem 3 of [11], and the second follows from surjectivity and comparison of ranks.

7. Applications

Combining Theorem 5.10 with [10, Thm 5.3], we obtain the following result.

Theorem 7.1. Let K denote a finite extension of Q_{ℓ} and k its residue field. Suppose that ρ : G_{Q} → GL_{2}(K) is a continuous odd representation ramified at only finitely many primes. Assume its reduction, \overline{ρ} : G_{Q} → GL_{2}(k) is irreducible and modular. Suppose also that ℓ ≠ 2, that \overline{ρ}|_{Q(\sqrt{(−1)^{(ℓ−1)/2}ℓ})} is absolutely irreducible, and that at least one of the following holds:

• The centraliser of \overline{ρ}(G_{ℓ}) consists only of scalars, ρ is potentially Barsotti-Tate and the type of W D(ρ) is strongly acceptable for \overline{ρ}|_{G_{ℓ}},

• or there are characters χ_{1} and χ_{2} of G_{ℓ} such that χ_{1}|_{I_{ℓ}} and χ_{2}|_{I_{ℓ}} have finite order, \overline{χ}_{1}\overline{ϵ} ≠ \overline{χ}_{2} and

ρ|_{G_{ℓ}} ≃ (\begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix}) .

Then ρ is modular.

Using the theorem, we will obtain the following strengthening of [10, Thm 5.4]:

Theorem 7.2. Let E_{/Q} be an elliptic curve whose conductor is not divisible by 27. Then E is modular.

Let us first recall some basic facts about an elliptic curve E over a finite extension F of Q_{ℓ}. If j(E) ∉ 𝒪_{F}, then E acquires multiplicative reduction over a quadratic extension of F. If j(E) ∈ 𝒪_{F}, then E acquires good reduction over a finite Galois extension F′/F with ramification degree dividing 6 if ℓ > 5, dividing 12 if ℓ = 3 and dividing 24 if ℓ = 2.

In the case of potentially good reduction, the j-invariant of the reduction of the Néron model of E over 𝒪_{F′} is the reduction of j(E). In particular, the notions of potentially good ordinary and potentially supersingular reduction are well-defined. Letting ρ_{E,ℓ} denote the representation of G_{F} on the Tate module of E, we have that ρ_{E,ℓ} ⊗_{Z_{ℓ}} Q_{ℓ} is potentially Barsotti-Tate. In fact it becomes Barsotti-Tate over F′ and the representations W D(ρ_{E,ℓ})|_{I_{F}} and ρ_{E,ℓ′}|_{I_{F}} for ℓ′ ≠ ℓ are defined and equivalent over \overline{Q}.

Lemma 7.3. Let ℓ be a prime and E an elliptic curve over Q_{ℓ}. Suppose that ℓ > 3 or that E has a twist over Q_{ℓ} which acquires semistable reduction over a tamely ramified extension. Then E has a quadratic twist E′ over Q_{ℓ} with discriminant Δ such that one of the following holds:

• E′ has semistable reduction over Q_{ℓ};

• ℓ ≠ 2 and E′ has good reduction over an extension F of Q_{ℓ} if and only if 4|e(F). In that case v_{ℓ}(Δ) ∈ {3, 9}, j ≡ 1728 mod ℓ and E is potentially ordinary if and only if ℓ ≡ 1 mod 4.

• ℓ ≠ 3 and E′ has good reduction over an extension F of Q_{ℓ} if and only if 3|e(F). In that case v_{ℓ}(Δ) ∈ {4, 8}, j ≡ 0 mod ℓ and E is potentially ordinary if and only if ℓ ≡ 1 mod 3.

We leave the proof to the reader. (See the tables in [12], for example, for the assertions concerning Δ in the case ℓ ≤ 3. for the assertions concerning j, use that (j − 1728)Δ is a square and jΔ is a cube.)

