HECKE RINGS AND UNIVERSAL DEFORMATION RINGS

EHUD DE SHALIT

1. Introduction

Wiles’ proof of the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves is based on the ”modularity” of certain universal deformation rings.

Fix an odd irreducible representation

ρ̄: G_{ℚ} → GL_{2}(k) (1.1)

from the absolute Galois group of ℚ to the group of 2 × 2 invertible matrices over a finite field k, and a deformation type D (see section 2 for precise definitions). One constructs then a certain complete noetherian local ring, the universal deformation ring R_{D} = R_{D}(ρ̄), and a universal deformation ρ_{D}^{univ} : G_{ℚ} → GL_{2}(R_{D}), whose specializations give all the deformations of ρ̄ of type D, up to strict equivalence (see [M2], [M3]). Here it is implicitly assumed that ρ̄ itself is of type D, to begin with.

If we assume in addition that \overline{ρ} is modular (in the sense that it comes from reduction mod λ of the λ-adic representation associated to some cusp form, see 2.1 below), one can also associate to D another complete noetherian local ring, the Hecke algebra T_{D}, and a canonical homomorphism φ_{D} : R_{D} → T_{D} of local rings. T_{D} is the (p-adic completion of the) Hecke algebra acting on all the modular forms whose associated λ-adic representation is of type D, and lifts ρ̄. The modularity of ρ̄ is needed to assure that there is at least one such form. The homomorphism φ_{D} is derived from the universality of (R_{D}, ρ_{D}^{univ}). Similarly, any deformation ρ of ρ̄ with values in a complete local ring R defines a homomorphism φ : R_{D} → R bringing ρ_{D}^{univ} to ρ, and ρ is called modular if and only if φ factors through φ_{D}. The assertion that every deformation of type D is modular is therefore equivalent to the assertion that φ_{D} is an isomorphism. In this set-up the main theorem to be proved is the following.

Theorem 1. Assume that D is a minimal deformation type. Then (i) φ_{D} is an isomorphism (ii) T_{D} is a local complete intersection (l.c.i.).

We shall follow the proof of this theorem given by R. Taylor and A. Wiles in the appendix to their paper^{1}. The proof gives (i), and the complete intersection property (ii) comes as a by-product, rather than a prerequisite, to (i). In the earlier proof of (i), given by Wiles in chapter 3 of [W], (ii) had to be known in advance, and its independent proof made up the main body of [T-W]. It was observed by Faltings that a slight modification of the arguments of Taylor and Wiles yields both (i) and (ii) simultaneously, and this observation was incorporated into the appendix of [T-W].

Date: February 16, 1996.

1Our exposition benefitted at several points also from the excellent survey paper by Darmon, Diamond and Taylor [D-D-T].

The minimality condition on D (see remark 2 in section 2 below for a precise definition) is essential for the proof to work. It means, roughly speaking, that a deformation of type D has ”no more ramification than what ρ̄ forces it to have”. While the main theorem remains true without the minimality assumption, different ideas are needed to pass from minimal to non-minimal D. Very roughly, one measures by how much R_{D} and T_{D} change when D is modified, and proves that the equality R_{D} = T_{D} propagates from the minimal deformation type to any D. The Gorenstein-ness of T_{D} plays a crucial role in establishing a tool to measure the ”change in T_{D}”. The passage from the minimal case to the general case is treated in Ribet’s article in this volume [Ri2].

We shall not work in greatest generality. For example, we shall assume throughout that the determinant of ρ is the cyclotomic character, and at l ≠ p (p is the characteristic of k) we shall assume that D is of ”type A” in Wiles’ terminology (Wiles himself considered types ”B” and ”C” as well, and Diamond [Di2] [Di3] completed the picture by allowing the restriction of ρ to the decomposition group at l to be arbitrary). However, this will be enough for the application to the Shimura-Taniyama-Weil conjecture, in the minimal case.

Before we turn to a detailed outline of the proof of theorem 1, let us explain how it implies the Shimura-Taniyama-Weil conjecture for semi-stable elliptic curves. By abuse of language one calls an elliptic curve E over ℚ semi-stable if it has good or semi-stable (multiplicative) reduction everywhere. This is equivalent to the fact that the conductor of E is a square-free integer. Let E be a semi-stable elliptic curve defined over ℚ, take p = 3, and assume that ρ̄ = ρ̄_{E,3}, the representation of G_{ℚ} on the 3-division points, is irreducible. (The reducible case is handled via a trick of Wiles which involves ρ̄_{E,5} as well. See chapter 5 of [W] and [Ru].) Then in fact ρ̄ satisfies all the technical conditions listed below, in section 2.1. Let us quickly check them.

First, GL_{2}(𝔽_{3}) can be lifted to a subgroup of GL_{2}(A) for some ring of integers A in a number field in which 3 splits completely, and therefore ρ̄ can be viewed as a complex representation. Since P GL_{2}(𝔽_{3}) ≅ S_{4}, ρ̄ is modular by the Langlands-Tunnell theorem. Second, the determinant of ρ̄ is the cyclotomic character mod 3 thanks to the Weil pairing. Third, ρ̄ remains absolutely irreducible when restricted to the absolute Galois group of L = ℚ(\sqrt{−3}). If ρ̄|G_{L} were reducible, it would have two invariant lines (in 𝔽̄_{3}^{2}), for if it had only one such line, that line would be invariant under G_{ℚ} as well. Thus ρ̄(G_{L}) is a torus, the splitting field of ρ̄ is an abelian extension M of L, and [M : L] is relatively prime to 3. Since E is semistable, M/L would be unramified outside 3. Class field theory tells us that there are no abelian extensions of L of degree relatively prime to 3 which are unramified outside 3, so M = L, contradicting the irreducibility of ρ̄. Finally, ρ̄ is ”type A” at every l ≠ 3 and ”Selmer” or ”flat” at 3 by the semi-stability of E, as follows from the l-adic analytic model of E as a Tate curve (see below).

For any p, ρ = ρ_{E,p}, the representation of G_{ℚ} on the p-adic Tate module of E, is a deformation of ρ̄ = ρ̄_{E,p}, the representation on the p-division points. It is a minimal deformation if and only if for every prime of bad reduction (including possibly p) the order of the minimal discriminant Δ_{E} of E at that prime is not divisible by p. This follows easily from the Tate parametrization of E at the bad primes. At a prime l of split multiplicative reduction one has E(ℚ̄_{l}) ≅ ℚ̄_{l}^{×}/ ⟨q_{E,l}⟩ as a Galois module. In particular one has a short exact sequence of Galois modules

0 → μ_{p} → E[p] → ℤ/pℤ → 0 (1.2)

and the splitting field of E[p] is ℚ_{l}(μ_{p}, q_{E,l}^{1/p}). If the order of Δ_{E}, hence also of the Tate period q_{E,l}, is divisible by p, ρ̄ is unramified (if l ≠ p) or flat (if l = p), but ρ is ramified or non-flat (resp.). On the other hand if the order of q_{E,l} is not divisible by p, already ρ̄ is ramified (if l ≠ p) or Selmer non-flat (if l = p). If l is a prime of non-split multiplicative reduction then the above analysis applies to the unramified quadratic twist of E. Since the notion of being unramified or flat is invariant under unramified twists, the same conclusion holds.

Now let D be the minimal deformation type described in section 2.2 below, and assume that the order of Δ_{E} at primes of bad reduction is not divisible by 3. Then ρ = ρ_{E,3} is a deformation of type D of ρ̄. The main theorem therefore implies that ρ factors through T_{D}, so there exists a ring homomorphism h : T_{D} → ℤ_{3}, such that h ∘ φ_{D} brings ρ_{D}^{univ} to ρ. But h defines a ℤ_{3}-valued weight-2 newform with trivial nebentypus, and for all but finitely many primes l, φ_{D}(tr(ρ_{D}^{univ}(F rob_{l}))) = T_{l} is the l-Hecke operator. It follows that tr(ρ(F rob_{l})) = h(T_{l}). Since also det(ρ(F rob_{l})) = l, and ρ is irreducible, ρ is the representation associated to the modular form h. If the level of h is N then the Isogeny Theorem (due in this case to Serre and in general to Faltings) implies that there exists a non-constant morphism from the modular curve X_{0}(N) to E. However, for many applications, such as the analytic continuation and functional equation of L(E/ℚ, s), it is enough to know that ρ is associated to a modular form.

Example 1. Take for E the curve y^{2} + xy = x^{3} − x^{2} − x, p = 3, and ρ̄ = ρ̄_{E,3}. Then

• Im(ρ̄) = GL_{2}(𝔽_{3}) (exercise !), det(ρ̄) = ω

• ρ̄ is flat non-ordinary at 3 (E has good, supersingular reduction there)

• ρ̄ is semi-stable ramified at 73 (E is a Tate curve at 73 and Δ_{E} = 73)

• E has good reduction outside 73.

The first point implies the irreducibility of ρ̄. The last two points imply that ρ_{E,3} is a minimal deformation of ρ̄. Thus the main theorem applies as it stands, and proves that E is modular. Indeed, it is the curve labelled 73A in the Antwerp tables.

