Ring Theoretic Properties of Certain Hecke Algebras

Richard Taylor D.P.M.M.S., Cambridge University, 16 Mill Lane, Cambridge, CB2 1SB, U.K.

Andrew Wiles Department of Mathematics, Princeton University, Washington Road, Princeton, NJ 08544, U.S.A.

7 October 1994

Introduction

In the work of one of us (A.W.) on the conjecture that all elliptic curves defined over ℚ are modular, the importance of knowing that certain Hecke algebras are complete intersections was established. The purpose of this article is to provide the missing ingredient in [W2] by establishing that the Hecke algebras considered there are complete intersections. As is recorded in [W2], a method going back to Mazur [M] allows one to show that these algebras are Gorenstein, but this seems to be too weak for the purposes of that paper. The methods of this paper are related to those of chapter 3 of [W2].

We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument and we would like to thank him for allowing us to reproduce this in the appendix to this paper. R.T. would like to thank A.W. for his invitation to collaborate on these problems and for sharing his many insights into the questions considered. R.T. would also like to thank Princeton University, Université de Paris 7 and Harvard University for their hospitality during some of the work on this paper. A.W. was supported by an NSF grant.

1 Notation

Let p denote an odd prime, let O denote the ring of integers of a finite extension K/ℚ_{p}, let λ denote its maximal ideal and let k = O/λ.

If L is a perfect field G_{L} will denote its absolute Galois group and if the characteristic of L is not p then ϵ : G_{L} → ℤ_{p}^{×} will denote the p-adic cyclotomic character. If L is a number field and ℘ a prime of its ring of integers then G_{℘} will denote a decomposition group at ℘ and I_{℘} the corresponding inertia group. We will denote by Frob_{℘} the arithmetic Frobenius element of G_{℘}/I_{℘}.

If G is a group and M a G-module we will let M^{G} and M_{G} denote respectively the invariants and coinvariants of G on M. If ρ is a representation of G into the automorphisms of some abelian group we shall let V_{ρ} denote the underlying G-module. If H is a normal subgroup of G then we shall let ρ^{H} and ρ_{H} denote the representation of G/H on respectively V_{ρ}^{H} and V_{ρ,H}.

We shall also fix a continuous representation

\overline{ρ} : G_{ℚ} ⟶ GL_{2}(k)

with the following properties.

• \overline{ρ} is modular in the sense that it is a modp representation associated to some modular newform of some weight and level.

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

• If c denotes complex conjugation then det \overline{ρ}(c) = −1.

• The restriction of \overline{ρ} to the decomposition group at p either has the form

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

with ψ_{1} and ψ_{2} distinct characters and with ψ_{2} unramified; or is induced from a character χ of the unramified quadratic extension of ℚ_{p} whose restriction to the inertia group is the fundamental character of level 2, I_{p} → → 𝔽_{p^{2}}^{×}.

• If l ≠ p then

– either \overline{ρ}|_{I_{l}} ∼ ( \begin{matrix}χ & 0 \\ 0 & 1 \end{matrix} ),

– or \overline{ρ}|_{I_{l}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ),

– or \overline{ρ}|_{G_{l}} is absolutely irreducible and in the case \overline{ρ}|_{I_{l}} is absolutely reducible we have l ≢ −1 mod p.

(This implies that \overline{ρ}|_{G_{l}} is either unramified or of type A, B or C as defined in chapter 1 of [W2]. On the other hand if \overline{ρ}|_{G_{l}} is of type A, B or C then some twist satisfies the condition above.)

In the case that \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) we will fix the pair of characters ψ_{1}, ψ_{2}.

Note that in some cases this may involve making a choice.

We will let Q denote a finite set of primes q with the properties

• \overline{ρ} is unramified at q,

• q ≡ 1 mod p,

• \overline{ρ}(Frob_{q}) has distinct eigenvalues, which we shall denote α_{q} and β_{q}.

Much of our notation will involve a subscript Q to denote dependence on Q, whenever Q = ∅ we may simply drop it from the notation.

For q ∈ Q we shall let Δ_{q} denote the Sylow p-subgroup of (ℤ/qℤ)^{×}. We shall let δ_{q} denote a generator. We will write Δ_{Q} for the product of the Δ_{q} with q ∈ Q. We will let 𝔞_{Q} denote the kernel of the map O[Δ_{Q}] → O which sends every element of Δ_{Q} to 1. Let χ_{q} denote the character

G_{ℚ} ⟶ Gal (ℚ(ζ_{q})/ℚ) ≅ (ℤ/qℤ)^{×} → → Δ_{q},

and let χ_{Q} = ∏_{q∈Q} χ_{q}.

We will denote by N_{Q} the product of the following quantities:

• the conductor of \overline{ρ};

• the primes in Q;

• p, if \overline{ρ} is not flat (i.e. \overline{ρ} does not arise from the action of G_{p} on the \overline{ℚ_{p}}-points of some finite flat group scheme over ℤ_{p}) or if det \overline{ρ}|_{I_{p}} ≠ ϵ. (We remark that if \overline{ρ}|_{G_{p}} is flat but det \overline{ρ}|_{I_{p}} ≠ ϵ then \overline{ρ}|_{G_{p}} arises from an etale group scheme over ℤ_{p}. We also note that in [W2] the term flat is not used when the group scheme is ordinary.)

We will let Γ_{Q} denote the inverse image under Γ_{0}(N_{Q}) → (ℤ/N_{Q}ℤ)^{×} of the product of the following subgroups:

• the Sylow p-subgroup of (ℤ/Mℤ)^{×}, where M denotes the conductor of \overline{ρ};

• for each q ∈ Q the unique maximal subgroup of (ℤ/qℤ)^{×} of order prime to p.

Let 𝕋(Γ_{Q}) denote the ℤ-subalgebra of the complex endomorphisms of the space of weight 2 cusp forms on Γ_{Q} which is generated by the Hecke operators T_{l} and ⟨l⟩ for l |̸pN_{Q}, by U_{q} for q ∈ Q and by U_{p} if p|N_{Q}. Let 𝔪 denote the ideal of 𝕋(Γ_{Q}) ⊗_{ℤ}O generated by λ, by tr \overline{ρ}(Frob_{l}) − T_{l} and det \overline{ρ}(Frob_{l}) − l⟨l⟩ for l |̸pN_{Q}, by U_{q} − α_{q} for q ∈ Q and by U_{p} − ψ_{2}(Frob_{p}) if p|N_{Q}. Note that if Q ≠ ∅ this definition only makes sense if O is sufficiently large that k contains the eigenvalues of \overline{ρ}(Frob_{q}) for all q ∈ Q. It is a deep result following from the work of many mathematicians that 𝔪 is a proper ideal (see [D]), and so maximal. We let 𝕋_{Q} denote the localisation of 𝕋(Γ_{Q}) ⊗_{ℤ} O at 𝔪. Note that 𝕋_{Q} is reduced because the operators T_{l} for l |̸N_{Q} act semi-simply on the space of cusp forms for Γ_{Q} and the U_{q} for q ∈ Q act semi-simply on all common eigenspaces for the T_{l} for which the corresponding p-adic representation τ is either ramified at q or for which τ(Frob_{q}) has distinct eigenvalues. There is a natural map O[Δ_{Q}] → 𝕋_{Q}, which sends x ∈ Δ_{Q} to ⟨y⟩ where y ∈ ℤ, y ≡ x mod q for all q ∈ Q and y ≡ 1 mod N_{∅}.

It follows from the discussion after theorem 2.1 of [W2] or from the work of Carayol [C2] that there is a continuous representation

ρ_{Q}^{mod} : G_{ℚ} ⟶ GL_{2}(𝕋_{Q});

such that if l |̸N_{Q}p then ρ_{Q}^{mod} is unramified at l and we have tr ρ_{Q}^{mod}(Frob_{l}) = T_{l} and det ρ_{Q}^{mod}(Frob_{l}) = l⟨l⟩. In particular the reduction of ρ_{Q}^{mod} modulo the maximal ideal of 𝕋_{Q} is \overline{ρ}. From [C1] we can deduce the following.