We also recall the following lemma:

Lemma 7.4. Let ℓ be a prime and E an elliptic curve over Q_{ℓ}. Let ρ = ρ_{E,ℓ}⊗_{Z_{ℓ}}Q_{ℓ}. 1. If E has potentially multiplicative reduction, then

ρ ∼ (\begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix})

for some quadratic character χ.

2. If E has potentially good ordinary reduction, then

ρ ≃ (\begin{matrix}ϵω^{j}χ & ∗ \\ 0 & ω^{−j}χ^{−1} \end{matrix})

for some character χ such that χ|_{I_{ℓ}} is trivial if ℓ ≥ 3, and quadratic if ℓ = 2.

3. If E has potentially supersingular reduction, then ρ is irreducible.

Proof. First, we consider the case in which E is potentially ordinary. Let F be a Galois extension of Q_{ℓ} over which E acquires good ordinary reduction. The ℓ-power torsion geometric points of the closed fiber of the Néron model of E_{F} give rise to an unramified quotient of ρ|_{F}, so ρ|_{I_{F}} is of the form

( \begin{matrix}ϵ & ∗ \\ 0 & 1 \end{matrix} ) .

Let v generate the line on which I_{F} acts via ϵ. Since I_{F} is normal in G_{ℓ}, we see that I_{F} acts via ϵ on ρ(g)v for any g ∈ G_{ℓ}, so ρ is reducible. Moreover the representation has the form

( \begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix} )

with χ_{1}|_{I_{F}} = χ_{2}|_{I_{F}} trivial. Since χ_{1}|_{I_{ℓ}} and χ_{2}|_{I_{ℓ}} have order dividing the number of roots of unity in Z_{ℓ}^{×}, they are of the required form.

The case of potentially multiplicative reduction is similar; multiplicative reduction is attained over a quadratic extension and the Tate model yields an unramified quotient.

There remains the case in which E is potentially supersingular. We will prove that if E has potentially good reduction and ρ is reducible, then E is potentially ordinary. Let F be a finite extension of Q_{ℓ} such that E_{F} has good reduction. Since ρ is reducible, we must have

ρ_{E,ℓ}|_{F} ∼ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}) .

Since ρ_{E,ℓ} arises from the Tate module of an ℓ-divisible group Γ over 𝒪_{F}, the same is true of χ_{1} and χ_{2} by [25, Prop 2.3.1]. Since the ℓ-divisible group over 𝒪_{F} corresponding to each χ_{i} has dimension 0 or 1, it follows from [25, Thm 4.2.1] that χ_{i}|_{I_{F}} is either trivial or ϵ|_{I_{F}}. Since χ_{1}χ_{2} = ϵ, it follows that one of the characters is unramified. By Tate’s full faithfulness theorem [32, Thm 4], Γ has non-trivial connected and étale parts, so E_{F} has ordinary reduction. ■

We now prove the following weaker version of Theorem 7.2:

Theorem 7.5. Let E_{/Q} be an elliptic curve such that \overline{ρ}_{E,3}|_{Q(\sqrt{−3})} is absolutely irreducible. If the conductor of E is not divisible by 27, then E is modular.

Proof. Recall that the modularity of \overline{ρ} = \overline{ρ}_{E,3} follows from results of Langlands and Tunnell. If E has a twist with semistable reduction over Q_{3}, then E is modular by Theorem 5.4 of [10], so suppose this is not the case. Since we assume the conductor of E is not divisible by 27, lemma 7.3 shows that E acquires good supersingular reduction over any extension L of Q_{3} with e(L) = 4, but not over any extension with e(L) = 2. It follows that τ = W D(ρ)|_{I_{3}} has the form ω̃_{2}^{2} ⊕ ω̃_{2}^{6}, where ρ = \overline{ρ}_{E,3}.