2. An outline of the proof

2.1. Set-up. Let p ≥ 3 be a prime number, k a finite field of characteristic p, and ρ̄: G_{ℚ} → GL_{2}(k) an odd irreducible continuous representation. Assume

• ρ̄ is modular - there exist (a) a newform f of weight κ, level N and nebentypus ψ (for some κ, N and ψ), (b) a prime λ in the field K_{f} generated over ℚ by the Fourier coefficients of f, dividing p, (c) an embedding of O_{K_{f}} /λ in k̄, such that for every l not dividing N

det(X − ρ̄(F rob.l)) ≡ X^{2} − a_{l}(f)X + ψ(l)l^{κ−1} mod λ (2.1)

• det(ρ̄) = ω, the cyclotomic character mod p

• ρ̄|G_{L} is absolutely irreducible, where L = ℚ (\sqrt{(\frac{−1}{p}) p})

• ρ̄|G_{l} ∼ \binom{ωχ^{−1} ∗}{χ} and χ is unramified, for l ≠ p (G_{l} is the decomposition

group at l)

• ρ̄|G_{p} is either flat - it is the Galois module attached to the generic fiber of a finite flat k-vector group scheme over ℤ_{p}, or it is not flat but Selmer - ρ̄|G_{p}

∼ \binom{ωψ^{−1} ∗}{ψ} for an unramified ψ. (In such a case, the ∗ has to be ”trés

ramifié” in the language of [Se], otherwise ρ̄|G_{p} would be both Selmer and flat.)

Remark 1. (i) N(ρ̄), the prime-to-p Artin conductor of ρ̄, is square free, and ρ̄ is ramified at p too.

(ii) In the Selmer non-flat case (at p), and in the ramified case (at l ≠ p), the unramified character ψ (resp. χ) is trivial or quadratic. This observation is due to Diamond ([Di1], 6.1 and 6.2).

For example, at l ≠ p, the proof of (ii) goes as follows. ρ̄|G_{l} factors through a two-step solvable extension of ℚ_{l}, with an unramified quotient and a tamely ramified submodule which is isomorphic to k. The unramified quotient acts by conjugation on the ramified submodule through the character ωχ^{−2}. On the other hand, we know from the structure theory of tamely ramified extensions, that this action should be through ω, hence χ^{2} = 1.

2.2. Deformations. Let O be the ring of integers in a finite extension of ℚ_{p} such that, denoting by λ the prime ideal of O, O/λ = k. Let Σ be the set of primes where ρ̄ is ramified (p ∈ Σ). The key technical point in the proof of theorem 1 is the introduction of an auxiliary set of primes Q = {q_{1}, ..., q_{r}} satisfying

• Q ∩ Σ = ∅, ∀q ∈ Q q ≡ 1 mod p

• ∀q ∈ Q ρ̄(F rob_{q}) has distinct eigenvalues {α_{q}, β_{q}} contained in k.

If k is too small to contain the eigenvalues of some Frobenius, replace it by its quadratic extension, and change O and λ accordingly. More assumptions on Q will be imposed along the way. The set Q will vary, but ultimately our interest lies in Q = ∅. By abuse of notation we shall write Q also for the product of the primes in Q. Let ℚ_{S} (where S is a set of primes) be the maximal extension of ℚ which is unramified outside the primes in S, and G_{S} = Gal(ℚ_{S}/ℚ). We are now ready to define the deformation type.

Definition 1. A deformation of type D_{Q} of ρ̄ is a continuous representation ρ : G_{Σ∪Q} → GL_{2}(R), where R is a local complete noetherian O-algebra, with maximal ideal 𝔪_{R} and residue field R/𝔪_{R} = k, satisfying

• ρ mod 𝔪_{R} = ρ̄

• det(ρ) = ϵ, the cyclotomic character

• ρ|G_{l} ∼ \binom{ϵχ^{−1} ∗}{χ} and χ is unramified, for l ∈ Σ, l ≠ p (ρ is ”type A”)

• If ρ̄|G_{p} is flat, so is ρ - meaning that for every ideal I ⊆ R with R/I finite, ρ mod I is the Galois module attached to the generic fiber of a finite flat group scheme over ℤ_{p}, endowed with an action of R/I making it free of rank 2 over R/I.

If ρ̄ is not flat but Selmer then ρ|G_{p} ∼ \binom{ωψ^{−1} ∗}{ψ} for an unramified

ψ.

Remark 2. (i) As above, in the Selmer non-flat case (at p), and in the ramified case (at l ∈ Σ, l ≠ p), the unramified character ψ (resp. χ) is trivial or quadratic. Since p ≠ 2, it follows that it is the same character as the one figuring in ρ̄.

(ii) When Q = ∅, D (= D_{∅}) is minimal: if ρ̄ is unramified (at l ≠ p) or flat (at p), so is ρ.

Although no condition is imposed at q ∈ Q, we have

Lemma 2. ([T-W], appendix, lemma 7) If q ∈ Q, then ρ|G_{q} ∼ ( \begin{matrix}ϕ_{1} \\ & ϕ_{2} \end{matrix} ) .

Proof : It is enough to prove the lemma when R is Artinian. Since ρ̄ is unramified at q, and R is a p-adic ring, ρ(I_{q}) is a pro-p group. Since q ≠ p, ρ is tamely ramified at q. Let f and t be topological generators of Gal(ℚ_{q}^{tr}/ℚ_{q}) such that f restricts to the Frobenius automorphism on the maximal unramified extension ℚ_{q}^{nr} of ℚ_{q}, and t fixes ℚ_{q}^{nr}. Using the fact that α_{q} ≠ β_{q} choose a basis for the

space of ρ in which ρ (f) = ( \begin{matrix}a \\ & b \end{matrix} ) is diagonal. Since ρ̄ is unramified at q,

ρ(t) ≡ 1 mod 𝔪_{R}. Now suppose that ρ(t) = ( \begin{matrix}μ_{1} \\ & μ_{2} \end{matrix} ) (1 + N) = M(1 + N)

and N ≡ 0 mod 𝔪_{R}^{n} (n ≥ 1). Since M and N commute modulo 𝔪_{R}^{n+1} we have

ρ(t)^{q} ≡ ( \begin{matrix}μ_{1}^{q} \\ & μ_{2}^{q} \end{matrix} ) (1 + qN) mod 𝔪_{R}^{n+1}. Using the relation ftf^{−1} = t^{q} one gets

( \begin{matrix}a \\ & b \end{matrix} ) ( \begin{matrix}μ_{1} \\ & μ_{2} \end{matrix} ) (1 + N) ( \begin{matrix}a^{−1} \\ & b^{−1} \end{matrix} ) ≡ ( \begin{matrix}μ_{1}^{q} \\ & μ_{2}^{q} \end{matrix} ) (1 + qN) mod 𝔪_{R}^{n+1}

which implies (since q ≡ 1 mod p) that N is diagonal mod𝔪_{R}^{n+1}, and the desired result follows by induction on the nilpotency degree of 𝔪_{R}.

Definition 2. Let Δ_{q} = the p-Sylow subgroup of (ℤ/qℤ)^{×},

χ_{q} : G_{ℚ} → Δ_{q} the composition of the cyclotomic character mod q and the projection from (ℤ/qℤ)^{×} to Δ_{q}. (We call χ_{q} the nebentypus character of conductor q.)

Δ_{Q} = ∏_{q∈Q} Δ_{q}, Λ_{Q} = O[Δ_{Q}] (a local ring), and χ_{Q} = ∏_{q∈Q} χ_{q}.

Let us also distinguish between ϕ_{1} and ϕ_{2} using the convention that ϕ_{1} mod 𝔪_{R} sends F rob_{q} to α_{q} (this is possible since α_{q} ≠ β_{q}).

Corollary 3. ϕ_{1}|I_{q} = (ϕ_{2}|I_{q})^{−1} factors through χ_{q}|I_{q} : ϕ_{1}|I_{q} = ϕ̃_{1} ∘ χ_{q}|I_{q} for a unique character ϕ̃_{1} : Δ_{q} → (1 + 𝔪_{R}).

Proof : The first equality follows simply from the fact that ϵ = ϕ_{1}ϕ_{2} is unramified at q. Since ϕ_{1} mod 𝔪_{R} is unramified too, ϕ_{1}(I_{q}) ⊂ 1+𝔪_{R}. Now ϕ_{1} factors through the maximal abelian extension of ℚ_{q}, which, by the local Kronecker-Weber theorem, is generated by roots of unity. In particular ϕ_{1}|I_{q} factors through the cyclotomic character into ℤ_{q}^{×}, and since 1 + 𝔪_{R} is a pro-p group, through the cyclotomic character mod q, and eventually through χ_{q}.

The local conditions defining the deformation type D_{Q} are ”conditions” in the technical sense of [M3]. It follows from the irreducibility of ρ̄ that the deformation problem is representable. Thus there exist a universal deformation ring R_{Q}, and a universal deformation ρ_{Q}^{univ} : G_{ℚ} → GL_{2}(R_{Q}), such that every deformation ρ : G_{Σ∪Q} → GL_{2}(R) of type D_{Q} is strictly equivalent to a unique specialization of ρ_{Q}^{univ} under a unique homomorphism R_{Q} → R.

Next, apply the above discussion, on the shape of ρ|I_{q}, to the universal deformation ρ_{Q}^{univ}. Denote the ϕ̃_{1} of the corollary in this case by ϕ̃_{1}^{univ}. We shall give R_{Q} the structure of a Λ_{Q}-algebra by mapping Δ_{Q} to it via the homomorphism (ϕ̃_{1}^{univ})^{2} (the reason for the square will become clear soon; note that R_{Q} is already an O-algebra). Let 𝔞_{Q} be the augmentation ideal in Λ_{Q}. Then from lemma 2 and corollary 3, ρ_{Q}^{univ} mod 𝔞_{Q}R_{Q} is unramified at the q’s dividing Q, hence it is of type D. The universality of R (= R_{∅}) implies that there is a unique homomorphism R → R_{Q}/𝔞_{Q}R_{Q} bringing ρ_{∅}^{univ} to ρ_{Q}^{univ} mod 𝔞_{Q}R_{Q}. On the other hand ρ_{∅}^{univ} is clearly of type D_{Q}, so the universality of R_{Q} implies that there is a unique homomorphism R_{Q} → R bringing ρ_{Q}^{univ} to ρ_{∅}^{univ}. We conclude that R_{Q}/𝔞_{Q}R_{Q} can be canonically identified with R.

