Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur

Ravi Ramakrishna Cornell University

January 29, 2000

\underline{Introduction} Let p be an odd prime, k a finite field of characteristic p, and W (k) the ring of Witt vectors of k. The study of Galois representations ρ̄ : G_{Q} = Gal(Q̄/Q) → GL_{2}(k) and their deformations to characteristic zero p-adic rings such as W (k) has progressed rapidly in recent years. In much of this work ρ̄ has been assumed to be modular, that is ρ̄ is the reduction of a p-adic Galois representation associated to a classical Hecke eigenform.

The assumption of modularity allows one to bring to bear the tools of algebraic geometry (e.g. Jacobians of modular curves, the theory of congruences). The work of [Ri], [W], [TW], [Di2], [CDT], and [BCDT] uses these tools to great success. The seminal work of [W] and [TW] is of the following flavor. Let ρ̄ : G_{Q} = Gal(Q̄/Q) → GL_{2}(k) be a modular mod p Galois representation. Then all deformations of ρ̄ to p-adic rings unramified outside a finite set of primes R whose restrictions to the decomposition groups G_{v} = Gal(Q̄_{v}/Q_{v}) for v ∈ R satisfy certain local conditions are themselves modular. Thus ρ factors through G_{R} = Gal(Q_{R}/Q) where R contains p and infinity and Q_{R} is the maximal extension of Q unramified outside R. The assumed modularity of ρ̄ implies that ρ̄ is odd, that is det(ρ̄(c)) = −1 where c is complex conjugation. Serre conjectured that all odd absolutely irreducible mod p two dimensional Galois representations are modular of prescribed weight, level and character.

Fontaine and Mazur have conjectured that all p-adic Galois representations unramified outside a finite set R satisfying the local condition of potential semistability (a generalization of the local conditions alluded to above) that are not Tate twists of even representations with finite image arise in the étale cohomology of some smooth variety. See [Fo] for the definition of potentially semistable, semistable, crystalline etc. and [FM] for the precise statement of their conjecture. A consequence of their conjecture (combined with other well known conjectures) is that all two dimensional p-adic Galois representations satisfying these local properties are modular. (It is known that representations coming from modular forms satisfy their local conditions.) This can be interpreted as a characteristic zero version of Serre’s conjecture. We refer to this two dimensional statement as the Fontaine-Mazur conjecture in this paper. It is interesting to note that there are no parity assumptions in this version of their conjecture. Indeed, part of the conjecture is that even two dimensional p-adic Galois representations cannot satisfy their local conditions unless they are Tate twists of representations with finite image.

Our approach is to start with a mod p Galois representation ρ̄ that is not assumed to be modular. Let Ad^{0}ρ̄ be the set of all trace zero two by two matrices over k with Galois action through ρ̄ and by conjugation. We may assume, after twisting, that ρ̄ and Ad^{0}ρ̄ are ramified at exactly the same primes. Denote by S the union of this set of primes with {p, ∞}. In [R2] it is proved, that subject to certain conditions, ρ̄ can be deformed to W (k) by allowing ramification at a finite additional set of primes. There we could say nothing about the local properties of this deformation. Drawing on the ideas there and some new ones we generalize the results of [R2].

Theorem 1 Let k be a finite field of characteristic p ≥ 7 and ρ̄ : G_{Q} → GL_{2}(k). Let χ be the cyclotomic character.

a) If ρ̄ is even and ρ̄ |_{G_{p}} is not twist equivalent to ( \begin{matrix}χ & 0 \\ 0 & 1 \end{matrix} ) or twist equivalent to the indecomposable

representation ( \begin{matrix}χ^{p−2} & ∗ \\ 0 & 1 \end{matrix} ) then there is a finite set R of primes containing p and infinity and a

deformation ρ : G_{R} → GL_{2}(W (k)) of ρ̄.

b) If ρ̄ is odd then there is a finite set R of primes containing p and infinity and a deformation ρ : G_{R} → GL_{2}(W (k)) of ρ̄. Suppose further that ρ̄ |_{G_{p}} is not twist equivalent to the trivial

representation or the indecomposable unramified representation given by ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). If ρ̄ |_{G_{p}} is a

twist of an ordinary representation then, up to a twist by a character of finite order, ρ can be taken to be ordinary. If inertia at p acts via fundamental characters of level 2 then, up to a twist by a character of finite order, ρ can be taken to be potentially crystalline.

We are providing Galois theoretic evidence for Serre’s conjecture. While (we hope) our results are of interest to arithmetic geometers our methods involve only Galois cohomology. Thus we cannot expect to use these techniques to actually prove cases of the conjectures of Serre and Fontaine-Mazur.

We now give some consequences of Theorem 1. C. Khare uses Theorem 1 above in [Kh2] to prove the compatibility of Serre’s conjecture with base change in certain cases.

Corollary 1 Suppose ρ̄ is odd, absolutely irreducible and is not an excluded case in Theorem 1 part b). If the Fontaine-Mazur Conjecture is true then Serre’s conjecture is true for ρ̄.

Proof: The Fontaine-Mazur conjecture implies the ρ of Theorem 1, part b) is modular. Thus ρ̄ is modular of some weight, level and character. That ρ̄ is modular of the optimal parameters follows from work of Ribet, Diamond etal. See Section 1 of [Di2]. We remark that the excluded cases of Theorem 1 part b) comprise a subset of those that, in the sense of [Se1], have weight p or have weight congruent to 1 mod p + 1.