We now claim that the centralizer of \overline{ρ}|_{G_{3}} consists only of scalars and that W D(ρ)|_{I_{ℓ}} is strongly acceptable for \overline{ρ}. Let F be a ramified quadratic extension of Q_{3} and consider the twist E′ of E_{F} by any ramified quadratic character ψ of G_{F}. Considering ρ_{E′,ℓ′} for any ℓ′ ≠ 3, we see that E′ has good reduction, which is supersingular since j_{E′} = j_{E}. Therefore \overline{ρ}|_{G_{F}} ⊗ ψ arises from a local-local finite flat group scheme over 𝒪_{F} and so satisfies the hypotheses in §2.8. The claim concerning the centralizer follows, and τ is acceptable by theorem 2.10. To conclude strong acceptability, we need to know that if \overline{ρ}|_{G_{3}} is reducible, then the splitting field of E[3](\overline{Q}_{3}) is Q_{3}(\sqrt{−3}, Δ^{1/3}), and \overline{ρ}|_{G_{3}} is peu ramifié since 3|v_{3}(Δ). ■

We shall use Wiles’ argument switching to ℓ = 5, where we have:

Theorem 7.6. Let E_{/Q} be an elliptic curve such that \overline{ρ}_{E,5}|_{Q(\sqrt{5})} is absolutely irreducible. If \overline{ρ}_{E,5} is modular, then E is modular.

Proof. Theorem 5.3 of [10] applies if E has a twist with semistable reduction or, in view of lemma 7.4, potentially ordinary reduction at 5. We will show that theorem 7.1 applies even if E has potentially supersingular reduction (but has no twist with good reduction). Lemma 7.3 shows that we can assume E acquires good reduction over a field F with e(F) = 3. Note then that τ = W D(ρ) must be of the form ω̃_{2}^{8} ⊕ ω̃_{2}^{16}, where ρ = ρ_{E,5}. Applying the results of §2.5 to conclude that the centralizer of \overline{ρ} consists only of scalars and τ is acceptable for \overline{ρ}. Moreover, the list of possibilities in §2.5 shows that \overline{ρ}|_{I_{5}} is isomorphic to one of the following:

• ω_{2}^{m} ⊕ ω_{2}^{5m} for some m ≡ 1 mod 8;

• ( \begin{matrix}ω^{1−m} & ∗ \\ 0 & ω^{m} \end{matrix} ) for some m ∈ \{2, 3\}, with ∗ peu ramifié if m = 2.

Appealing to Theorem 5.3 of [4], we can rule out the possibility that \overline{ρ}|_{I_{5}} ≅ ω_{2} ⊕ ω_{2}^{5} and conclude that τ is strongly acceptable for \overline{ρ}. ■

To remove the irreducibility hypothesis in Theorem 7.5, we need the following lemma. We are grateful to Elkies for providing part of the proof.

Lemma 7.7. Suppose that E is an elliptic curve over Q such that neither \overline{ρ}_{E,5}|_{Q(\sqrt{5})} nor \overline{ρ}_{E,3}|_{Q(\sqrt{−3})} is absolutely irreducible. Then j_{E} ∈ \{0, (11/2)^{3}, 5(29)^{3}/2^{5}\}, and E is modular.

Proof. We divide the proof into four cases, according to whether the representations \overline{ρ}_{E,3} and \overline{ρ}_{E,5} are reducible.

Suppose first that both \overline{ρ}_{E,3} and \overline{ρ}_{E,5} are reducible. Then E gives rise to rational points on X_{0}(15), and as noted in [34], such points are accounted for elliptic curves with conductor 50 (and j = 5(29)^{3}/2^{5}), known to be modular.

Now suppose that one of the representations, say \overline{ρ}_{E,p}, is irreducible. In the case of p = 3, we see (taking into account complex conjugation) that the projective image of \overline{ρ}_{E,3} in PGL_{2}(F_{3}) ≅ S_{4} is isomorphic to Z/2Z × Z/2Z. It follows that the image of \overline{ρ}_{E,3} has order 8 and that \overline{ρ}_{E,3}|_{K} is in fact reducible for some quadratic extension K of Q. In the case of p = 5, we see (again using complex conjugation) that \overline{ρ}_{E,5}|_{Q(\sqrt{5})} is reducible and the image of \overline{ρ}_{E,5} has order 16.