2.3. The Hecke ring. Put N = N(ρ̄) if ρ̄|G_{p} is flat, and N = N(ρ̄)p if ρ̄|G_{p} is non-flat but Selmer. (This will turn out to be the minimal level at which one can find a weight 2 newform whose associated Galois representation lifts ρ̄.) Let

Γ_{Q} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(NQ) | the order of d mod Q in (ℤ/Qℤ)^{×} is prime to p\} (2.2)

so that Γ_{0}(NQ)/Γ_{Q} = Δ_{Q}. Let S_{2}(Γ_{Q}, O) be the space of O-valued weight-2 cuspforms on Γ_{Q}, and denote by T (Γ_{Q}) the subalgebra of End(S_{2}(Γ_{Q}, O)) generated over O by the Hecke operators T_{l} and ⟨l⟩ for primes l not dividing NQ, and U_{l} for l|NQ. Since ⟨l⟩ depends only on the image of l in Δ_{Q}, the diamond operators make T (Γ_{Q}) a Λ_{Q}-algebra.

Let 𝔪_{Q} be the ideal of T (Γ_{Q}) generated by λ, T_{l} − tr(ρ̄(F rob_{l})) and l⟨l⟩ − det(ρ̄(F rob_{l})) for (l, NQ) = 1, U_{l} − χ(F rob_{l}) for l|N(ρ̄), U_{p} − ψ(F rob_{p}) if ρ̄ is Selmer non-flat, and U_{q} − β_{q} for q|Q. Here χ and ψ are the unramified characters figuring in section 2.1, and β_{q} is the eigenvalue of ρ̄(F rob_{q}) which coincides with ϕ̄_{2}^{univ}(F rob_{q}) (recall that we chose ϕ_{1}^{univ} to define the action of Δ_{Q} on R_{Q}). The expressions of the form X − x, where X ∈ T (Γ_{Q}) and x ∈ k, are shorthand for X − x̃, where x̃ is a lifting of x to O. Since λ ∈ 𝔪_{Q}, it does not matter which lifting we choose. If 𝔪_{Q} is a proper ideal (i.e. not equal to the whole ring), it is clearly maximal, and the homomorphism from T (Γ_{Q}) to k defined by it is a k-valued eigenform, whose associated representation is ρ̄.

The following theorem is very deep and contains Ribet’s theorem on ”lowering the level” [Ri1], as well as improvements due to Wiles and Diamond. (See [Di1] theorem 6.4).

Theorem 4. 𝔪_{Q} is a proper maximal ideal in T (Γ_{Q}).

Note that 𝔪_{Q}, if proper, contains (at least one) minimal prime, which corresponds to a weight 2 newform on Γ_{Q} whose associated Galois representation lifts ρ̄. Thus the theorem is equivalent to the statement that ρ̄ is modular of weight 2 and the minimal possible level. Note that this is a variant of Serre’s ”Conjecture ϵ”, because Serre stipulated a level which was always prime-to-p. In the Selmer non-flat case considered here (where p|N) Serre would ”pay” for the omission of p from the level in allowing the newform to be of weight p + 1 rather than 2. Note also that here, and only here, is the crucial hypothesis that ρ̄ is modular (of some weight and level), being used. Sketching the proof of this theorem would take us outside the scope of our survey. Let us indicate only how it follows, in the form stated here, from [Di1], theorem 6.4.

Consider first the case Q = ∅. It then follows from [Di1], theorem 6.4, that there is a newform f of weight 2 and level N, a prime λ′ dividing p in the field K_{f}, and an embedding of O_{K_{f}} /λ′ in k̄ such that for every prime l not dividing Np, a_{l}(f) ≡ tr(ρ̄(F rob_{l})) mod λ′. Parts (2) and (3) of theorem 6 below, describing the restriction of the λ′-adic representation associated to f to the decomposition groups at bad l (those dividing N(ρ̄)) or at l = p, imply that a_{l}(f) ≡ χ(F rob_{l}) mod λ′ for l|N(ρ̄), and a_{p}(f) ≡ ψ(F rob_{p}) mod λ′ in the Selmer non-flat case. The nebentypus of f has order prime to p, so can be read from the determinant of ρ̄ = ρ_{f,λ′} mod λ′. But det(ρ̄) = ω, and the weight is 2, so the nebentypus is trivial, and f is on Γ_{0}(N). The kernel of the homomorphism of the Hecke algebra (into k̄) defined by f mod λ′ is the desired maximal ideal.

When Q is not empty, one resorts to the theory of old-forms. For simplicity assume that Q = {q} consists of a single prime (the general case is handled similarly). If α and β are the two eigenvalues of ρ_{f,λ′}(F rob_{q}), then αβ = q and α + β = a_{q}(f). It follows that in the two-dimensional space of old-forms spanned by f_{1}(z) = f(z)

and f_{2}(z) = f(qz), the matrix of U_{q} is ( \begin{matrix}a_{q} & −1 \\ q & 0 \end{matrix} ), and there exists a unique

linear combination g of f_{1} and f_{2} satisfying U_{q}g = βg. This g is an old eigenform on Γ_{0}(NQ), hence on Γ_{Q}, and defines a homomorphism from T (Γ_{Q}) to k̄ whose kernel is the desired 𝔪_{Q}. This completes the proof. An important point to bear in mind is that since q ≡ 1 mod p, there are other newforms on Γ_{Q} whose λ-adic representation lifts ρ̄, i.e. which are congruent to g. These newforms have non-trivial nebentypus factoring through Δ_{Q}. This will become clear once we prove, in section 3, the main theorem about the structure of T_{Q}.

Definition 3. Let T_{Q} be the localization of T (Γ_{Q}) in 𝔪_{Q}. Let X_{Q} be the modular curve over ℚ corresponding to the congruence group Γ_{Q}, and J_{Q} = Jac(X_{Q}).

The Hecke operators act as correspondences on X_{Q}, and as endomorphisms on J_{Q}. Let Ta_{p}(J_{Q}) be the p-adic Tate module of J_{Q}, which becomes a T (Γ_{Q})-module after we tensor it over ℤ_{p} with O. We can therefore localize it at 𝔪_{Q}. The resulting module, Ta_{p}(J_{Q})_{𝔪_{Q}}, is a T_{Q} -module. Thanks to the assumption that the residual representation of 𝔪_{Q}, namely ρ̄, is irreducible, we know the following result, which again is very deep, and is related to the Gorenstein-ness of T_{Q} (see [M1] and [Ti]).

Theorem 5. Ta_{p}(J_{Q})_{𝔪_{Q}} is free of rank 2 over T_{Q}.

Definition 4. Let ρ = ρ_{Q}^{mod} : G_{ℚ} → GL_{2}(T_{Q}) be the Galois representation on Ta_{p}(J_{Q})_{𝔪_{Q}}.

Remark 3. It is easy to see that Ta_{p}(J_{Q})_{𝔪_{Q}} ⊗_{ℤ_{p}} ℚ_{p} is free of rank 2 over T_{Q} ⊗_{ℤ_{p}} ℚ_{p}. One therefore obtains ρ, but with entries in T_{Q} ⊗_{ℤ_{p}} ℚ_{p}, by ”gluing” the p-adic representations associated to the individual newforms. From there it is possible to find ρ with entries in T_{Q} by the method of ”pseudo-representations” (see the discussion following [W], 2.1). However, theorem 5 will be needed later again.

The first part of the next theorem is ”classical” - it is the Eichler-Shimura relation. Parts (2) and (3) are more recent.

Theorem 6. (1) (Eichler-Shimura-Igusa) ρ is unramified outside Σ ∪ Q, and for (l, NQ) = 1, l ≠ p, the characteristic polynomial of ρ(F rob_{l}) is X^{2} − T_{l}X + ⟨l⟩l.

(2) (Carayol, following Deligne and Langlands) For l|N(ρ̄)

ρ|G_{l} ∼ ( \begin{matrix}χ^{−1}χ_{Q}ϵ & ∗ \\ 0 & χ \end{matrix} )

with an unramified character χ, χ(F rob_{l}) = U_{l}, and χ^{2} = χ_{Q}. The ∗ in the upper right corner is ramified.

For q|Q

ρ|G_{q} ∼ ( \begin{matrix}χ^{−1}χ_{Q}ϵ & ∗ \\ 0 & χ \end{matrix} )

with an unramified character χ, χ(F rob_{q}) = U_{q}.

(3) (Unpublished correspondence of Fontaine-Serre, Wiles) If N(ρ̄) = N, ρ is flat at p. If, on the other hand, p|N, then

ρ|G_{p} ∼ ( \begin{matrix}ψ^{−1}χ_{Q}ϵ & ∗ \\ 0 & ψ \end{matrix} )

with an unramified character ψ, and ψ(F rob_{p}) = U_{p}.

In case (2), the automorphic representation of GL_{2}(𝔸_{ℚ}) associated to any eigenform of level NQ whose λ-adic representation lifts ρ̄, is ”special” at l|N(ρ̄), and ”principal series” at q|Q.

Corollary 7. The representation ρ′_{Q} = ρ_{Q}^{mod} ⊗ χ_{Q}^{−1/2} is of type D_{Q}, and there exists a unique surjective homomorphism of Λ_{Q}-algebras R_{Q} → T_{Q}, bringing ρ_{Q}^{univ} to ρ′_{Q}.