• If q ∈ Q then ρ_{Q}^{mod}|_{G_{q}} = ϕ_{1} ⊕ ϕ_{2} where ϕ_{1} is unramified and ϕ_{1}(Frob_{q}) = U_{q}, and where ϕ_{2}|_{I_{q}} = χ_{q}|_{I_{q}}.

• If l ≠ p and \overline{ρ}|_{I_{l}} is non-trivial but unipotent then ρ_{Q}^{mod}|_{I_{l}} is unipotent.

• If l ∉ Q ∪ \{p\} and either \overline{ρ}|_{I_{l}} = χ ⊕ 1 or \overline{ρ}|_{G_{l}} is absolutely irreducible

then ρ_{Q}^{mod}(I_{l}) →^{∼} \overline{ρ}(I_{l}).

• det ρ_{Q}^{mod} = χ_{Q}ϵϕ where ϕ is a character of order prime to p.

Moreover if \overline{ρ}|_{G_{p}} is flat and if det \overline{ρ}|_{I_{p}} = ϵ then p |̸N_{Q} so ρ_{Q}^{mod}|_{G_{p}} is flat (i.e. the reduction modulo every ideal of finite index is flat). If \overline{ρ}|_{G_{p}} is not flat or if det \overline{ρ}|_{I_{p}} ≠ ϵ then p|N_{Q}, \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) and U_{p} is a unit in 𝕋_{Q}. It

follows from theorem 2 of [W1] (or more directly in the case ψ_{1}|_{I_{q}} ≠ ϵ from

proposition 12.9 of [G]) that ρ_{Q}^{mod}|_{G_{p}} ∼ ( \begin{matrix}χ_{1}ϵ & ∗ \\ 0 & χ_{2} \end{matrix} ), where χ_{2} is unramified

and χ_{1}(I_{p}) has order prime to p. In the case that χ_{1} is unramified we know further that χ_{1} = χ_{2} (see proposition 1.1 of [W2]) and that this character has finite order. It will be convenient to introduce the twist ρ′_{Q} = ρ_{Q}^{mod} ⊗ χ_{Q}^{−1/2} of ρ_{Q}^{mod}. In particular we see that det ρ′_{Q} is valued in O^{×}.

The main theorem of this paper is as follows. Recall that we may write 𝕋 for 𝕋_{∅}.

Theorem 1 The ring 𝕋 is a complete intersection.

We note that if O′ is the ring of integers of a finite extension K′/K then the ring constructed using O′ in place of O is just 𝕋_{Q} ⊗_{O} O′. Also 𝕋 is a complete intersection if and only if 𝕋 ⊗_{O} O′ is (using for instance corollary 2.8 on page 209 of [K2]). Thus we may and we shall assume that O is sufficiently large that the eigenvalues of every element of \overline{ρ} are rational over k and that there is a homomorphism π : 𝕋 → → O. In particular the definition of 𝕋_{Q} makes sense for all Q. There is an induced map π_{Q} : 𝕋_{Q} → 𝕋 → O. The map 𝕋_{Q} → 𝕋 takes the operators T_{l} and ⟨l⟩ to themselves and the operator U_{q} to the unique root of U^{2} − T_{q}U + q⟨q⟩ in 𝕋 above α_{q}. We will let ℘_{Q} denote the kernel of π_{Q} and will let η_{Q} denote the ideal π_{Q}(Ann_{𝕋_{Q}}(℘_{Q})). Then it is known that ∞ > #℘_{Q}/℘_{Q}^{2} ≥ #O/η_{Q} with equality if and only if 𝕋_{Q} is a complete intersection (see the appendix of [W2] or [L], we are using the fact that 𝕋_{Q} is reduced).

2 Generalisation of a Result of de Shalit

In this section we shall use the methods of de Shalit (see [dS]) to prove the following theorem.

Theorem 2 The ring 𝕋_{Q} is a free O[Δ_{Q}] module of O[Δ_{Q}]-rank equal to the O-rank of 𝕋.

By lemma 3 of [DT] we may choose a prime R with the following properties:

• R |̸6N_{Q}p;

• R ≢ 1 mod p; • \overline{ρ}(Frob_{R}) has distinct eigenvalues α_{R} and β_{R};

• (1 + R)^{2} det \overline{ρ}(Frob_{R}) ≠ R(tr \overline{ρ}(Frob_{R}))^{2}.

Let Γ_{Q−} be defined in the same way as Γ_{Q} but with (ℤ/qℤ)^{×} replacing its maximal subgroup of order prime to p in the definition for each q ∈ Q. Let Γ′_{Q} = Γ_{Q} ∩ Γ_{1}(R) and let Γ′_{Q−} = Γ_{Q−} ∩ Γ_{1}(R). The purpose of introducing the auxiliary prime R is to make these groups act freely on the upper half complex plane. Let 𝕋′(Γ′_{Q}) denote the ℤ-subalgebra of the complex endomorphism ring of the space of weight two modular (not necessarily cusp) forms generated by the operators T_{l} and ⟨l⟩ for l |̸N_{Q}R and by U_{l} for l|N_{Q}R. Let 𝔪′_{Q} denote the maximal ideal of 𝕋′(Γ′_{Q}) ⊗_{ℤ} O generated by the following elements:

• λ;

• T_{l} − tr \overline{ρ}(Frob_{l}) and l⟨l⟩ − det \overline{ρ}(Frob_{l}) for l |̸N_{Q}Rp;

• U_{q} − α_{q} for q ∈ Q or q = R;

• U_{l} − tr \overline{ρ}_{I_{l}}(Frob_{l}) if l|N_{∅} and l ≠ p;

• U_{p} − ψ_{2}(Frob_{p}) if p|N_{∅};

• T_{p} − tr \overline{ρ}_{I_{p}}(Frob_{p}) if p |̸N_{∅}.

Let 𝕋′_{Q} denote the localisation of 𝕋′(Γ′_{Q}) ⊗_{ℤ} O at 𝔪′_{Q}. Let Y′_{Q} denote the quotient of the upper half complex plane by Γ′_{Q} and let X′_{Q} denote its standard compactification. Complex conjugation c acts continuously on these Riemann surfaces. We let H^{1}(Y′_{Q}, O)^{±} and H^{1}(X′_{Q}, O)^{±} denote the ±1 eigenspaces of c on H^{1}(Y′_{Q}, O) and H^{1}(X′_{Q}, O). All these definitions go over verbatim, but with Q− replacing Q.

Lemma 1 𝕋′_{Q} ≅ 𝕋_{Q} and 𝕋′_{Q−} ≅ 𝕋.

This is a standard argument which we will only sketch. First observe that because \overline{ρ} is irreducible 𝕋′_{Q} and 𝕋′_{Q−} can be defined using the ring generated by the Hecke operators on the spaces of weight two cusp forms S_{2}(Γ′_{Q}) and S_{2}(Γ′_{Q−}) (rather than spaces of all modular forms). The same arguments as in the proof of proposition 2.15 of [W2] show that we can drop the Hecke operators T_{p} if p |̸N_{Q} and the Hecke operators U_{l} for l ≠ p and l|N_{∅} from the definition without changing the Hecke algebra. Next we will show that we need only consider the algebras generated in the endomorphisms of S_{2}(Γ_{Q})^{2} ⊂ S_{2}(Γ′_{Q}) and S_{2}(Γ)^{2^{#Q+1}} ⊂ S_{2}(Γ′_{Q−}). This follows from the following facts.

• As R ≢ 1 mod p and det \overline{ρ} is unramified at R, no component of 𝕋′_{Q} nor of 𝕋′_{Q−} can correspond to an eigenform with a non-trivial action of (ℤ/Rℤ)^{×}.