Now suppose that the other of the two representations \overline{ρ}_{E,q} is reducible. The case of p = 3, q = 5 is discussed in the final remark of [34], and the details are given in [10, lemma 5.5]. In that case one finds that E is isomorphic to an modular elliptic curve of conductor 338, with j = (11/2)^{3}. We now analyze the situation with the roles of 3 and 5 interchanged. For clarity, we repeat the argument of [10, Lemma 5.5] with two arbitrary distinct odd primes p and q.

Thus, our elliptic curve E_{/Q} satisfies the properties that there is subgroup of order q defined over Q and E[p](\overline{Q}) contains two lines which are interchanged by the action of G_{Q}. We will now exhibit all such E_{/Q} as Q-rational points on a suitable curve and will thereby check directly that all such E_{/Q} are modular. Define the curve Y_{/Q} to be the quotient of the smooth connected affine curve Y (pq)_{/Q} (in the sense of [22, §3.1]) by the subgroup of elements

g = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(Z/pq)

for which c ≡ 0 mod p, and a ≡ d ≡ 0 mod q or b ≡ c ≡ 0 mod q. This is the coarse moduli scheme attached to the functor “isomorphism classes of elliptic curves E over S with a cyclic order q subgroup C and an unordered pair of cyclic order p subgroups {L_{1}, L_{2}} such that the natural map of S-group schemes L_{1} ×_{S} L_{2} → E[p] is an isomorphism,” for variable Q-schemes S.

The base change Y_{/C} is a smooth connected open Riemann surface which is naturally identified with the quotient of the upper half plane in C by the action of the group of elements g ∈ SL_{2}(Z) whose image in GL_{2}(Z/pq) satisfies the above congruences. The elements of Y (Q) correspond to \overline{Q}-isomorphism classes of triples (E, C, \{L_{1}, L_{2}\}) with E_{/Q} an elliptic curve, C a G_{Q}-stable subgroup of E[q](\overline{Q}) with order q, and \{L_{1}, L_{2}\} a non-ordered set of distinct lines in E[p](\overline{Q}) such that the set \{L_{1}, L_{2}\} is stable under the action of Gal(\overline{Q}/Q) on lines in E[p](\overline{Q}).

In order to determine the Q-rational points on Y, we first identify it with another curve. Let Y_{0}(N)_{/Q} denote the smooth geometrically connected curve which is the coarse moduli scheme for the functor “isomorphism classes of elliptic curves E over S with a cyclic subgroup C of order N” for variable Q-schemes S. If d|N and (d, N/d) = 1, then there is a natural involution W_{d} : Y_{0}(N) → Y_{0}(N) which on geometric points is given by sending (E, C) to (E/C[d], (E[d] + C)/C[d]). This is compatible with the involution W_{d} : Y_{0}(d) → Y_{0}(d) via the natural projection Y_{0}(N) → Y_{0}(d) (we should really write W_{d,N} for accuracy). We also note that if e|(N/d) and (e, N/de) = 1, then the operators W_{d} and W_{e} commute, with composite W_{de}.

There is a natural map Y_{0}(p^{2}q) → Y arising from the map

(E, C) → (E/C[p], C[pq]/C[p], {C[p^{2}]/C[p], E[p]/C[p]})

on ‘points’. This is visibly W_{p^{2}}-invariant, so we get a natural map of smooth geometrically connected curves Y_{0}(p^{2}q)/W_{p^{2}} → Y. One can check that the map is an isomorphism by noting that the resulting map on complex points is a bijection.