Proof: After twisting, det(ρ′_{Q}) = ϵ. Parts (2) and (3) of the theorem now imply that ρ′_{Q} is of type D_{Q}. By the universality of R_{Q} we obtain the desired map to T_{Q}. Since this map brings ϕ_{1}^{univ}|I_{q} to \sqrt{χ_{q}|I_{q}}, it respects the Δ_{q} action (this finally justifies the peculiar square in the definition of the Δ_{q}-action on R_{Q}). It remains to prove that the map is surjective, or that every Hecke operator is contained in the image. That T_{l} and ⟨l⟩ are in the image ((l, NQ) = 1) follows from the relations tr(ρ_{Q}^{mod}(F rob_{l})) = T_{l}, and det(ρ_{Q}^{mod}(F rob_{l})) = l⟨l⟩. Parts (2) and (3) of the theorem show directly that U_{l} (l|N(ρ̄)), U_{q} (q|Q) and U_{p} (if ρ̄ is Selmer non-flat) are in the image.

When Q = ∅ we drop the subscript and write R (resp. T) for R_{∅} (resp. T_{∅}). The following is the main theorem (theorem 1).

Theorem 8. The map R → T is an isomorphism, and T is a l.c.i.

2.4. The Taylor-Wiles-Faltings criterion. The following commutative-algebra criterion lies at the basis of the proof of the main theorem. See [T-W], appendix, or [Sc].

Lemma 9. Let ϕ : R → T be a surjective homomorphism of local complete noetherian O-algebras, and assume that T is finite and flat over O. Suppose that for some r ≥ 1 and for every n ≥ 1 there exist local complete noetherian O-algebras R_{Q} and T_{Q} and a commutative diagram

\begin{aligned}O[[S_{1}, ..., S_{r}]] & → & R_{Q} & ↠ & R \\ & & ↓ & & ↓ \\ & & T_{Q} & ↠ & T \end{aligned} (2.3)

where all four maps in the square on the right of the diagram are surjective, and (i) (S_{1}, ..., S_{r})R_{Q} = Ker(R_{Q} → R) (ii) (S_{1}, ..., S_{r})T_{Q} = Ker(T_{Q} → T) (iii) 𝔟 = Ker(O[[S_{1}, ..., S_{r}]] → T_{Q}) ⊆ ((1 + S_{1})^{p^{n}} − 1, ..., (1 + S_{r})^{p^{n}} − 1), and T_{Q}

is free of finite rank over O[[S_{1}, ..., S_{r}]]/𝔟.

(iv) R_{Q} is topologically generated as an O-algebra by r elements.

Then R → T is an isomorphism, and they are l.c.i.

One should note that there are two types of assumptions here. Points (ii)-(iii) mean that the {T_{Q}} are large and ”regularly controlled” by r parameters, namely the S_{i}’s. As n increases these parameters consume an increasingly significant part of T_{Q}. The subscript Q here has no meaning, and simply hints for the upcoming application. The R_{Q} and T_{Q} are not canonical, and are not assumed even to relate to each other as we vary n. Indeed, the first step in the proof of the lemma is to show, by a ”Mittag-Leffler” argument (after we reduce the picture mod λ), that it is possible to make an inverse system of diagrams as above, and pass to a limit

O[[S_{1}, ..., S_{r}]] → R_{∞} ↠ R
↓  ↓
T_{∞} ↠ T

where T_{∞} is now finite free over O[[S_{1}, ..., S_{r}]].

On the other hand, point (iv) means that the R_{Q} are uniformly small, so in the limit diagram R_{∞} will still be generated topologically as an O-algebra by r elements. Considerations of Krull dimension then yield R_{∞} = T_{∞} ≅ O[[X_{1}, ..., X_{r}]], from which the desired equality R = T = O[[X_{1}, ..., X_{r}]]/(S_{1}, ...S_{r}) is finally deduced.

To apply the lemma Taylor and Wiles find an r depending only on ρ̄, and for every n ≥ 1 they choose a set Q as above, containing precisely r primes, all congruent to 1 modulo p^{n} (and not merely p), so that (iv) will hold. The R_{Q}, the T_{Q} and the maps between them are chosen as in the previous sections. The map O[[S_{1}, ..., S_{r}]] → R_{Q} is taken to be the map that sends 1 + S_{i} to a fixed generator of Δ_{q_{i}}. If we let p^{n_{i}} be the order of Δ_{q_{i}}, the p-Sylow subgroup of (ℤ/q_{i}ℤ)^{×} (so that n_{i} ≥ n), then Λ_{Q} is identified with O[[S_{1}, ..., S_{r}]]/𝔟, where 𝔟 = ((1 + S_{1})^{p^{n_{1}}} − 1, ..., (1 + S_{r})^{p^{nr}} − 1), and the augmentation ideal 𝔞_{Q} is identified with (S_{1}, ..., S_{r})/𝔟. Point (i) was already noticed at the end of section 2.2. Points (ii) and (iii) are guaranteed by proposition 10 below, to be proved in section 3, and by the fact that all the q_{i} ≡ 1 mod p^{n}. The proof of proposition 10 is essentially topological. By an extension of a result of Mazur, a certain piece of the (singular) cohomology of the modular curve, with coefficients in O, is free of rank one over T_{Q}. This allows to replace the question on the structure of T_{Q} over Λ_{Q} by a similar question on the structure of the cohomology.

The greatest difficulty lies in point (iv). We have to choose r and the Q’s so that the number r of primes in Q (which is the number of parameters S_{i}) is just the minimal number of generators of R_{Q}. This is guaranteed by proposition 11 below, to be proved in section 4. The proof uses in a deep way Tate’s global duality theorem in Galois cohomology. As will become clear, this is where the assumption that the deformation type is minimal enters.

We are thus left with the task of proving the following two propositions. Together with the commutative-algebra criterion of lemma 9, they imply the main theorem.

Proposition 10. Let Q be a set of r primes as in section 2.2. Then (i) T_{Q} is finite and free over Λ_{Q} (ii) rank_{Λ_{Q}}T_{Q} = rank_{O}T. Equivalently, T_{Q}/𝔞_{Q}T_{Q} = T.

Proposition 11. There exists an r (namely, dim_{k} H_{D∗}^{1}(G_{Σ}, Symm^{2}ρ̄), see section 4) such that for every n ≥ 1 there exists a set Q of r primes, disjoint from Σ, satisfying

• For every q ∈ Q, q ≡ 1 mod p^{n}

• For every q ∈ Q, ρ̄(F rob_{q}) has distinct eigenvalues contained in k

• R_{Q} can be topologically generated as an O-algebra by r elements.

3. Proof of proposition 10 - on the structure of the Hecke algebra

3.1. New and old. Let Γ̃_{Q} = Γ_{0}(NQ), and define the Hecke algebra T̃_{Q} as in section 2.3, but with respect to Γ̃_{Q} instead of Γ_{Q}, omitting the diamond operators. In other words, we first let T (Γ̃_{Q}) be the Hecke algebra on S_{2}(Γ̃_{Q}, O) (over O). The inclusion S_{2}(Γ̃_{Q}, O) ⊂ S_{2}(Γ_{Q}, O) induces a surjective homomorphism T (Γ_{Q}) ↠ T (Γ̃_{Q}). The image of the maximal ideal 𝔪_{Q} in T (Γ̃_{Q}) is also a proper maximal ideal 𝔪̃_{Q} (cf the proof of theorem 4), and we set T̃_{Q} = T (Γ̃_{Q})𝔪̃_{Q}. Localizing the above restriction map between the two Hecke algebras we get a surjective homomorphism T_{Q} ↠ T̃_{Q}. Note that since T (Γ̃_{Q}) is pro-artinian, it is a product of its localizations at maximal ideals, and every T (Γ̃_{Q})-module is the direct sum of its localizations.

The next theorem (see also theorem 5) was first proved, for Γ_{0}(N) and N prime, by Mazur, and then generalized by Tilouine, Ribet, Gross, Edixhoven, and Wiles (see [W], theorem 2.1, and [Ti]). It is a ”multiplicity one” result for certain finite Hecke modules (killed by p). The Gorenstein-ness of the Hecke algebra, which is a consequence of the complete intersection property in theorem 1, is known to follow from it. However, the Gorenstein property itself is not directly used in the Taylor-Wiles proof of theorem 1 in the minimal case (although it is heavily used in the passage from minimal to non-minimal D). Let Y_{Q} ⊂ X_{Q} be the open modular curve which is obtained from X_{Q} when we delete the cusps.

Theorem 12. The following modules are finite free over T_{Q} :

• Ta_{p}(J_{Q})_{𝔪_{Q}} = (Ta_{p}(J_{Q}) ⊗_{ℤ_{p}} O) ⊗_{T (Γ_{Q})} T_{Q} - free of rank 2

• H^{1}(X_{Q}, O)_{𝔪_{Q}}^{±} = H^{1}(Y_{Q}, O)_{𝔪_{Q}}^{±} (± referes to the action of complex conjugation on the modular curve) - each free of rank 1.

Similar statements hold for T̃_{Q}.

The theorem, as well as the fact that we can replace the closed curve X by the open curve Y, rely on the irreducibility of ρ̄ (so 𝔪_{Q} is not an ”Eisenstein” prime).

Recall that T = T_{∅} = T̃_{∅} is the localization of T (Γ_{0}(N)) at 𝔪 = 𝔪_{∅}. While there is no map from T (Γ_{0}(NQ)) to T (Γ_{0}(N)) (the U_{q} for q|Q do not preserve S_{2}(Γ_{0}(N), O) ⊂ S_{2}(Γ_{0}(NQ), O)), the next lemma shows that after we localize at 𝔪̃_{Q} and 𝔪, such a map exists, and in fact is an isomorphism.

Lemma 13. There exists an isomorphism T̃_{Q} ≅ T, mapping the Hecke operators T_{l} (l |̸N) and U_{l} (l|N) in T̃_{Q} to the corresponding operators in T.