• As α_{R}/β_{R} ≠ R^{±1} in k, no component of 𝕋′_{Q} nor of 𝕋′_{Q−} can correspond to an eigenform which is special at R (i.e. an eigenform which corresponds to a cuspidal automorphic representation of GL_{2}(𝔸) whose component at R is special).

• As for each prime q ∈ Q, α_{q}/β_{q} ≠ q^{±1} in k, no component of 𝕋′_{Q−} can correspond to an eigenform which is special at q.

The ring generated by the Hecke operators T_{l} and ⟨l⟩ for l |̸pN_{Q}R, by U_{q} for q ∈ Q ∪ {R} and by U_{p} if p|N_{Q} on S_{2}(Γ_{Q})^{2} is isomorphic to 𝕋(Γ_{Q})[u_{R}]/(u_{R}^{2} − T_{R}u_{R} + R⟨R⟩). In fact U_{R} acts by the matrix

( \begin{matrix}T_{R} & 1 \\ −R⟨R⟩ & 0 \end{matrix} )

on S_{2}(Γ_{Q})^{2}. Similarly the ring generated by the Hecke operators T_{l} and ⟨l⟩ for l |̸pN_{Q}R, by U_{q} for q ∈ Q ∪ {R} and by U_{p} if p|N_{Q} on S_{2}(Γ)^{2^{#Q+1}} is isomorphic to 𝕋(Γ)[u_{q} : q ∈ Q ∪ {R}]/(u_{q}^{2} − T_{q}u_{q} + q⟨q⟩ : q ∈ Q ∪ {R}). Tensoring with O and localising at the appropriate maximal ideal we get the desired isomorphism. We have to use the fact that u_{R}^{2} − T_{R}u_{R} + R⟨R⟩ has two roots in 𝕋_{Q} with distinct reductions modulo the maximal ideal and the similar facts over 𝕋 for u_{q}^{2} − T_{q}u_{q} + q⟨q⟩ with q ∈ Q ∪ {R}.

Because \overline{ρ} is irreducible we see that H^{1}(Y′_{Q}, O)_{𝔪′_{Q}} = H^{1}(X′_{Q}, O)_{𝔪′_{Q}} and that H^{1}(Y′_{Q−}, O)_{𝔪′_{Q−}} = H^{1}(X′_{Q−}, O)_{𝔪′_{Q−}}. By corollary 1 of theorem 2.1 of [W2] we see that H^{1}(X′_{Q}, O)_{𝔪′_{Q}}^{±} are free rank one 𝕋′_{Q}-modules and that H^{1}(X′_{Q−}, O)_{𝔪′_{Q−}}^{±} are free rank one 𝕋′_{Q−}-modules. Hence it will suffice to prove the following proposition.

Proposition 1 H^{1}(Y′_{Q}, O)^{−} is a free O[Δ_{Q}]-module, with O[Δ_{Q}]-rank equal to the O-rank of H^{1}(Y′_{Q−}, O)^{−}.

Because H^{1}(Y′_{Q−}, K) = H^{1}(Y′_{Q}, K)^{Δ_{Q}} we need only show that H^{1}(Y′_{Q}, O)^{−} is a free O[Δ_{Q}]-module. Because R ≥ 5, Γ′_{Q−} acts freely on the upper half complex plane and so may be identified with the fundamental group of Y′_{Q−}. In particular we see that Γ′_{Q−} is a free group. Similarly Γ′_{Q} acts freely on the upper half complex plane and we get identifications

H^{1}(Y′_{Q}, O) ≅ H^{1}(Γ′_{Q}, O) ≅ H^{1}(Γ′_{Q−}, O[Δ_{Q}]),

the latter arising from Shapiro’s lemma. Under these identifications complex conjugation goes over to the involution induced by conjugation by ξ =

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) and trivial action on the coefficients. (This follows because the

action of c on Y′_{Q} is induced by the map z ↦ −\overline{z} of the upper half complex plane to itself.)

Because Γ′_{Q−} is a free group, the cocycles Z^{1}(Γ′_{Q−}, O[Δ_{Q}]) are a free O[Δ_{Q}]- module. (If γ_{1}, ..., γ_{a} are free generators of Γ′_{Q−} then we have an isomorphism

\begin{matrix}Z^{1}(Γ′_{Q−}, O[Δ_{Q}]) & →^{∼} & O[Δ_{Q}]^{a} \\ ψ & ↦ & (ψ(γ_{1}), ..., ψ(γ_{a})).) \end{matrix}

On the other hand ξ acts trivially on Δ_{Q} and so the coboundaries are contained in Z^{1}(Γ′_{Q−}, O[Δ_{Q}])^{+}. Thus H^{1}(Y′_{Q}, O)^{−} ≅ Z^{1}(Γ′_{Q−}, O[Δ_{Q}])^{−} is a free O[Δ_{Q}]- module, as desired.

Before leaving this section we remark the following corollaries of theorem 2.

Corollary 1 If q ∉ Q then 𝕋_{Q∪{q}}/(δ_{q} − 1) →^{∼} 𝕋_{Q}. Moreover (δ_{q} − 1)𝕋_{Q∪{q}} and (1 + δ_{q} + ... + δ_{q}^{#Δ_{q}−1})𝕋_{Q∪{q}} are annihilators of each other in 𝕋_{Q∪{q}}.

Corollary 2 If for some Q the ring 𝕋_{Q} is a complete intersection then so is 𝕋.

The proof is by showing that under the assumption that 𝕋_{Q∪{q}} is a complete intersection, so is 𝕋_{Q}. The argument in section 2 of [K1] shows that if a complete local noetherian ring S is a complete intersection, if f ∈ S and if Hom_{S}(S/(f), S) = Hom_{S}(S/(f), Ann_{S}(f)) is a free S/(f) module then S/(f) is a complete intersection. We apply this result with S = 𝕋_{Q∪{q}} and f = δ_{q}−1. The final condition is met because the annihilator of δ_{q} − 1 in 𝕋_{Q∪{q}} is a free rank one 𝕋_{Q} module by the last corollary.

Corollary 3 η_{Q} = η#Δ_{Q}.

We remind the reader that η_{Q} is defined at the end of section one and that η = η_{∅}. The proof of the corollary is by showing that η_{Q∪{q}} = η_{Q}#Δ_{q} if q ∉ Q. Write Q′ = Q ∪ {q} and let θ denote the natural surjection 𝕋_{Q′} → → 𝕋_{Q}. It suffices to prove that in 𝕋_{Q′} we have the equation Ann ℘_{Q′} = (1 + δ_{q} + ... + δ_{q}^{#Δ_{q}−1})θ^{−1}(Ann_{𝕋_{Q}}(℘_{Q})). However θ^{−1}(Ann_{𝕋_{Q}}(℘_{Q})) is just the set of elements t of 𝕋_{Q′} for which t℘_{Q′} ⊂ (δ_{q} − 1)𝕋_{Q′}. The inclusion Ann ℘_{Q′} ⊃ (1 + δ_{q} + ... + δ_{q}^{#Δ_{q}−1})θ^{−1}(Ann_{𝕋_{Q}}(℘_{Q})) is now clear. Conversly if t ∈ Ann ℘_{Q′} then t annihilates ker θ and hence t = (1 + δ_{q} + ... + δ_{q}^{#Δ_{q}−1})s. Then we see that s℘_{Q′} ⊂ (δ_{q} − 1)𝕋_{Q′}, i.e. that s ∈ θ^{−1}(Ann_{𝕋_{Q}}(℘_{Q})). The other inclusion now follows.

Corollary 4 #(𝔞_{Q}𝕋_{Q}/℘_{Q}𝔞_{Q}𝕋_{Q})#(O/η) = #(O/η_{Q}).