We will study Q-rational points on Y_{0}(p^{2}q)/W_{p^{2}}, and even its ‘compactification’ X_{0}(p^{2}q)/W_{p^{2}}, with W_{p^{2}} acting on the smooth connected proper curve X_{0}(p^{2}q)_{/Q} in the unique way extending the above action on Y_{0}(p^{2}q). For p = 3, q = 5, one finds that X_{0}(45)/W_{9} is an elliptic curve of conductor 15, and has at most four rational points, all accounted for by modular elliptic curves with j = (11/2)^{3}. For p = 5, q = 3, the resulting curve X = X_{0}(75)/W_{25} is a curve of genus whose rational points were determined by Elkies as follows. The quotient E_{0} = X/W_{3} has genus one and exactly one rational cusp. Elkies found an explicit Weierstrass equation for E_{0} and concluded it is isomorphic to the elliptic curve of conductor 15 denoted 15-A3(B) [1, 1, 1, −5, 2] in the tables of Cremona [6]. This curve has rank 0 and a torsion subgroup of order 8. One need only look in the fibers of X → E_{0} over the 7 non-cuspidal points in E_{0}(Q) in order to find the rational points on Y. By writing the function field Q(X) as Q(E_{0})[T]/(T^{2} − f) for an explicit f ∈ Q(E_{0}), Elkies showed there are no rational points of X in 6 of the fibers, and that there is a single point in the geometric fiber on X over the remaining point. It follows that there is a unique non-cuspidal point in X(Q), and since it is fixed by W_{3}, it must arise from an elliptic curve with complex multiplication. One can also check that j = 0 for such a curve.

Finally we rule out the possibility that both \overline{ρ}_{E,3} and \overline{ρ}_{E,5} are irreducible. First suppose that E has potentially multiplicative or good ordinary reduction. In that case lemma 7.4 shows that \overline{ρ}_{E,5}|_{G_{5}} is reducible, so its semisimplification is isomorphic to ω\overline{χ} ⊕ \overline{χ}^{−1} for some character χ. On the other hand, since \overline{ρ}_{E,5} is induced from G_{Q(\sqrt{5})}, the ratio of the above characters on G_{5} must be the quadratic character trivial on G_{Q_{5}(\sqrt{5})}. This gives ω\overline{χ}^{2} = ω^{2}, contradicting the fact that ω is not a square. We can therefore assume that E has potentially supersingular reduction. If E has a quadratic twist with good supersingular reduction, then the order of \overline{ρ}_{E,5}(I_{5}) is divisible by 24, contradicting that \overline{ρ}_{E,5}(G_{Q}) has order 16. Otherwise, lemma 7.3 shows that the order of \overline{ρ}_{E,3}(I_{5}) is divisible by 3, contradicting that \overline{ρ}_{E,3}(G_{Q}) has order 8. ■

We now complete the proof of Theorem 7.2. According to theorem 7.5, we may suppose that \overline{ρ}_{E,3}|_{Q(\sqrt{−3})} is not absolutely irreducible. By lemma 7.7, we may assume \overline{ρ}_{E,5}|_{Q(\sqrt{5})} is absolutely irreducible. Wiles’ argument using the Hilbert Irreducibility Theorem shows that there is an elliptic curve E′ over Q such that

• \overline{ρ}_{E′,5} ∼ \overline{ρ}_{E,5};

• \overline{ρ}_{E′,3}|_{Q(\sqrt{−3})} is absolutely irreducible.

Since \overline{ρ}_{E′,5} ∼ \overline{ρ}_{E,5}, the conductor of E′ is not divisible by 27. Therefore E′ is modular by 7.5, so \overline{ρ}_{E,5} ∼ \overline{ρ}_{E′,5} is modular. Therefore E is modular by theorem 7.6.

Finally, we record the following strengthening of theorem 7.6, immediate from theorem 7.2:

Theorem 7.8. Let E_{/Q} be an elliptic curve. If \overline{ρ}_{E,5} is modular or \overline{ρ}_{E,5}|_{Q(\sqrt{5})} is not absolutely irreducible, then E is modular.

Appendix A. Deformation theory

We recall some general facts from the deformation theory of representations of profinite groups. The basic results are due to Mazur [23], with improvements by Ramakrishna, Faltings, deSmit and Lenstra [24], [8].