Proof: (1) In the first step one uses the fact that α_{q} ≠ β_{q} and q ≡ 1 mod p, to show that Ta_{p}(J̃_{Q})𝔪̃_{Q} is ”Q-old”. By ”Q-old” we mean that Ta_{p}(J̃_{Q})𝔪̃_{Q}, which is a direct summand of Ta_{p}(J̃_{Q}) ⊗_{ℤ_{p}} O, is contained in the Tate module of the Q-old subvariety of J̃_{Q}. This Q-old subvariety is isogenous to a product of 2^{r} copies of J = J_{0}(N).

One can prove (1) by computing the module of fusion between the Q-old and Qnew parts of the Jacobian as in [W], proposition 2.4’ (p. 503, see the remark at the end of the section there). For simplicity let us illustrate the proof when Q = {q} consists of one prime (the proof of the general case is the same, proceeding one q at a time). Let J̃_{Q}^{old} and J̃_{Q}^{new} be the old and new subvarieties of J̃_{Q}. Thus, if μ : J^{2} → J̃_{Q} is the map derived by P ic functoriality from the two degeneracy maps X_{0}(Nq) → X_{0}(N), and μ̂ : J̃_{Q} → J^{2} is the dual map, J̃_{Q}^{old} is the image of μ, and J̃_{Q}^{new} is the connected component of Ker(μ̂). Both J̃_{Q}^{new} and J̃_{Q}^{old} are stable under the Hecke algebra, including the U_{q}-operator. Consider J[p^{∞}]_{𝔪} = (J[p^{∞}] ⊗_{ℤ_{p}} O) ⊗_{T (Γ)} T (recall T = T (Γ)_{𝔪} is the localization of the Hecke algebra at 𝔪) and suppose that

J[p^{∞}]_{𝔪}^{2} ∩ Ker(μ̂ ∘ μ) = {0}.

Then the relevant module of fusion F = J̃_{Q}^{old}[p^{∞}]𝔪̃_{Q} ∩ J̃_{Q}^{new}[p^{∞}]𝔪̃_{Q} = {0}, for it is contained in μ(J[p^{∞}]_{𝔪}^{2}) ∩ Ker(μ̂). In particular J̃_{Q}^{old}[𝔪̃_{Q}] ∩ J̃_{Q}^{new}[𝔪̃_{Q}] = {0} (where J[𝔪] is the kernel of 𝔪 in J[p] ⊗ k). However, it follows from theorem 12 that dim_{k} J̃_{Q}[𝔪̃_{Q}] = 2. Equivalently, the multiplicity of ρ̄ in the semisimplification of this Galois module is 1. Since ρ̄ is a constituent (i.e. a subquotient Galois module) of J̃_{Q}^{old}[p^{∞}]𝔪̃_{Q}, hence of J̃_{Q}^{old}[𝔪̃_{Q}], it is not a constituent of J̃_{Q}^{new}[𝔪̃_{Q}] or of J̃_{Q}^{new}[p^{∞}]𝔪̃_{Q}, which means that J̃_{Q}^{new}[p^{∞}]𝔪̃_{Q} = 0, as desired.

Clearly J[p^{∞}]_{𝔪}^{2} ∩ Ker(μ̂ ∘ μ) is a finite group, and has a decomposition series all of whose factors are killed by 𝔪, so to prove J[p^{∞}]_{𝔪}^{2} ∩ Ker(μ̂ ∘ μ) = {0}, it will be enough to show that J[𝔪]^{2} ∩ Ker(μ̂ ∘ μ) = 0. An explicit calculation gives

μ̂ ∘ μ = ( \begin{matrix}q + 1 & T_{q} \\ T_{q} & q + 1 \end{matrix} ) ∈ End(J^{2}) = M_{2}(End(J)).

Since q ≡ 1 mod 𝔪 and T_{q} ≡ α_{q} +β_{q} mod 𝔪, the matrix representing μ̂∘μ on J[𝔪]^{2}

is ( \begin{matrix}2 & t \\ t & 2 \end{matrix} ) where t = α_{q} + β_{q}. This matrix is non-singular since α_{q}β_{q} = q = 1

and α_{q} ≠ β_{q} imply t ≠ ±2.

An alternative proof goes as follows. A newform g which is new at q must be ”special” there (because q divides the level to the first power, compare theorem 6 (2)). The local representation ρ_{g,λ}|G_{q} therefore has the form described in theorem 6. Since q ≡ 1 mod p, the character ω is trivial on G_{q}, and we find that ρ̄_{g,λ}|G_{q} ∼

\binom{χ ∗}{χ}. On the other hand if g belongs to 𝔪̃_{Q} then ρ̄_{g,λ} = ρ̄, and the assumption

was that the two eigenvalues of ρ̄(F rob_{q}) are distinct. This contradiction proves that all the newforms belonging to 𝔪̃_{Q} are Q-old.

(2) On the Q-old forms the Hecke algebra is simply T (Γ_{0}(N))[u_{q}]/(u_{q}^{2} − T_{q}u_{q} + ⟨q⟩q) (one variable for each q in Q). Since the roots of u_{q}^{2} − T_{q}u_{q} + ⟨q⟩q are distinct modulo 𝔪_{∅}, and since U_{q} − β_{q} is contained in 𝔪̃_{Q}, Hensel’s lemma shows that after we localize at 𝔪̃_{Q} we get T_{∅}.

3.2. Freeness of the Hecke algebra over Λ_{Q}. Let Ỹ_{Q} be the open modular curve associated to Γ̃_{Q}. As we shall prove below, H^{1}(Y_{Q}, O)^{−} is a free Λ_{Q}-module of rank equal to the O-rank of H^{1}(Ỹ_{Q}, O)^{−}. It follows from theorem 12 that T_{Q} is a free Λ_{Q}-module, whose rank is equal to the O-rank of T̃_{Q}, and invoking lemma 13 we get proposition 10.

Proposition 14. H^{1}(Y_{Q}, O)^{−} is a free Λ_{Q}-module of rank equal to the O-rank of H^{1}(Ỹ_{Q}, O)^{−}.

Proof: First, let us pretend that Γ̃_{Q} had no elliptic elements. Then Γ̃_{Q} = π_{1}(Ỹ_{Q}) is free as the fundamental group of the incomplete curve. By Shapiro’s lemma one may identify

H^{1}(Y_{Q}, O) = H^{1}(Γ_{Q}, O) = H^{1}(Γ̃_{Q}, O[Δ_{Q}]). (3.1)

The action of Δ_{Q}, the group of deck-transformations of Y_{Q}/Ỹ_{Q}, on H^{1}(Y_{Q}, O), corresponds in this isomorphism to the action of Γ̃_{Q}/Γ_{Q} on H^{1}(Γ_{Q}, O) through conjugation. Via the Shapiro isomorphism this, in turn, gets translated to the action of Δ_{Q} on the coefficients of H^{1}(Γ̃_{Q}, O[Δ_{Q}]). Since Γ̃_{Q} is free, the module of 1cocycles Z^{1}(Γ̃_{Q}, O[Δ_{Q}]) is a free O[Δ_{Q}]-module. In fact, choosing g free generators for Γ̃_{Q}, say γ_{1}, ..., γ_{g}, the map

c ⟼ (c(γ_{1}), ..., c(γ_{g})) (3.2)

is an isomorphism of Z^{1}(Γ̃_{Q}, O[Δ_{Q}]) onto O[Δ_{Q}]^{g}, because a cocycle can be fixed arbitrarily on the generators, and then it is determined on any word.

The action of complex conjugation on H^{1}(Y_{Q}, O) is deduced from the map z ↦ −z̄ on the upper half plane. If γ ∈ Γ_{Q} gets mapped in π_{1}(Y_{Q}) to the path [γ] which is the projection (in Y_{Q}) of the geodesic {i, γ(i)} connecting i to γ(i) in the upper half plane, then the geodesic \{−ī, −\overline{γ(i)}\} = \{i, ξγξ^{−1}(i)\} projects to

the path [ξγξ^{−1}], which is its complex conjugate. Here ξ = ( \begin{matrix}−1 \\ & 1 \end{matrix} ), and as

a base point for the fundamental group we took the image of i. The action of complex conjugation on H^{1}(Y_{Q}, O) therefore corresponds to the action of ξ on H^{1}(Γ_{Q}, O) through conjugation. Via the Shapiro isomorphism it gets translated to an action of ξ on H^{1}(Γ̃_{Q}, O[Δ_{Q}]), where ξ is still acting by conjugation on the group, and trivially on the coefficients. This action can be defined already at the level of 1-cocycles and 1-coboundaries. Clearly Z^{1}(Γ̃_{Q}, O[Δ_{Q}])^{−} is free over O[Δ_{Q}], as a direct factor of a free module. About B^{1}(Γ̃_{Q}, O[Δ_{Q}]) we need not worry because the 1-coboundaries are in the + eigenspace for the action of ξ (since

ξ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ξ^{−1} = ( \begin{matrix}a & −b \\ −c & d \end{matrix} )). It follows that H^{1}(Γ̃_{Q}, O[Δ_{Q}])^{−} is free over

Λ_{Q} = O[Δ_{Q}], and

H^{1}(Γ̃_{Q}, O[Δ_{Q}])^{−}/𝔞_{Q}H^{1}(Γ̃_{Q}, O[Δ_{Q}])^{−} = H^{1}(Γ̃_{Q}, O)^{−} = H^{1}(Ỹ_{Q}, O)^{−} (3.3)

from where the statement about the rank follows.