To prove this corollary note that 𝔞_{Q}/𝔞_{Q}^{2} ≅ ⨁_{q∈Q} O/#Δ_{q}. Thus it follows from theorem 2 that 𝔞_{Q}𝕋_{Q}/𝔞_{Q}^{2}𝕋_{Q} ≅ 𝕋_{Q} ⊗_{O[Δ_{Q}]} 𝔞_{Q}/𝔞_{Q}^{2} ≅ ⨁_{q∈Q} 𝕋/#Δ_{q} and so we deduce that 𝔞_{Q}𝕋_{Q}/℘_{Q}𝔞_{Q}𝕋_{Q} ≅ ⨁_{q∈Q} O/#Δ_{q}. This corollary now follows from the last one.

3 Some Algebra

In this section we shall establish certain criteria for rings to be complete intersections. We shall rely on the numerical criterion established in the appendix of [W2]. In that appendix there is a Gorenstein hypothesis which can be checked in the cases where we will apply the results of this section. However in order to state the results of this section in somewhat greater generality we shall reference the paper [L], where the Gorenstein hypothesis in the appendix of [W2] is removed, rather than the appendix of [W2] directly.

Fix a finite flat reduced local O algebra T with a section π : T → → O. We will consider complete local noetherian O algebras R together with maps R → → T. We will denote by J_{R} the kernel of the map R → → T, by π_{R} the induced map R → → O, by ℘_{R} the kernel of π_{R} and by η_{R} the image under π_{R} of the annihilator in R of ℘_{R}. We will let Ψ_{R} = (℘_{R}^{2} ∩ J_{R})/℘_{R}J_{R}.

If S → → R → → T then ℘_{S}^{2} → → ℘_{R}^{2}, J_{S} is the pre-image of J_{R} and so Ψ_{S} → → Ψ_{R}. We have an exact sequence

(0) → Ψ_{R} → J_{R}/℘_{R}J_{R} → ℘_{R}/℘_{R}^{2} → ℘_{T} /℘_{T}^{2} → (0).

From this we deduce the following facts.

• #Ψ_{R} < ∞. (To see this it suffices to show that (Ψ_{R})_{℘_{R}} = (0). However as T is reduced (J_{R})_{℘_{R}} = (℘_{R})_{℘_{R}} and so the map (J_{R}/℘_{R}J_{R})_{℘_{R}} → (℘_{R}/℘_{R}^{2})_{℘_{R}} is an isomorphism. The result follows.)

• #Ψ_{R}#(℘_{R}/℘_{R}^{2}) = #(℘_{T} /℘_{T}^{2} )#(J_{R}/℘_{R}J_{R}).

Lemma 2 Suppose we have the inequalities #(℘_{T} /℘_{T}^{2} ) ≤ #(O/η_{T})#Ψ_{R} and #(O/η_{T})#(J_{R}/℘_{R}J_{R}) ≤ #(O/η_{R}) < ∞ and suppose R is a finite flat Oalgebra. Then R is a complete intersection.

To show this first note that we have the inequalities

#(℘_{R}/℘_{R}^{2}) = #(℘_{T} /℘_{T}^{2} )#(J_{R}/℘_{R}J_{R})/#Ψ_{R}
≤ #(℘_{T} /℘_{T}^{2} )#(O/η_{R})/(#Ψ_{R}#(O/η_{T}))
≤ #(O/η_{R}).

Now applying the criterion of [L] we see that R is a complete intersection.

Lemma 3 We have the inequality:

#(O/η_{R}) ≤ #(J_{R}/℘_{R}J_{R})#(O/η_{T}).

As Fitt_{R}(J_{R}) ⊂ Ann_{R}(J_{R}) we see that Fitt_{O}(J_{R}/℘_{R}J_{R}) ⊂ π_{R}Ann_{R}(J_{R}). On the other hand it is easy to see that

Ann_{R}(℘_{R}) ⊃ {s ∈ R|s℘_{R} ⊂ J_{R}}Ann_{R}(J_{R}).

Applying π_{R} we see that

η_{R} ⊃ η_{T} Fitt_{O}(J_{R}/℘_{R}J_{R}),

and the lemma follows.

Lemma 4 If R is a complete intersection which is finite and flat over O and η_{R} ≠ (0) then #(℘_{T} /℘_{T}^{2} ) ≤ #(O/η_{T})#Ψ_{R}. If R is a power series ring the same result is true without the assumptions that it is finite over O and that η_{R} ≠ (0).

For the first part we see that, as R is a complete intersection, #(℘_{R}/℘_{R}^{2}) = #(O/η_{R}) (see [L]). Thus we see that

#Ψ_{R}#(O/η_{R}) = #(℘_{T} /℘_{T}^{2} )#(J_{R}/℘_{R}J_{R}) ≥ #(℘_{T} /℘_{T}^{2} )#(O/η_{R})/#(O/η_{T}).

The first result follows. For the second result note that we can factor R → → T as R → → R′ → → T with R′ a complete intersection which is finite and flat over O and for which ℘_{R′}/℘_{R′}^{2} →^{∼} ℘_{T} /℘_{T}^{2} (by the proof of lemma 9 of [L]). Then #O/η_{R′} = #℘_{R′}/℘_{R′}^{2} < ∞ and so #℘_{T} /℘_{T}^{2} ≤ #(O/η_{T})#Ψ_{R′}. However Ψ_{R} → → Ψ_{R′}, so the result follows.

We now return to the notation of the first section. We will let Ψ_{Q} denote Ψ_{𝕋_{Q}} and J_{Q} denote J_{𝕋_{Q}}.

Proposition 2 Suppose that for a series of sets Q_{n} we have ideals I_{n} in 𝕋_{Q_{n}} with the following properties.

1. I_{n} is contained in 𝔪_{𝕋_{Qn}}^{2} and 𝕋_{Q_{n}}/I_{n} has finite cardinality.

2. I_{n+1}𝕋 ⊂ I_{n}𝕋 and ⋂_{n} I_{n}𝕋 = (0).

3. There is a surjective map of O-algebras 𝕋_{Q_{n+1}}/I_{n+1} → → 𝕋_{Q_{n}}/I_{n} such that

the diagram

𝕋_{Q_{n+1}}/I_{n+1} ⟶ 𝕋_{Q_{n}}/I_{n}
↓  ↓
𝕋/I_{n+1}𝕋 ⟶ 𝕋/I_{n}𝕋

commutes. (Note that this map is not assumed to take a given Hecke operator to itself.)

4. lim_{←} 𝕋_{Q_{n}}/I_{n} is a power series ring.

Then for n sufficiently large 𝕋_{Q_{n}} is a complete intersection, and hence 𝕋 is a complete intersection (by corollary 2 of theorem 2).

Let P denote lim_{←} 𝕋_{Q_{n}}/I_{n}. We get a natural map P → → 𝕋 and can choose maps P → 𝕋_{Q_{n}} compatible with the maps 𝕋_{Q_{n}} → → 𝕋 and 𝕋_{Q_{n}} → → 𝕋_{Q_{n}}/I_{n}. Because I_{n} ⊂ 𝔪_{𝕋_{Qn}}^{2} we see that the map P → 𝕋_{Q_{n}} is surjective. We have a sequence

Ψ_{P} ⟶ ⟶ Ψ_{Q_{n}} ⟶ ((J_{Q_{n}} + I_{n}) ∩ (℘_{Q_{n}}^{2} + I_{n}))/(J_{Q_{n}}℘_{Q_{n}} + I_{n}).

(Note that although the maps P → 𝕋_{Q_{n}} and Ψ_{P} → Ψ_{Q_{n}} are not compatible as n varies the composite map above is.) Moreover Ψ_{P} = lim_{←}((J_{Q_{n}}+I_{n})∩(℘_{Q_{n}}^{2}+ I_{n}))/(J_{Q_{n}}℘_{Q_{n}} +I_{n}) (using the fact that 𝕋_{Q_{n}}/I_{n} is finite for all n) and so as Ψ_{P} is finite we have that the map Ψ_{P} → ((J_{Q_{n}} + I_{n}) ∩ (℘_{Q_{n}}^{2} + I_{n}))/(J_{Q_{n}}℘_{Q_{n}} + I_{n}) is injective for n sufficiently large. Thus for n sufficiently large Ψ_{P} →^{∼} Ψ_{Q_{n}}. We deduce the inequality

#(℘/℘^{2}) ≤ #(O/η)#Ψ_{P} = #(O/η)#Ψ_{Q_{n}},

where the first inequality follows from lemma 4. The proposition follows on applying corollary 4 of theorem 2 and lemma 2.