Let G be a profinite group, and let 𝒪 be a local Noetherian ring with residue field k. We give k the discrete topology. Suppose that V is a finite-dimensional k-vector space with a continuous action of G. We assume that End_{k[G]} V = k, and we consider deformations of the representation

G ⟶ End_{k} V

to certain 𝒪-algebras.

We let 𝒞_{𝒪}^{∗} denote the category of local topological 𝒪-algebras A such that the natural map

A → proj lim_{𝔞∈𝒰 _{A}} A/𝔞

is a topological isomorphism, where 𝒰_{A} is the set of open ideals 𝔞 ≠ A such that A/𝔞 is Artinian. (See [20, Exp. VII_{B}] for the basic theory of such rings.) We remark that if A is an object of 𝒞_{𝒪}^{∗}, then 𝒰_{A} is simply the set of open ideals. We let 𝔪_{A} denote the maximal ideal of A and k_{A} the residue field. Note that we do not assume that k_{A} = k. (In the applications, 𝒪 will be a complete discrete valuation ring, and A will be a complete local Noetherian 𝒪-algebra.)

Definition A.1. A deformation of V to A (an object of 𝒞_{𝒪}^{∗}) is an isomorphism class of A[G]-modules D such that D is free over A, k_{A} ⊗_{A} D is isomorphic to k_{A} ⊗_{k} V and G → End_{A}(D) is continuous.

We let F_{V} (A) denote the set of deformations of V to A. If A → B is a morphism in 𝒞_{𝒪}^{∗}, then extension of scalars defines a map F_{V} (A) → F_{V} (B), allowing us to regard F_{V} as a functor from 𝒞_{𝒪}^{∗} to the category of sets. According to Theorem 7.1 of [8], F_{V} is representable on the full subcategory of 𝒞_{𝒪}^{∗} whose objects have residue field k. The proof actually shows that F_{V} is representable on 𝒞_{𝒪}^{∗} by an object R_{V,𝒪} with residue field k. We call R_{V,𝒪} the universal 𝒪-deformation ring of V, and the canonical element of F_{V} (R_{V,𝒪} ) is called the universal 𝒪-deformation of V.

If R_{V,𝒪} is Noetherian, then it represents F_{V} on the category 𝒞_{𝒪} of complete local Noetherian 𝒪-algebras, because 𝒞_{𝒪} is a full subcategory of 𝒞_{𝒪}^{∗} (as shown by the proof of Proposition 2.4 of [8]). This holds, for instance, if G is topologically finitely generated.

A.2. Change of rings. Suppose we are given another local Noetherian ring 𝒪′ with maximal ideal 𝔪′ and residue field k′, and a local map 𝒪 → 𝒪′. Let V′ = V ⊗_{k} k′ and note that End_{k[G]} V = k if and only if End_{k′[G]} V′ = k′. For an object B in 𝒞_{𝒪′}^{∗} , we can identify F_{V} (B) with F_{V′}(B), so we have a canonical bijection

Hom_{𝒞} _{𝒪}^{∗} (R_{V,𝒪} , B) → Hom_{𝒞} _{𝒪′}^{∗} (R_{V′,𝒪′} , B).

For an object A of 𝒞_{𝒪}^{∗} with residue field k, consider

𝒪′ ˆ⊗_{𝒪} A = proj lim_{n>0,𝔞∈𝒰 _{A}}(𝒪′/(𝔪′)^{n}) ⊗_{𝒪} (A/𝔞).

Each ring (𝒪′/(𝔪′)^{n}) ⊗_{𝒪} (A/𝔞) is a local Artinian ring with residue field k′ because A has residue field k. Thus by Exp. VII_{B}, 0.2 of [20], 𝒪′ ˆ⊗_{𝒪} A is an object of 𝒞_{𝒪′}^{∗} . The natural map A → 𝒪′ ˆ⊗_{𝒪} A is continuous, and for any object B of 𝒞_{𝒪}^{∗}, it induces a bijection

Hom_{𝒞} _{𝒪′}^{∗} (𝒪′ ˆ⊗_{𝒪} A, B) ⟶ Hom_{𝒞} _{𝒪}^{∗} (A, B).