Unfortunately, Γ̃_{Q} might have elliptic elements. This is remedied by introducing an auxiliary prime R > 3, and replacing Γ_{Q} and Γ̃_{Q} by Γ′_{Q} = Γ_{Q} ∩ Γ_{1}(R) and Γ̃′_{Q} = Γ̃_{Q} ∩ Γ_{1}(R) respectively. We contend ourselves with a brief sketch of how one modifies the arguments above, which now apply verbatim to Γ′_{Q} and Γ̃′_{Q}. To deduce proposition 10 for the original Γ_{Q} and Γ̃_{Q}, we need to find a maximal ideal 𝔪′_{Q} (resp. 𝔪̃′_{Q}) in T (Γ′_{Q}) (resp. T (Γ̃′_{Q})) such that localization at 𝔪′_{Q} yields an isomorphism T′_{Q} ≅ T_{Q} (resp. T̃′_{Q} ≅ T̃_{Q}). In the language of congruences between modular forms we want to make sure that there are no congruences between R-new and R-old forms that belong to 𝔪′_{Q}. The computations of Ribet and Wiles, similar to those in lemma 13, show that this will be the case as long as

• {1, α_{R}, β_{R}, α_{R}β_{R} = R} are 4 distinct elements of k, where as before {α_{R}, β_{R}} are the two eigenvalues of F rob_{R}.

The existence of an R as above follows from the Čebotarev density theorem and the irreducibility of ρ̄ (see [D-T], lemma 3). For p = 3 it can be checked directly, especially if Im(ρ̄) is the full GL_{2}(𝔽_{3}).

For full details on the modifications needed to take care of the elliptic elements in Γ̃_{Q}, see section 2 of [T-W].

4. Proof of proposition 11 - on the structure of the universal deformation ring

4.1. The Selmer group. For the concepts defined in this section, see also Washington’s paper in this volume [Wa]. Let ρ̄ be as in 2.1, and define

(4.1) W = ad^{0}(ρ̄) (trace-0 matrices in the adjoint representation of ρ̄)

W^{∗} = Hom(W, μ_{p}) ≅ W (1) ≅ Symm^{2}(ρ̄) (4.2)

(the last isomorphism stems from ad(ρ̄) ≅ ρ̄⊗ρ̄^{∨} ≅ ρ̄⊗ρ̄(−1) ≅ Symm^{2}(ρ̄)(−1)⊕k, hence W ≅ Symm^{2}(ρ̄)(−1)). If ρ̄ is ”Selmer”, U is the space of ρ̄, and 0 → U^{0} → U → U^{1} → 0 is the filtration defining the Selmer condition at p (so that U^{i} are modules for ρ̄|G_{p} and U^{0} is the ω-eigenspace for I_{p}), put

W^{0} = Hom(U/U^{0}, U^{0}) ⊂ W^{1} = Hom^{0}((U, U^{0}), (U, U^{0})) ⊂ W. (4.3)

Thus W^{1} is the submodule of upper-triangular, trace-0 matrices, and W^{0} consists of those whose diagonal vanishes. Note that G_{p} acts on W^{0} through ωψ^{−2} = ω (see remark 2.1(ii)), trivially on W^{1}/W^{0}, and through ω^{−1}ψ^{2} = ω^{−1} on W/W^{1}.

We shall next define the ”local conditions”, which are subgroups L_{v} of H^{1}(G_{v}, W) for the various decomposition groups G_{v}. We shall later consider global cohomology classes whose restriction to every G_{v} falls into L_{v}. Since p ≥ 3, H^{1}(G_{∞}, W) = 0, and we only have to define these local conditions at the finite places. Let

• L_{l} = H^{1}(G_{l}/I_{l}, W^{I_{l}}) if l|N(ρ̄)

• L_{p} = H_{Se}^{1}(G_{p}, W) := Ker(H^{1}(G_{p}, W) → H^{1}(I_{p}, W/W^{0})) if ρ̄ is Selmer but

not flat

L_{p} = H_{fl}^{1}(G_{p}, W) if ρ̄ is flat.

• L_{q} = H^{1}(G_{q}, W) if q|Q.

Some explanations are in order. First, we did not define L_{l} for primes l ∉ Σ ∪ Q, but they could be defined there too as H^{1}(G_{l}/I_{l}, W^{I_{l}}) = H^{1}(G_{l}/I_{l}, W). Instead, we shall assume in (4.5) below that our cohomology classes are unramified outside Σ ∪ Q, which, of course, is a synonimous condition.

At l|N(ρ̄) our local condition confirms with [W], p.461, because H^{1}(G_{l}/I_{l}, W^{I_{l}}) = Ker(H^{1}(G_{l}, W) → H^{1}(G_{l}, W/W^{I_{l}})). To check this use the exact diagram

\begin{aligned}H^{1}(G_{l}/I_{l}, W^{I_{l}}) & ↪ & H^{1}(G_{l}, W^{I_{l}}) & → & H^{1}(I_{l}, W^{I_{l}}) \\ ∥ & & ↓ & & ↓ 0 \\ H^{1}(G_{l}/I_{l}, W^{I_{l}}) & ↪ & H^{1}(G_{l}, W) & → & H^{1}(I_{l}, W) \\ & & ↓ & & ↓ \\ & & H^{1}(G_{l}, W/W^{I_{l}}) & → & H^{1}(I_{l}, W/W^{I_{l}}) \end{aligned} (4.4)

To justify the arrow labeled with a 0 note that it is part of the long exact sequence

0 → W^{I_{l}} ≃ W^{I_{l}} → (W/W^{I_{l}})^{I_{l}} → H^{1}(I_{l}, W^{I_{l}}) → H^{1}(I_{l}, W)

and both (W/W^{I_{l}})^{I_{l}} and H^{1}(I_{l}, W^{I_{l}}) = Hom(I_{l}, k) are one-dimensional over k.

At the auxiliary q′s we simply took L_{q} to be the whole H^{1}, which means that there is no local condition imposed there.

The most subtle condition is the one imposed at p. If ρ̄ is Selmer non-flat the definition is analogous to that of L_{l} for l|N(ρ̄). If ρ̄ is flat, then H_{fl}^{1}(G_{p}, W) has the same meaning as in [Co]. Let us recall that there is a canonical isomorphism between H^{1}(G_{p}, ad(ρ̄)) and Ext_{k[G_{p}]}^{1}(ρ̄, ρ̄), the Ext group being computed in the category of k[G_{p}]-modules. Since ρ̄|G_{p} is assumed to be the representation attached to a finite flat group scheme Γ (say), one may ask whether a specific Galois module, which is an extension of ρ̄ by ρ̄, in fact arises from a finite flat group scheme over ℤ_{p} which is an extension of Γ by Γ in the category of group schemes over ℤ_{p}. This need not be the case in general, and one denotes H_{fl}^{1}(G_{p}, ad(ρ̄)) those cohomology classes whose corresponding extension arises in such a manner. Then H_{fl}^{1}(G_{p}, W) = H_{fl}^{1}(G_{p}, ad(ρ̄)) ∩ H^{1}(G_{p}, W). While this definition of the local condition at p is appealing intuitively, it is not directly amenable to computations in Galois cohomology. For computational purposes it is important to have a ”linear-algebra” criterion for an extension class to be ”flat”, and this is supplied by Fontaine’s work on Honda systems (as explained in [Co]), or by later developments due to Fontaine and Lafaille (as used in [W]).

Definition 5. The ”Selmer group” is

H_{D_{Q}}^{1}(ℚ, W) = Ker (H^{1}(G_{Σ∪Q}, W) → ∏_{v∈Σ∪Q} H^{1}(G_{v}, W)/L_{v}) (4.5)

In other words, it is the group of cohomology classes which at every place v satisfy the ”local condition” L_{v}.

The importance of the Selmer group stems from the fact that it is canonically isomorphic to the reduced tangent space of the deformation problem. Recall that we denoted by R_{Q} the universal deformation ring for deformations of ρ̄ of type D_{Q}, and that, for notational convenience, we drop the subscript Q if Q is empty.

Theorem 15. ([M2], [M3]) There is a canonical isomorphism

H_{D}^{1}(ℚ, W) ≅ Hom(𝔪_{R}/(λ, 𝔪_{R}^{2}), k). (4.6)

A similar identity holds with R_{Q} and D_{Q}. In particular, R_{Q} can be topologically generated as an O-algebra by r(Q) = dim_{k} H_{D_{Q}}^{1}(ℚ, W) elements, and this is the minimal number of generators.

Proof: The last assertion, about the number of generators needed to generate R_{Q} as a topological O-algebra, is a consequence of (4.6) and Nakayama’s lemma. Let us prove (4.6). A k-linear homomorphism from 𝔪_{R}/(λ, 𝔪_{R}^{2}) to k defines a local O-algebra homomorphism from R to the ring of dual numbers k[ϵ] (ϵ^{2} = 0), and vice versa. It therefore corresponds to ρ, an infinitesimal deformation of ρ̄ (i.e. a deformation with values in GL_{2}(k[ϵ])), which is unique up to strict equivalence. Writing

ρ(g)ρ̄(g)^{−1} = 1 + ϵc_{ρ}(g) (4.7)

c_{ρ}(g) becomes a 1-cocycle in W for the adjoint action, namely

c_{ρ}(gh) = c_{ρ}(g) + ad(ρ̄)(g)c_{ρ}(h) (4.8)

(c_{ρ}(g) has trace zero because det(ρ) = det(ρ̄) = ω). Replacing ρ by a strictly equivalent deformation has the effect of changing c_{ρ} by a coboundary, and every cohomology class is obtained in such a way from an infinitesimal deformation of ρ̄. This is the construction which associates to an infinitesimal deformation of ρ̄ a class [c_{ρ}] in H^{1}(ℚ, W). It remains to check that ρ is of type D if and only if [c_{ρ}] ∈ H_{D}^{1}(ℚ, W).