Corollary 1 Suppose that we have an integer r and a series of sets Q_{m} with the following properties:

1. if q ∈ Q_{m} then q ≡ 1 mod p^{m};

2. \overline{ρ} is unramified at q and \overline{ρ}(Frob_{q}) has distinct eigenvalues; 3. #Q_{m} = r;

4. 𝕋_{Q_{m}} can be generated as an O-algebra by r elements.

Then 𝕋 is a complete intersection.

To prove this corollary it is useful to have the following definition. By a level n structure we shall mean a quadruple B = (A, α, β, γ), where

• A is an O-algebra,

• α : O[[T_{1}, ..., T_{r}]] → → A,

• β : O[[S_{1}, ..., S_{r}]]/(p^{n}, (S_{1} + 1)^{p^{n}} − 1, ..., (S_{r} + 1)^{p^{n}} − 1) → A makes A a free module over O[[S_{1}, ..., S_{r}]]/(p^{n}, (S_{1} + 1)^{p^{n}} − 1, ..., (S_{r} + 1)^{p^{n}} − 1),

• and γ : A/(S_{1}, ..., S_{r}) →^{∼} 𝕋/p^{n}.

If B is a structure of level n and n′ ≤ n then it induces a structure of level n′ by reducing mod(p^{n′}, (S_{1} + 1)^{p^{n′}} − 1, ..., (S_{r} + 1)^{p^{n′}} − 1).

Let A_{m} = 𝕋_{Q_{m}}/(p^{m}, δ_{q}^{p^{m}} − 1|q ∈ Q_{m}). This extends to a level m structure that we will denote B_{m}. For n ≤ m we will let B_{m,n} denote the level n structure induced by B_{m}. There are only finitely many isomorphism classes of structures of level n and so we may choose recursively integers m(n) with the following two properties.

1. B_{m(n),n−1} ≅ B_{m(n−1),n−1}.

2. B_{m(n),n} ≅ B_{m,n} for infinitely many integers m.

Let I_{n} denote the kernel of the map from 𝕋_{Q_{m(n)}} to the ring underlying B_{m(n),n}. We claim that the pairs (Q_{m(n)}, I_{n}) for n ≥ 2 satisfy the requirements of proposition 2 (we use n ≥ 2 to ensure that I_{n} ⊂ 𝔪_{Q_{n}}^{2}). We need only check that lim_{←} B_{m(n),n} is a power series ring. On the one hand it is a finite free O[[S_{1}, ..., S_{r}]]-module, and so has Krull dimension r + 1. On the other hand it is a quotient of O[[T_{1}, ..., T_{r}]] and so must in fact equal O[[T_{1}, ..., T_{r}]].

4 Galois Cohomology

It remains to find a sequence of sets Q_{m} with the properties of corollary 1 of proposition 2. We must recall some definitions in Galois cohomology. We define H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}).

1. If l ≠ p then H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}) = H^{1}(𝔽_{l}, (ad^{0}\overline{ρ})^{I_{l}}) = ker (H^{1}(ℚ_{l}, ad^{0}\overline{ρ}) → H^{1}(I_{l}, ad^{0}\overline{ρ})).

2. If \overline{ρ}|_{G_{p}} is flat and det \overline{ρ}|_{I_{p}} = ϵ then we will let H_{f}^{1}(ℚ_{p}, ad^{0}\overline{ρ}) denote those elements in H^{1}(ℚ_{p}, ad^{0}\overline{ρ}) ⊂ Ext_{k[G_{p}]}^{1}(V\overline{ρ}, V\overline{ρ}) which correspond to extensions which can be realised as the \overline{ℚ_{p}}-points on the generic fibre of a finite flat group scheme over ℤ_{p}.

3. If \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) with ψ_{1}|_{I_{p}} ≠ ϵ then we let H_{f}^{1}(ℚ_{p}, ad^{0}\overline{ρ}) denote

the kernel of H^{1}(ℚ_{p}, ad^{0}\overline{ρ}) → H^{1}(I_{p}, (ad^{0}\overline{ρ})/Hom_{k}(V\overline{ρ}/F, F)), where F denotes the line in V\overline{ρ} where G_{p} acts by the character ψ_{1}.

4. Finally if \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) with ψ_{1}|_{I_{p}} = ϵ but \overline{ρ} is not flat then

we will let H_{f}^{1}(ℚ_{p}, ad^{0}\overline{ρ}) denote the kernel of the map H^{1}(ℚ_{p}, ad^{0}\overline{ρ}) → H^{1}(ℚ_{p}, (ad^{0}\overline{ρ})/Hom_{k}(V\overline{ρ}/F, F)), where F denotes the line in V\overline{ρ} where G_{p} acts by the character ψ_{1}.

We define H_{Q}^{1}(ℚ, ad^{0}\overline{ρ}) to be the inverse image under

H^{1}(ℚ, ad^{0}\overline{ρ}) ⟶ ∏_{l∉Q} H^{1}(ℚ_{l}, ad^{0}\overline{ρ})

of ∏_{l∉Q} H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}).

We also define H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}(1)) to be the annihilator of H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}) under the pairing of Tate local duality H^{1}(ℚ_{l}, ad^{0}\overline{ρ})×H^{1}(ℚ_{l}, ad^{0}\overline{ρ}(1)) → k. We then define H_{Q^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)) to be the inverse image under

H^{1}(ℚ, ad^{0}\overline{ρ}(1)) ⟶ ∏_{l∉Q} H^{1}(ℚ_{l}, ad^{0}\overline{ρ}(1))

of ∏_{l∉Q} H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}(1)).

Lemma 5 dim_{k} H_{Q}^{1}(ℚ, ad^{0}\overline{ρ}) ≤ dim_{k} H_{Q^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)) + #Q

To see this we apply proposition 1.6 of [W2]. For l ∉ Q and l ≠ p we see that h_{l} = 1 because the index of H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}) in H^{1}(ℚ_{l}, ad^{0}\overline{ρ}) is equal to #H^{1}(I_{l}, ad^{0}\overline{ρ})^{G_{𝔽_{l}}} which in turn equals

#((ad^{0}\overline{ρ}(−1))_{I_{l}})^{G_{𝔽_{l}}} = #((ad^{0}\overline{ρ}(1))^{I_{l}})_{G_{𝔽_{l}}} = #H^{0}(ℚ_{l}, ad^{0}\overline{ρ}(1)).

For q ∈ Q we have that h_{q} = #H^{0}(ℚ_{q}, ad^{0}\overline{ρ}(1)) = #k. It remains to check that h_{p}h_{∞} ≤ 1. In the case that \overline{ρ}|_{G_{p}} is not flat or det \overline{ρ}|_{I_{p}} ≠ ϵ this is proved in parts (iii) and (iv) of proposition 1.9 of [W2]. Thus suppose that \overline{ρ}|_{G_{p}} is flat and det \overline{ρ}|_{I_{p}} = ϵ. We must show that dim_{k} H_{f}^{1}(ℚ_{p}, ad^{0}\overline{ρ}) ≤ 1 + dim_{k} H^{0}(ℚ_{p}, ad^{0}\overline{ρ}) (= 1 if \overline{ρ}|_{G_{p}} is indecomposable and = 2 otherwise).

Following [FL] let M denote the abelian category of k vector spaces M with a distinguished subspace M^{1} and a k-linear isomorphism ϕ : M/M^{1} ⊕ M^{1} →^{∼} M. Then there are equivalences of categories between:

• M^{op};

• finite flat group schemes A/ℤ_{p} with an action of k;

• k[G_{p}]-modules which are isomorphic as modules over 𝔽_{p}[G_{p}] to the \overline{ℚ_{p}} points of some finite flat group scheme over ℤ_{p}.