It follows that 𝒪′ ˆ⊗_{𝒪} R_{V,𝒪} is the universal 𝒪′-deformation ring of V′. Furthermore, the universal deformation is obtained by extending scalars from R_{V,𝒪} .

A.3. Restricted deformations. Suppose that for each A in 𝒞_{𝒪}^{∗}, we are given a subset S_{V} (A) of F_{V} (A). We then have the following necessary and sufficient condition for S_{V} to be a functor represented by R_{V,𝒪} /I for some closed ideal I of R_{V,𝒪} : For all A ∈ 𝒞_{𝒪}^{∗} and D ∈ F_{V} (A), we have:

1. D ∈ S_{V} (A) if and only if D/𝔞 ∈ S_{V} (A/𝔞) for all 𝔞 ∈ 𝒰_{A};

2. if 𝔞, 𝔟 ∈ 𝒰_{A}, D/𝔞 ∈ S_{V} (A/𝔞) and D/𝔟 ∈ S_{V} (A/𝔟), then

D/(𝔞 ∩ 𝔟) ∈ S_{V} (A/(𝔞 ∩ 𝔟));

3. if A → A′ is an inclusion of Artinian rings in 𝒞_{𝒪}^{∗}, then D ∈ S_{V} (A) if and only

if D ⊗_{A} A′ ∈ S_{V} (A′). The necessity of (1), (2) and (3) is easily verified. The sufficiency is proved exactly as in Proposition 6.1 of [8].

Suppose we are given a local Noetherian 𝒪-algebra 𝒪′ as in §A.2. Suppose that S_{V} is a restriction on deformations represented by R_{𝒪,V} /I for some closed ideal I, and that S_{V′}(B) = S_{V} (B) for B ∈ 𝒞_{𝒪′}^{∗} . Then S_{V′} is represented by 𝒪′ ˆ⊗_{𝒪} R_{𝒪,V} /I, which is naturally isomorphic to the quotient of R_{𝒪′,V′} by the closure of the ideal generated by the image of I.

Example A.4. This example is based on an observation of Ramakrishna [24]. Suppose that k has positive characteristic and P is a property of finite discrete Gmodules which is preserved under taking submodules, quotients and finite products. Suppose there is a finite subfield k_{0} of k such that V = k ⊗_{k_{0}} V_{0} for some k_{0}[G]-module V_{0}. For D ∈ F_{V} (A) = F_{V_{0}}(A), we let A_{D,0} denote the the image of R_{V_{0},W (k_{0})} → A. We let D_{0} the corresponding element of F_{V_{0}}(A_{D,0}). (Given the quotient topology, A_{D,0} is an object of 𝒞_{W (k_{0})}^{∗} with residue field k_{0}. Since the map A_{D,0} → A is continuous, both spaces are Hausdorff and A_{D,0} is compact, the topology is the same as the subspace topology. This means that the set of 𝔞 ∩ A_{D,0} for 𝔞 ∈ 𝒰_{A} is a base of open ideals in A_{D,0}.)

Define S_{V}^{P} (A) as the set of D such that D_{0}/𝔞 has property P for all 𝔞 in 𝒰_{A_{D,0}}. One checks that S_{V}^{P} is independent of the choice of k_{0} and satisfies (1), (2) and (3). If A itself is finite, then S_{V}^{P} (A) is simply the set of D having property P. Note also that given 𝒪 → 𝒪′ as in §A.2, we have S_{V}^{P} (B) = S_{V′}^{P}(B) for B in 𝒞_{𝒪′}^{∗} .

References

[1] K. Brown, Cohomology of Groups, Springer-Verlag, 1982.

[2] H. Carayol, Sur les représentations ℓ-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Sup. 19 (1986), 409–468.

[3] H. Carayol, Sur les représentations galoisiennes modulo ℓ attachées aux formes modulaires, Duke Math. J. 59 (1989), 785–801.