Assume first that l|N(ρ̄), and ρ is of type D. Then for g ∈ I_{l} we have c_{ρ}(g) ∈

\{( \begin{matrix}0 & b \\ 0 & 0 \end{matrix} ) | b ∈ k\} = W^{I_{l}}, so the image of [c_{ρ}] in H^{1}(I_{l}, W/W^{I_{l}}) is 0. In view

of (4.4) (the injectivity of H^{1}(I_{l}, W) → H^{1}(I_{l}, W/W^{I_{l}})) , [c_{ρ}] satisfies the local condition at l. Conversely, if this condition is satisfied ρ|I_{l} is upper triangular, hence so is ρ|G_{l} (I_{l} being normal in G_{l}), and it follows that it has the shape of definition 2.1.

The argument at p, in the case that ρ̄ is Selmer non-flat, is entirely analogous, W^{0} replacing W^{I_{l}}. In the flat case, the definition of H_{fl}^{1}(G_{p}, W) makes the desired result a tautology. Finally at l ∉ Σ, ρ is unramified if and only if [c_{ρ}] is unramified.

A similar proof works when Q is not necessarily empty.

4.2. The dual Selmer group. By local Tate duality, at each place v, H^{1}(G_{v}, W) and H^{1}(G_{v}, W^{∗}) are dual abelian groups. We let L_{v}^{∗} be the exact annihilator of L_{v}. At l|N(ρ̄) we have L_{l}^{∗} = H^{1}(G_{l}/I_{l}, W^{∗I_{l}}), and at q|Q we simply have L_{q}^{∗} = 0. The dual Selmer group is defined by

H_{D_{Q}∗}^{1}(ℚ, W^{∗}) = Ker (H^{1}(G_{Σ∪Q}, W^{∗}) → ∏_{v∈Σ∪Q} H^{1}(G_{v}, W^{∗})/L_{v}^{∗}) . (4.9)

The central observation concerning the Selmer group and the dual Selmer group is that while their orders can be pretty hard to compute, their ratios are expressible as a product of local terms. The next theorem is a corollary of Tate’s global duality (the Tate-Poitou exact sequence). See [Gr] and [Wa] in this volume for its derivation.

Theorem 16. Notation as above

\frac{#H_{D}^{1}(ℚ, W)}{#H_{D∗}^{1}(ℚ, W^{∗})} = \frac{#H^{0}(ℚ, W)}{#H^{0}(ℚ, W^{∗})} ∏_{v∈Σ∞} \frac{#L_{v}}{#H^{0}(G_{v}, W)} (4.10)

A similar result holds with D_{Q} replacing D, and Σ ∪ Q replacing Σ. Wiles’ idea was that with a cleverly chosen set Q one can make H_{D_{Q}∗}^{1}(ℚ, W^{∗}) vanish, and so control H_{D_{Q}}^{1}(ℚ, W), hence r(Q). To this end we have to compute the local terms that intervene in the right hand side of the formula.

4.3. Computation of the local terms.

• #L_{p} = #H^{0}(G_{p}, W) · #k

In the flat case this computation is quite delicate. The main point is that if ρ̄|G_{p} is absolutely irreducible, dim H_{fl}^{1}(G_{p}, ad(ρ̄)) = 2. This was proved by Ramakrishna in his thesis [Ra] using the theory of Fontaine-Lafaille (alternatively, one can use Fontaine’s theory of ”Honda systems”, see [Co]). It follows that L_{p} = H_{fl}^{1}(G_{p}, ad^{0}(ρ̄)) is 1-dimensional, which is the desired result, since if ρ̄|G_{p} is irreducible, H^{0}(G_{p}, W) vanishes. See [Ra], [Co] or [D-D-T] (section 2.5) for full details, and a treatment of the case where ρ̄|G_{p} is reducible as well.

In the Selmer non-flat case we have the filtration 0 ⊂ W^{0} ⊂ W^{1} ⊂ W as explained in 4.1 above, so H^{0}(G_{p}, W) = 0, and we have to show that L_{p} = Ker(H^{1}(G_{p}, W) → H^{1}(I_{p}, W/W^{0})) is one-dimensional. The key point is that L_{p} = Ker(H^{1}(G_{p}, W) → H^{1}(G_{p}, W/W^{0})) (compare the situation at l|N(ρ̄), diagram (4.4)). Denote for the moment the latter group by L′_{p} ⊂ L_{p}. L_{p} is the tangent space to the (local) deformation problem classifying deformations of ρ̄|G_{p} of the

form ( ωψ̃^{−1} ∗ ; ψ̃ ) where ψ̃ is an unramified character of G_{p}. On the other hand

L′_{p} is the tangent space to the (local) deformation problem classifying deformations

of ρ̄|G_{p} of the form \binom{ωψ^{−1} ∗}{ψ} where ψ is the character appearing in ρ̄. As

explained in remark 2.2(i), the assumption that ρ̄ is non-flat implies that in any deformation ψ̃ = ψ. (In the language of [W], any Selmer deformation is strict, see [W], proposition 1.1, p.459, and [Di1], 6.1.) Hence the two tangent spaces are the same, and L′_{p} = L_{p}. Alternatively, one can use a diagram similar to (4.4)

\begin{aligned}H^{1}(G_{p}, W^{0}) & → & H^{1}(I_{p}, W^{0})^{G_{p}/I_{p}} \\ ↓ & & ↓ \\ H^{1}(G_{p}, W) & → & H^{1}(I_{p}, W)^{G_{p}/I_{p}} \\ ↓ & & ↓ \\ H^{1}(G_{p}, W/W^{0}) & → & H^{1}(I_{p}, W/W^{0}). \end{aligned} (4.11)

Since W^{I_{p}} = 0, the horizontal maps on the top and in the middle are isomorphisms. This immediately implies L_{p} = L′_{p}. Now compute H^{0}(G_{p}, W/W^{0}) = W^{1}/W^{0} ≅ k, H^{1}(G_{p}, W^{0}) = H^{1}(G_{p}, μ_{p} ⊗ k) = ℚ_{p}^{×}/ℚ_{p}^{×p} ⊗ k ≅ k^{2}, and the desired result comes out from the long exact sequence in cohomology associated to 0 → W^{0} → W → W/W^{0} → 0.

• #L_{∞} = #H^{0}(G_{∞}, W)/#k = 1 Here L_{∞} = H^{1}(G_{∞}, W) = (0) by definition. Since the eigenvalues of complex

conjugation on ρ̄ are ±1, the eigenvalues on W are {−1, −1, +1}, and the second equality follows.

• #L_{l} = #H^{0}(G_{l}, W) for l|N(ρ̄)

This follows from the exact sequence

0 → H^{0}(G_{l}, W) → H^{0}(I_{l}, W) → H^{0}(I_{l}, W) → H^{1}(G_{l}/I_{l}, W^{I_{l}}) → 0 (4.12)

where the arrow in the middle is F rob_{l} − 1 : W^{I_{l}} → W^{I_{l}}.

• H^{0}(ℚ, W) = H^{0}(ℚ, W^{∗}) = 0

For W this follows from the absolute irreducibility of ρ̄, since by Schur’s lemma the only endomorphisms commuting with the Galois action are the scalars, which are missing from W.

For W^{∗} the argument is a little different. A Galois-invariant vector in W^{∗} is, by (4.2), an invariant symmetric bilinear form. If this bilinear form is degenerate, then its kernel is invariant, contradicting the irreducibility of ρ̄. If it is non-degenerate, this means that the image of ρ̄ is contained in some orthogonal group. This already contradicts the fact that det(ρ̄) = ω, unless p = 3 (when ω^{2} = 1), but in this case it contradicts the absolute irreducibility of ρ̄|G_{L}, L as in 2.1. Indeed, over k̄ (which is of odd characteristic) the invariant quadratic form could be taken to be the standard one x^{2} + y^{2}, and its orthogonal group the

group O(2, k̄) = \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) | a^{2} + b^{2} = 1\} ∪ \{( \begin{matrix}a & b \\ b & −a \end{matrix} ) | a^{2} + b^{2} = 1\} . Then

ρ̄(G_{L}) ⊂ SO(2, k̄) = \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) | a^{2} + b^{2} = 1\} , which is diagonalizable.

• #H^{0}(G_{q}, W) = #k for q ∈ Q

The eigenvalues of F rob_{q} on W are α_{q}β_{q} = q = 1 (since q ≡ 1 mod p), α_{q}^{2} and β_{q}^{2}. The latter two are not equal to 1 by our assumption that α_{q} ≠ β_{q}.

• #H^{1}(G_{q}, W) = #k^{2}

Here, at q ∈ Q, H^{1}(G_{q}/I_{q}, W) = W/(F rob_{q} − 1)W is one-dimensional (by the assumptions on α_{q} and β_{q}), H^{1}(I_{q}, W)^{G_{q}/I_{q}} = Hom(ℤ_{p}(1), W)^{F rob_{q}} = W [F rob_{q} − q] = W [F rob_{q} −1] = W^{F rob_{q}} is again one-dimensional, and H^{2}(G_{q}/I_{q}, W) = 0 since G_{q}/I_{q} ≅ ℤ̂. The desired result follows from the inflation-restriction exact sequence.

These computations allow us to deduce the following corollary from theorem 16.

Corollary 17. (i) r(Q) = dim_{k} H_{D_{Q}}^{1}(ℚ, W) = dim_{k} H_{D_{Q}∗}^{1}(ℚ, W^{∗}) + #(Q)

(ii) When we add a new prime q as above to a given set of primes Q, either dim_{k} H_{D_{Q}∗}^{1}(ℚ, W^{∗}) drops by 1, or dim_{k} H_{D_{Q}}^{1}(ℚ, W) grows by 1.

To prove proposition 11 it is therefore enough to prove the following

Proposition 18. Let r = dim_{k} H_{D∗}^{1}(ℚ, W^{∗}). Then for every n it is possible to choose a set Q of r primes, disjoint from Σ, satisfying

• For every q ∈ Q, q ≡ 1 mod p^{n}

• For every q ∈ Q, ρ̄(F rob_{q}) has distinct eigenvalues in k

• H_{D_{Q}∗}^{1}(ℚ, W^{∗}) = 0.