See section 9 of [FL] for details. The only point here is that an action of k on the generic fibre of a finite flat group scheme over ℤ_{p} extends uniquely to an action on the whole scheme. Let M(\overline{ρ}) denote the object of M corresponding to \overline{ρ}. Then dim_{k} M(\overline{ρ}) = 2 and dim_{k} M(\overline{ρ})^{1} = 1 (since det \overline{ρ}|_{I_{p}} = ϵ). We get an embedding Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})) ↪ H^{1}(ℚ_{p}, ad\overline{ρ}). We will show that

1. dim_{k} Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})) = 2 if \overline{ρ}|_{G_{p}} is indecomposable and = 3 otherwise;

2. the composite map Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})) ↪ H^{1}(ℚ_{p}, ad\overline{ρ}) →^{tr} H^{1}(ℚ_{p}, k) is non-trivial, where tr denotes the map induced by the trace.

The lemma will then follow.

For the first point it is explained in lemma 4.4 of [R] how to calculate Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})). Let \{e_{0}, e_{1}\} be a basis of M(\overline{ρ}) with e_{1} ∈ M(\overline{ρ})^{1}. Let ϕ(e_{0}, 0) = αe_{0} + βe_{1} and ϕ(0, e_{1}) = γe_{0} + δe_{1}. Then Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})) can be identified as a k-vector space with M_{2}(k) modulo the subspace of matrices of the form

( \begin{matrix}r & 0 \\ s & t \end{matrix} ) ( \begin{matrix}α & γ \\ β & δ \end{matrix} ) − ( \begin{matrix}α & γ \\ β & δ \end{matrix} ) ( \begin{matrix}r & 0 \\ 0 & t \end{matrix} ) = ( \begin{matrix}0 & (r − t)γ \\ sα + (t − r)β & sγ \end{matrix} ) ,

for any r, s, t ∈ k. Thus dim_{k} Ext_{M}^{1}(M(\overline{ρ}), M(\overline{ρ})) = 2 if γ ≠ 0 and = 3 if γ = 0. However γ = 0 if and only if M(\overline{ρ})^{1} is a subobject of M(\overline{ρ}) in M. This is true if and only if \overline{ρ} has a one dimensional quotient on which inertia acts by ϵ which itself is true if and only if \overline{ρ}|_{G_{p}} is decomposable.

For the second point consider the k[G_{p}]-module \overline{ρ} ⊗ τ where τ is the unramified representation

Frob_{p} ⟼ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Then \overline{ρ} ⊗ τ is an extension of \overline{ρ} by itself. Moreover its extension class maps to the element of H^{1}(ℚ_{p}, k) = Hom(ℚ_{p}^{×}, k) which is trivial on ℤ_{p}^{×} and takes p ↦ 2. Finally it is isomorphic to the action of G_{p} on the \overline{ℚ_{p}} points of a finite flat group scheme over ℤ_{p}, because this is true over an unramified extension.

Lemma 6 𝕋_{Q} can be generated as an O algebra by dim_{k} H_{Q}^{1}(ℚ, ad^{0}\overline{ρ}) elements.

Let 𝔪_{Q} denote the maximal ideal of 𝕋_{Q}. It will suffice to show that there is an embedding of k-vector spaces

κ : Hom_{k}(𝔪_{Q}/(𝔪_{Q}^{2}, λ), k) ↪ H_{Q}^{1}(ℚ, ad^{0}\overline{ρ}).

We first define

κ : Hom_{k}(𝔪_{Q}/(𝔪_{Q}^{2}, λ), k) ⟶ H^{1}(ℚ, ad\overline{ρ}).

If θ is a non-zero element of the left hand group we may extend it uniquely to a map of local O-algebras θ̃: 𝕋_{Q} → → k[ϵ] where ϵ^{2} = 0. Let ρ_{θ} = θ̃ ∘ ρ′_{Q}. We get an exact sequence

(0) ⟶ V\overline{ρ} ⟶ V_{ρ_{θ}} ⟶ V\overline{ρ} ⟶ (0),

and hence a class κ(θ) in Ext_{k[G_{ℚ}]}^{1}(\overline{ρ}, \overline{ρ}) ≅ H^{1}(ℚ, ad\overline{ρ}). Because det ρ_{θ} is valued in k ⊂ k[ϵ] we see that κ(θ) actually lies H^{1}(ℚ, ad^{0}\overline{ρ}).

We claim that res_{l}κ(θ) lies in H_{f}^{1}(ℚ_{l}, ad^{0}\overline{ρ}) for l ∉ Q. This computation is very similar to some in [W2], but is not actually carried out there, so we give an argument here. First suppose that l ≠ p and that either p |̸#\overline{ρ}(I_{l}) or \overline{ρ}|_{G_{l}} is absolutely irreducible. In this case ρ′_{Q}(I_{l}) →^{∼} \overline{ρ}(I_{l}) and det ρ′_{Q}|_{I_{l}} has order prime to l. Because either p |̸#\overline{ρ}(I_{l}) or p = 3 and ad\overline{ρ}(I_{l}) ≅ A_{4} we have that H^{1}(\overline{ρ}(I_{l}), ad^{0}\overline{ρ}) = (0) and so ρ_{θ}|_{I_{l}} ≅ \overline{ρ}|_{I_{l}} ⊗_{k} k[ϵ]. The result follows in this case. Secondly suppose that \overline{ρ}|_{I_{l}} is unipotent and nontrivial. Then the same is true for ρ′_{Q}|_{I_{l}} and then also for ρ_{θ}. However the Sylow p-subgroup of I_{l} is pro-cyclic and so ρ_{θ}|_{I_{l}} must also be of the form \overline{ρ} ⊗_{k} k[ϵ] and res_{l}κ(θ) ∈ H^{1}(I_{l}, ad^{0}(\overline{ρ})) must vanish. In the case l = p, \overline{ρ} is flat and det \overline{ρ}|_{I_{p}} = ϵ the claim is immediate

from the definitions. In the case l = p and \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) with ψ_{1}|_{I_{p}} ≠ ϵ use the fact that ρ′_{Q}|_{I_{p}} ∼ ( \begin{matrix}ψ̃_{1} & ∗ \\ 0 & 1 \end{matrix} ) where ψ̃_{1} denotes the Teichmuller lifting

of ψ_{1}|_{I_{p}}. Finally in the case l = p, \overline{ρ}|_{G_{p}} ∼ ( \begin{matrix}ψ_{1} & ∗ \\ 0 & ψ_{2} \end{matrix} ) with ψ_{1}|_{I_{p}} = ϵ but \overline{ρ}

not flat use the fact that

ρ′_{Q}|_{G_{p}} ∼ ( \begin{matrix}δϵ & ∗ \\ 0 & δ \end{matrix} )

where δ is an unramified character of order prime to p.

It remains to show that κ is injective. Suppose it were not. Then we could find a non-zero θ such that ρ_{θ} ∼ \overline{ρ} ⊗_{k} k[ϵ]. Thus tr ρ′_{Q} is valued in O + ker θ̃ and in particular 𝕋_{Q} is not generated as an O-algebra by tr ρ′_{Q}. We will show this is not the case. If q ∈ Q and δ ∈ Δ_{q} then we can find σ ∈ G_{q} such that (U_{q}δ)^{2} −(tr ρ′_{Q}(σ))(U_{q}δ)+det ρ′_{Q}(σ) = 0. (σ will in fact lie above Frob_{q}.) This polynomial has distinct roots in k and so both its roots in 𝕋_{Q} lie in the sub-O-algebra T generated by the image of tr ρ′_{Q}. Thus for all δ ∈ Δ_{q}, U_{q}δ ∈ T. Hence U_{q} ∈ T, and as U_{q} is a unit, δ ∈ T. Moreover for all l ∉ Q for which \overline{ρ} is unramified we see that T_{l}χ_{Q}(Frob_{l})^{−1/2} ∈ T and hence T_{l} ∈ T. If p|N_{Q} then U_{p}χ_{Q}(Frob_{p})^{−1/2} is a root of the polynomial X^{2}−(tr ρ′_{Q}(σ))X +det ρ′_{Q}(σ) for any element σ of G_{p} which lies above Frob_{p}. For some σ over Frob_{p} this polynomial has two distinct roots in k and so U_{p} ∈ T. Thus T = 𝕋_{Q} as we required.