[4] B. Conrad, Finite group schemes over bases with low ramification, in preparation.

[5] B. Conrad, Ramified deformation problems, in preparation.

[6] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, 1992.

[7] H. Darmon, F. Diamond, R. Taylor, Fermat’s Last Theorem, in Current Developments in Mathematics, 1995, International Press, 1996, pp. 1–154.

[8] B. de Smit, H. Lenstra, Explicit construction of universal deformation rings, in Modular Forms and Fermat’s Last Theorem (Boston, 1995), Springer-Verlag, 1997.

[9] F. Diamond, The refined conjecture of Serre, in Elliptic Curves, Modular Forms and Fermat’s Last Theorem (Hong Kong, 1993), International Press, 1995, pp. 22–37.

[10] F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144 (1996), 137–166.

[11] F. Diamond, The Taylor-Wiles construction and multiplicity one, Inv. Math. 128 (1997), 379–391.

[12] F. Diamond, K. Kramer, Classification of \overline{ρ}_{E,ℓ} by j-invariant of E, in Modular Forms and Fermat’s Last Theorem (Boston, 1995), Springer-Verlag, 1997.

[13] F. Diamond, R. Taylor, Non-optimal levels for mod ℓ modular representations of Gal(\overline{Q}/Q), Inv. Math. 115 (1994), 435–462.

[14] F. Diamond, R. Taylor, Lifting modular mod ℓ representations, Duke Math. J. 74 (1994), 253–269.

[15] J.-M. Fontaine, Représentations p-adiques semi-stables, in Périodes p-adiques, Astérisque 223, pp. 113–184.

[16] J.-M. Fontaine, B. Mazur, Geometric Galois representations, in Elliptic Curves, Modular Forms and Fermat’s Last Theorem (Hong Kong, 1993), International Press, 1995, pp. 41–78.

[17] K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.

[18] P. Gérardin, Facteurs locaux des algebres simples de rang 4. I, in Groupes Réductifs et Formes Automorphes, I (Paris, 1976–77) Univ. Paris VII, 1978, pp. 37–77.

[19] A. Grothendieck, Groupes de Monodromie en Géometrie Algébrique (SGA 7), Lecture Notes in Math. 288, Springer-Verlag, 1972.

[20] A. Grothendieck, Schémas en Groupes (SGA3), Lecture notes in Math. 151, Springer-Verlag, 1970.

[21] Y. Ihara, On modular curves over finite fields, Discrete Subgroups of Lie Groups and Applications to Moduli (Bombay, 1973), Oxford Univ. Press, 1975, pp. 161–202.

[22] N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Princeton Univ. Press, 1985.

[23] B. Mazur, Deforming Galois representations, in Galois Groups over Q (MSRI, 1987) Springer-Verlag, 1989, pp. 385–437.

[24] R. Ramakrishna, On a variation of Mazur’s deformation functor, Comp. Math. 87 (1993), 269–286.

[25] M. Raynaud, Schémas en groupes de type (p, p, …, p), Bull. Soc. Math. France 102 (1974), 241–280.

[26] T. Saito, Modular forms and p-adic Hodge theory, Inv. Math., to appear.

[27] J.-P. Serre, Le problème des groupes de congruence pour SL_{2}, Ann. Math. 92 (1970), 489–527.

[28] J.-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, 1977.

[29] J.-P. Serre, Trees, Springer-Verlag, 1980.

[30] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\overline{Q}/Q), Duke Math. J. 54 (1987), 179–230.

[31] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton Univ. Press, 1971.

[32] J. Tate, p-divisible groups in Proceedings of a Conference on Local Fields (Dreibergen, 1966), Springer, 1967, pp. 158-183.

[33] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572.

[34] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995), 443–551.

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

E-mail address: bconrad@math.harvard.edu

Department of Mathematics, M.I.T., Cambridge, MA 02139, USA

E-mail address: fdiamond@math.mit.edu

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

E-mail address: rtaylor@math.harvard.edu