It will then follow from the corollary that r = r(Q), and from theorem 4.1, that R_{Q} can be topologically generated as an O-algebra by r elements. Discussion. The ”miracle” occuring in the minimal case is that all the local terms in the product expressing the ratio #H_{D}^{1}(ℚ, W)/#H_{D∗}^{1}(ℚ, W^{∗}) are 1, hence we can achieve an equality r = r(Q). In the numerical criterion of lemma 9 it is crucial to know that r, the number of variables S_{i} (which turns out to be the number of primes in Q, see the discussion preceding proposition 10), is actually equal to r(Q), the minimal number of generators of R_{Q}, and is not smaller. In the non-minimal case some of the local terms in (4.10) would be bigger than 1, and corollary 17(i) would look like r(Q) = dim_{k} H_{D_{Q}}^{1}(ℚ, W) = dim_{k} H_{D_{Q}∗}^{1}(ℚ, W^{∗})+t+#(Q) for some t > 0, so we would never be able to achieve r(Q) = r.

What remains to be checked is that we can ”kill” the dual Selmer group by a clever choice of the set Q, while maintaining the other restrictions imposed on the q′s in Q. The proof of this fact, given in the next section, is group-theoretical in nature, based on the Čebotarev density theorem.

5. Conclusion of the proof : some group theory

5.1. A reduction step. Since

H_{D_{Q}∗}^{1}(ℚ, W^{∗}) = Ker (H_{D∗}^{1}(ℚ, W^{∗}) → ∏_{q∈Q} H^{1}(G_{q}/I_{q}, W^{∗})) ,

it is enough to prove that for every cohomology class 0 ≠ [ψ] ∈ H_{D∗}^{1}(ℚ, W^{∗}) (ψ is a cocycle representing the cohomology class) there are infinitely many primes q not in Σ such that

• q ≡ 1 mod p^{n}

• the two eigenvalues of ρ̄(F rob_{q}) are distinct

• res_{q}([ψ]) ≠ 0.

We can then choose the q′s successively, each time decreasing dim H_{D_{Q}∗}^{1}(ℚ, W^{∗}) by 1, until we annihilate it completely after r steps.

By Čebotarev’s density theorem it is enough to find a σ ∈ G_{ℚ} such that

• σ|ℚ(ζ_{p^{n}}) = 1

• the eigenvalues of ρ̄(σ) are distinct

• ψ_{σ} ∉ (σ − 1)W^{∗}.

If we find such a σ, let q be a prime with σ|L = (\frac{L/ℚ}{𝔔}), where L is a Galois extension of ℚ containing ℚ(ζ_{p^{n}}) and the splitting fields of ρ̄ and ψ, and 𝔔 is a suitable prime above q in L. Then σ belongs to the decomposition group of q in L, so the last condition clearly implies res_{q}([ψ]) ≠ 0, and the first two obviously imply the other two restrictions imposed on q.

Consider the following diagram of fields

F  = extension of ℚ(ζ_{p^{n}}) cut by Ker(ρ̄)
|
K  = extension of ℚ(ζ_{p^{n}}) cut by Ker(ad^{0}(ρ̄))
|  H

ℚ(ζ_{p^{n}})

− |_{H̃ ⊂ G̃ = Im(ρ̄)} | −

Since Ker (Im(ρ̄) → Im(ad^{0}(ρ̄))) are the scalars in Im(ρ̄), H = H̃k^{×}/k^{×} ⊂ G = Im(ad^{0}(ρ̄)) ⊂ P GL_{2}(k).

Lemma 19. H^{1}(K/ℚ, W^{∗}) = 0.

Let us postpone the proof of the lemma for the moment, and see how to use it to find a σ as above. From the lemma, 0 ≠ ψ|G_{K} ∈ Hom(G_{K}, W^{∗})^{Gal(K/ℚ)}. But ψ(G_{K}) is a G_{ℚ}-submodule of W^{∗}, hence by the irreducibility of ρ̄|G_{L}, which implies the irreducibility of W^{∗} (see the argument in section 4.3), it is all of W^{∗}.

Next we claim that it is possible to find σ_{0} ∈ G_{ℚ(ζ_{p^{n}})} such that ρ̄(σ_{0}) has distinct eigenvalues {α, β}. If this is not the case, ρ̄(G_{ℚ(ζ_{p^{n}})}) is contained in the upper-triangular matrices, by an elementary computation with 2×2 matrices. But then, either ρ̄(G_{ℚ(ζ_{p^{n}})}) is contained in a torus, in which case ρ̄ is dihedral, and therefore ρ̄|G_{L} is not irreducible, or ρ̄|G_{ℚ(ζ_{p^{n}})} has a unique invariant line, in which case it must be invariant under ρ̄, contradicting its irreducibility.

The eigenvalues of σ_{0} on W are α/β, 1, and β/α. These are also its eigenvalues on W^{∗} because σ_{0} fixes ζ_{p}. Thus (σ_{0} − 1)W^{∗} ≠ W^{∗}, and ψ(G_{K}) ⊄ (σ_{0} − 1)W^{∗}.

Now look for σ = τσ_{0} with τ ∈ G_{K}. Note that τ acts trivially on W and W^{∗}. Since ρ̄(τ) is a scalar matrix, ρ̄(σ) will still have distict eigenvalues, and clearly σ leaves the p^{n} roots of unity invariant. All that remains to check is that

ψ_{σ} = τψ_{σ_{0}} + ψ_{τ} = ψ_{σ_{0}} + ψ_{τ} ∉ (σ − 1)W^{∗} = (σ_{0} − 1)W^{∗}. (5.1)

If σ = σ_{0} is not good, then by the above we can find a τ ∈ G_{K} such that adding ψ_{τ} will move ψ_{σ_{0}} out of (σ_{0} − 1)W^{∗}.

Proof of the lemma : We need the following classification theorem, due to Dickson ([Hu], II.8.27).

Theorem 20. Any finite subgroup of P GL_{2}(k̄) is one of the following (1) a subgroup of a Borel group

(2) conjugate to P GL_{2}(k′) or P SL_{2}(k′) for a finite field k′

(3) isomorphic to the dihedral group D_{2n} for (n, p) = 1

(4) isomorphic to A_{4}, S_{4}, or A_{5}.

Let G̃ = Im(ρ̄) = Gal(M/ℚ), where M is the splitting field of ρ̄. Let G = Im(ad^{0}(ρ̄)) = Gal(N/ℚ), where N is the splitting field of ad^{0}(ρ̄). Let Z = Ker(G̃ → G) = Gal(M/N), the scalar matrices in Im(ρ̄). Case 1. Z ≠ {±1}. Then det(Z) ≠ 1, so W^{Z} = W implies that W^{∗Z} = 0. Now Z is cyclic of order prime to p, and F/M is a p-extension since ℚ(ζ_{p}) ⊂ M ⊂ F = M(ζ_{p^{n}}), so Z lifts to a subgroup (still denoted Z) of Gal(F/ℚ). ¿From the inflationrestriction exact sequence we get in this case H^{1}(F/ℚ, W^{∗}) = H^{1}(F^{Z}/ℚ, W^{∗Z}) = 0. A fortiori H^{1}(K/ℚ, W^{∗}) = 0. Case 2. Z = {±1}, p > 3. Then Z fixes ℚ(ζ_{p}) and therefore Δ = Gal(ℚ(ζ_{p})/ℚ) is a quotient of G ⊂ P GL_{2}(k). Using the classification theorem we see that G is contained in a Borel, or has order prime to p (the other groups don’t have a cyclic quotient of order p − 1). The first option contradicts the irreducibility of ρ̄. The second implies that H = Gal(K/ℚ(ζ_{p^{n}})) also has order prime to p, so H^{1}(K/ℚ, W^{∗}) = H^{1}(ℚ(ζ_{p^{n}})/ℚ, W^{∗H}). But W^{∗H} = 0, or else W^{∗} is reducible, contradicting the absolute irreducibility of ρ̄|G_{L}. Case 3. Z = {±1}, p = 3. We may assume that 3|#G, and G is not contained in a Borel, otherwise we finish the proof as in step 2. Again, Δ = Gal(ℚ(ζ_{3})/ℚ) is a quotient of G, so this rules out G ≅ A_{5}, and we are left with the possibility G = P GL_{2}(k′) (P SL_{2}(k′) is simple if k′ ≠ 𝔽_{3}, and for 𝔽_{3} it is A_{4}, which does not have a normal subgroup of index 2. Also S_{4} = P GL_{2}(𝔽_{3})).

If k′ = 𝔽_{3} then S_{4} = P GL_{2}(𝔽_{3}) = Gal(N/ℚ) has a normal subgroup V_{4} ⊂ Gal(N/L) (its unique 2-Sylow subgroup). Since Gal(K/N) is a 3-group, this V_{4} lifts to a normal subgroup of Gal(K/L). Since ρ̄|G_{L} is absolutely irreducible, W^{∗V_{4}} = 0, and inflation-restriction finishes the proof as before.

There remains the case where k′ = 𝔽_{3^{n}} and n > 1. Then H = Gal(K/ℚ(ζ_{p^{n}})) ≅ Gal(N/ℚ(ζ_{p})) ≅ P SL_{2}(k′), and W = W^{∗} is the standard adjoint representation of P SL_{2}(k′) on trace-0 matrices. Wiles relies here on a result of Cline Parshall and Scott [CPS] that H^{1}(H, W^{∗}) = 0. With this the lemma is settled, and with it are concluded also the proofs of propositions 18 and 11, and of the main theorem.

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HECKE RINGS AND UNIVERSAL DEFORMATION RINGS 21

Hebrew University, Jerusalem, 91904, Israel

E-mail address: deshalit@math.huji.ac.il