Finally we turn to the proof of the main theorem. As in [W2] (after equation (3.8)) we may find a set of primes Q_{m} with the following properties:

1. if q ∈ Q_{m} then q ≡ 1 mod p^{m};

2. if q ∈ Q_{m} then \overline{ρ} is unramified at q and \overline{ρ}(Frob_{q}) has distinct eigenvalues;

3. H_{∅^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)) ↪ ⨁_{q∈Q_{m}} H^{1}(𝔽_{q}, ad^{0}\overline{ρ}(1)).

As for each such q, H^{1}(𝔽_{q}, ad^{0}\overline{ρ}) = k we see that by shrinking Q_{m} we may suppose that the latter map is an isomorphism. Then we have that #Q_{m} = dim_{k} H_{∅^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)). Also H_{Q_{m}^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)) is the kernel of the map in 3. above and so is trivial. Thus by lemma 5 we see that dim_{k} H_{Q_{m}}^{1}(ℚ, ad^{0}\overline{ρ}) ≤ #Q_{m} and so 𝕋_{Q_{m}} can be generated by #Q_{m} = dim_{k} H_{∅^{∗}}^{1}(ℚ, ad^{0}\overline{ρ}(1)) elements. The main theorem now follows from corollary 1.

References

[C1] H.Carayol, Sur les représentations p-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Ec. Norm. Super. 19 (1986) 409-468.

[C2] H.Carayol, Formes modulaires et représentations Galoisiennes à valeurs dans un anneau local complet, in “p-adic monodromy and the Birch-Swinnerton-Dyer conjecture” (eds. B.Mazur and G.Stevens), Contemporary Math. 165 (1994).

[D] F.Diamond, The refined conjecture of Serre, to appear in the proceedings of the 1993 Hong Kong conference on modular forms and elliptic curves.

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

[FL] J.-M.Fontaine and G.Lafaille, Construction de représentations p-adiques, Ann. Sci. Ec. Norm. Super. 15 (1982) 547-608.

[G] B.Gross, A tameness criterion for Galois representations associated to modular forms modp, Duke Math. J. 61 (1990), 445-517.

[K1] E.Kunz, Almost complete intersections are not Gorenstein, J. of Algebra 28 (1974), 111-115.

[K2] E.Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser 1985.

[L] H.Lenstra, Complete intersections and Gorenstein rings, to appear in the proceedings of the 1993 Hong Kong conference on modular forms and elliptic curves.

[M] B.Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 133-186.

[dS] E. de Shalit, On certain Galois representations related to the modular curve X_{1}(p), to appear in Compositio Math.

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

[W1] A.Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math. 94 (1988), 529-573.

[W2] A.Wiles, Modular elliptic curves and Fermat’s last theorem, preprint (1994).

Appendix

The purpose of this appendix is to explain certain simplifications to the arguments of chapter 3 of [W2] and to section 3 of this paper. These simplifications were found by G.Faltings and we would like to thank him for allowing us to include them here. We should make it clear that the arguments of this appendix (just as those of chapter 3 of [W2] and section 3 of this paper) apply only to proving conjecture 2.16 of [W2] for the minimal Hecke ring and minimal deformation problem. In order to prove theorem 3.3 of [W2] one needs to invoke theorem 2.17 and the arguments of chapter 2 of [W2].

We will keep the notation and assumptions of the main body of this paper. Let Q denote a finite set of primes as described in section 1 of this paper. By a deformation of \overline{ρ} of type Q we shall mean a complete noetherian local Oalgebra A with residue field k together with an equivalence class of continuous representations ρ : G_{ℚ} → GL_{2}(A) with the following properties:

• ρ mod 𝔪_{A} = \overline{ρ};

• ϵ^{−1} det ρ is a character of finite order prime to p;

• if l ∉ Q ∪ \{p\} and \overline{ρ}|_{I_{l}} is semi-simple then ρ(I_{l}) →^{∼} \overline{ρ}(I_{l});

• if l ∉ Q ∪ \{p\} and \overline{ρ}|_{I_{l}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) then ρ|_{I_{l}} ∼ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} );

• if \overline{ρ} is flat and det \overline{ρ}|_{I_{p}} = ϵ then ρ is flat;

• if either \overline{ρ} is not flat or if det \overline{ρ}|_{I_{p}} ≠ ϵ then ρ|_{G_{p}} ∼ ( \begin{matrix}ϕ_{1} & ∗ \\ 0 & ϕ_{2} \end{matrix} ) where ϕ_{2}

is unramified and ϕ_{2} mod 𝔪_{A} = ψ_{2}.

As in chapter 1 of [W2] there is a universal lift ρ_{Q}^{univ} : G_{ℚ} → GL_{2}(R_{Q}) of type Q. Recall that the universal property is for lifts up to conjugation. Moreover one checks (c.f. the second paragraph of the proof of lemma 6) that there is a natural isomorphism

Hom_{k}(𝔪_{R_{Q}}/(λ, 𝔪_{R_{Q}}^{2}), k) ≅ H_{Q}^{1}(ℚ, ad^{0}\overline{ρ}).

There is also a natural map R_{Q} → 𝕋_{Q} so that ρ_{Q}^{univ} pushes forward to a conjugate of ρ′_{Q}.

Recall that if Q = ∅ we shall often drop it from the notation. In this appendix we shall reprove the following result.

Theorem 3 R →^{∼} 𝕋 and these rings are complete intersections.

We note that if O′ is the ring of integers of a finite extension K′/K then the rings 𝕋′_{Q} and R′_{Q} constructed using O′ in place of O are just 𝕋_{Q} ⊗_{O} O′ and R_{Q} ⊗_{O} O′. Also 𝕋_{Q} is a complete intersection if and only if 𝕋_{Q} ⊗_{O} O′ is (using for instance corollary 2.8 on page 209 of [K2]). Thus we may and we shall assume that O is sufficiently large that the eigenvalues of every element of \overline{ρ}(G_{ℚ}) are rational over k.

We recall that in the penultimate paragraph of section 4 of this paper we showed that 𝕋_{Q} is generated as an O-algebra by tr ρ′_{Q}(G_{ℚ}). Thus we see that the map R_{Q} → 𝕋_{Q} is a surjection.

We will need the following result.

Lemma 7 If q ∈ Q then ρ_{Q}^{univ}|_{G_{q}} ∼ ( \begin{matrix}ϕ_{1} & 0 \\ 0 & ϕ_{2} \end{matrix} ) where ϕ_{1}|_{I_{q}} = ϕ_{2}|_{I_{q}}^{−1} and both

these characters factor through χ_{q} : I_{q} → → Δ_{q}.

It suffices to check the first assertion. As \overline{ρ} is unramified at q, ρ_{Q}^{univ}|_{G_{q}} factors through ℤ̂ ⋉ ℤ_{p}(1), where ℤ̂ is topologically generated by some lift f of Frob_{q}, ℤ_{p}(1) is topologically generated by some element σ and where fσf^{−1} = σ^{q}. As \overline{ρ}(Frob_{q}) has distinct eigenvalues it is easy to see that after

conjugation we may assume that ρ_{Q}^{univ}(f) = ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) where a ≢ b mod 𝔪_{R_{Q}}.

We will show that ρ_{Q}^{univ}(σ) is a diagonal matrix with entries congruent to 1 mod 𝔪_{R_{Q}}. We will in fact prove this mod 𝔪_{R_{Q}}^{n} for all n by induction on n. For n = 1 there is nothing to prove. So suppose this is true modulo 𝔪_{R_{Q}}^{n} with n > 0. Then

ρ_{Q}^{univ}(σ) ≡ ( \begin{matrix}μ_{1} & 0 \\ 0 & μ_{2} \end{matrix} ) (1_{2} + N) mod 𝔪_{R_{Q}}^{n+1},

where each μ_{i} ≡ 1 mod 𝔪_{R_{Q}} and where N ≡ 0 mod 𝔪_{R_{Q}}^{n}. We see that mod 𝔪_{R_{Q}}^{n+1} we have

1 + ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) N ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )^{−1}

≡ ( \begin{matrix}μ_{1}^{q−1} & 0 \\ 0 & μ_{2}^{q−1} \end{matrix} ) (1 + N)^{q}
≡ ( \begin{matrix}μ_{1}^{q−1} & 0 \\ 0 & μ_{2}^{q−1} \end{matrix} ) (1 + qN)
≡ ( \begin{matrix}μ_{1}^{q−1} & 0 \\ 0 & μ_{2}^{q−1} \end{matrix} ) + N,

and as a ≢ b mod 𝔪_{R_{Q}} we deduce that N is diagonal mod𝔪_{R_{Q}}^{n+1} as required.

We can choose ϕ_{2} so that ϕ_{2}(f) ≡ β_{q} mod 𝔪_{R_{Q}}. Then we can define a map Δ_{q} → R_{Q}^{×} to be ϕ_{2}|_{I_{q}}^{2}. This makes R_{Q} into an O[Δ_{Q}]-algebra. Using the last lemma and the universal properties of R_{Q} and of R it is easy to see that R_{Q}/𝔞_{Q} →^{∼} R. It moreover follows from the discussion preceding theorem 1 of this paper that the map R_{Q} → → T_{Q} is a map of O[Δ_{Q}]-algebras.

The key observation is the following ring theoretic proposition. Theorem 3 follows on applying it to the rings R_{Q_{n}} and 𝕋_{Q_{n}} for the sets Q_{n} constructed in section 4 of this paper. Note that there is a map O[[S_{1}, ..., S_{r}]] → → O[Δ_{Q_{n}}] with kernel the ideal ((1 + S_{1})^{#Δ_{q_{1}}} − 1, ..., (1 + S_{r})^{#Δ_{qr}} − 1), where Q_{n} = {q_{1}, ..., q_{r}}. Note also that by the displayed isomorphism a couple of lines before theorem 3 there exists a surjection of O-algebras O[[X_{1}, ..., X_{r}]] → → R_{Q_{n}}.

Proposition 3 Suppose r is a non-negative integer and that we have a map of O-algebras R → → T with T finite and flat over O. Suppose for each positive integer n we have a map of O-algebras R_{n} → → T_{n} and a commutative diagram of O-algebras

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

where

1. there is a surjection of O-algebras O[[X_{1}, ..., X_{r}]] → → R_{n},

2. (S_{1}, ..., S_{r})R_{n} ⊂ ker (R_{n} → → R),

3. (S_{1}, ..., S_{r})T_{n} = ker (T_{n} → → T),

4. if 𝔟_{n} denotes the kernel of O[[S_{1}, ..., S_{r}]] → T_{n} then 𝔟_{n} ⊂ ((1 + S_{1})^{p^{n}} − 1, ..., (1 + S_{r})^{p^{n}} − 1) and T_{n} is a finite free O[[S_{1}, ..., S_{r}]]/𝔟_{n}-module.

Then R →^{∼} T and these rings are complete intersections.

Reducing mod λ we see that it suffices to prove this result with k replacing O everywhere. In this case we see that the last condition becomes 𝔟_{n} ⊂ (S_{1}^{p^{n}}, ..., S_{r}^{p^{n}}). Further we may replace R by its reduction modulo 𝔪_{R}ker (R → → T), and so we may assume that R is finite over k. We may replace T_{n} by T_{n}/(S_{1}^{p^{n}}, ..., S_{r}^{p^{n}}) and so assume that 𝔟_{n} = (S_{1}^{p^{n}}, ..., S_{r}^{p^{n}}). Finally we may replace R_{n} by its image in R ⊕ T_{n}.

Now define an n-structure to be a pair of k-algebras B → → A together with a commutative diagram of k-algebras

k[[S_{1}, ..., S_{r}]]
↓
k[[X_{1}, ..., X_{r}]] → → B → → R
↓  ↓
A  → → T,

such that

1. B ↪ R ⊕ A,

2. (S_{1}, ..., S_{r})B ⊂ ker (B → → R),

3. (S_{1}, ..., S_{r})A = ker (A → → T),

4. A is a finite free k[[S_{1}, ..., S_{r}]]/(S_{1}^{p^{n}}, ..., S_{r}^{p^{n}})-module.

Note that #B ≤ (#T)^{p^{nr}}#R and so we see that there are only finitely many isomorphism classes of n-structures. If S is an n-structure and if m ≤ n then we may obtain an m-structure S^{(m)} by replacing A by A/(S_{1}^{p^{m}}, ..., S_{r}^{p^{m}}) and B by its image in R ⊕ (A/(S_{1}^{p^{m}}, ..., S_{r}^{p^{m}})).

As explained above, it follows from the hypotheses of the proposition that an n-structure S_{n} exists for each n. We next claim that we can find, for each n, n-structures S′_{n} such that for m ≤ n we have S′_{m} ≅ (S′_{n})^{(m)}. To prove this observe that we can find recursively integers n(m) with the following properties

• S_{n(m)}^{(m)} ≅ S_{n}^{(m)} for infinitely many n

• and for m > 1, S_{n(m)}^{(m−1)} ≅ S_{n(m−1)}^{(m−1)}.

Then set S′_{m} = S_{n(m)}^{(m)}.

Thus we obtain a commutative diagram

... R′_{n} ... R′_{2} → → R′_{1} → → R
↓ ↓ ↓ ↓
... T′_{n} ... T′_{2} → → T′_{1} → → T

of k[[X_{1}, ..., X_{r}, S_{1}, ..., S_{r}]]-algebras. Moreover we have that

• k[[X_{1}, ..., X_{r}]] → → R′_{n} → → T′_{n},

• T′_{n} is a finite free k[[S_{1}, ..., S_{r}]]/(S_{1}^{p^{n}}, ..., S_{r}^{p^{n}})-module,

• R′_{n}/(S_{1}, ..., S_{r}) → → R and T′_{n}/(S_{1}, ..., S_{r}) →^{∼} T.

Let R′_{∞} denote the quotient of k[[X_{1}, ..., X_{r}]] by the intersection of the ideals ker (k[[X_{1}, ..., X_{r}]] → → R′_{n}) and let T′_{∞} denote the quotient of k[[X_{1}, ..., X_{r}]] by the intersection of the ideals ker (k[[X_{1}, ..., X_{r}]] → → T′_{n}). Then we have a commutative diagram

k[[S_{1}, ..., S_{r}]]
↓
k[[X_{1}, ..., X_{r}]] → → R′_{∞} → → R
↓  ↓
T′_{∞}  → → T,

such that

• R′_{∞} → → T′_{∞},

• T′_{∞} is a finite free k[[S_{1}, ..., S_{r}]]-module,

• R′_{∞}/(S_{1}, ..., S_{r}) → → R and T′_{∞}/(S_{1}, ..., S_{r}) →^{∼} T.

We deduce that T′_{∞} has Krull dimension r and hence we deduce that the map k[[X_{1}, ..., X_{r}]] → → T′_{∞} has trivial kernel. That is we have isomorphisms k[[X_{1}, ..., X_{r}]] →^{∼} R′_{∞} →^{∼} T′_{∞}. Thus R →^{∼} T. As T has Krull dimension 0 and T ≅ k[[X_{1}, ..., X_{r}]]/(S_{1}, ..., S_{r}) we see that T is a complete intersection, and the proposition is proved.