Suppose for the moment ρ̄ is not an exceptional case of Theorem 1 part b) and that ρ̄ is modular with associated eigenform f of minimal weight and level and prescribed character. Darmon has pointed out that the above corollary implies there is an eigenform g congruent to f with eigenvalues in W (k). A priori this does not seem obvious.

Corollary 2 a) Let ρ̄ : G_{Q} → SL_{2}(k) be surjective. There is a finite set of primes R and a deformation ρ : G_{R} → SL_{2}(W (k)) of ρ̄ unramified outside the finite set R.

b) There exist infinitely many even surjective representations ρ : G_{Q} → SL_{2}(Z_{7}). Each such representation is unramified outside a finite set of primes.

Proof: For part a), since ρ̄ is even by hypotheses, we must check that ρ̄ |_{G_{p}} is not an excluded case described in Theorem 1, part a). If this were the case we know ρ̄ |_{G_{p}} is a twist by a character ϕ of G_{p} of a representation in one of the two forms Theorem 1, part a). Taking determinants of the restriction to G_{p}, and keeping in mind detρ̄ = 1, we see χ = ϕ^{−2} or χ = ϕ^{2}. In either case the mod p character χ must be the square of another character. It is an easy exercise in local class field theory to see this is not the case. Theorem 1 part a) applies so ρ̄ deforms to W (k). If the determinant of the deformation ρ is not 1 we may twist by a character to guarantee it will be 1. (Recall p is odd). For part b) we recall that Mestre has shown in [Me] that SL_{2}(F_{7}) is a regular extension of Q(T). Specialization gives us an infinite supply of SL_{2}(F_{7}) extensions of Q. By part a) these deform to SL_{2}(Z_{7}). The surjectivity follows from Chapter IV, Lemma 3 of [Se3].

We remark that the ρ’s of Theorem 1 are unique. That is, when we insist that, up to a twist,

ρ |_{G_{q}}= ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ) for q ∈ R − S there will be only one such ρ (subject to local conditions at primes

in S described later in this paper). Note that ρ |_{G_{q}} could possibly be decomposable.

Finally, we would like to acknowledge that our use of the auxiliary sets of primes contained in R are chosen to essentially annihilate a dual Selmer group as in [W] and [TW]. Many of our computations mimic ideas of [W], [TW], [Di1], [CDT], [Bö1] and others. We are also grateful to Gebhard Böckle for explaining to us the details of the excluded cases in Theorem 1 part a) and we thank Chandrashekhar Khare for his many helpful suggestions.

\underline{Notation} p: A prime ≥ 5. k: A finite field of characteristic p ≥ 5. k̃: A subfield of k which is the minimal field of definition of Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗}. k(ϕ): The one dimensional space k with Galois action by the character ϕ. W (k): The ring of Witt vectors of k. S: The union of {p}, {∞}, and the finite set of places at which Ad^{0}ρ̄ (or ρ̄) is ramified. Q: A finite set of primes disjoint from S, whose elements lie in a certain Chebotarev classes. Q_{v}: The completion of Q at the place v. Q_{S}: The maximal separable extension of Q unramified outside the places of S. G_{Q}: Gal(Q̄/Q). G_{S}: Gal(Q_{S}/Q). G_{v}: Gal(Q̄_{v}/Q_{v}). I_{v}: The inertia subgroup of G_{v}. 1: The two by two identity matrix over W (k) or W (k)/p^{m} for suitable m. χ: The cyclotomic character, either in characteristic zero or mod p^{m}. ρ̄: A continuous representation from G_{S} or G_{v} to GL_{2}(k). N_{v}: A subspace of H^{1}(G_{v}, Ad^{0}ρ̄). C_{v}: Deformations of ρ̄ |_{G_{v}} to W (k) whose mod p^{n} reductions are stable under the action N_{v}. Ad^{0}ρ̄: The trace zero two by two matrices over k with G_{S} action through ρ̄ and by conjugation. (Ad^{0}ρ̄)^{∗}: The Cartier dual of Ad^{0}ρ̄. Ãd^{0}ρ̄: A descent of Ad^{0}ρ̄ to its minimal field of definition k̃. (Ãd^{0}ρ̄)^{∗}: A descent of (Ad^{0}ρ̄)^{∗} to its minimal field of definition k̃.

Throughout this paper we use Fact to denote a previously known result.

\underline{Deformation Theory} In this section we recall some less well known facts about deformation theory. We refer the reader to [M1] or [M2] for standard details of the theory. We only remark here that we will always fix the determinant of the deformations we consider. This corresponds to the fact that we must study the cohomology of Ad^{0}ρ̄ as opposed to that of all two by two matrices over k.

We suppose ρ̄ : G → GL_{2}(k) is a representation of a profinite group G whose image is in the upper triangular subgroup of GL_{2}(k). Let U^{0} and U^{1} respectively be the subgroups of Ad^{0}ρ̄

consisting of matrices of the form ( \begin{matrix}0 & x \\ 0 & 0 \end{matrix} ) and ( \begin{matrix}x & y \\ 0 & −x \end{matrix} ). Note U^{0} and U^{1} are stable under

the action of G_{p}.

Lemma 1 Suppose for ρ̄ as above H^{2}(G_{p}, U^{0}) = 0. Let ϕ_{1} and ϕ_{2} be the diagonal characters of ρ̄ with Teichmüller lifts ϕ̃_{1} and ϕ̃_{2}. Denote by ϕ_{1,n} and ϕ_{2,n} the mod p^{n} reductions of ϕ̃_{1} and ϕ̃_{2}. For i = 1, 2 let γ_{i} : G_{p} → (W (k))^{∗} be any characters congruent to 1 mod p with mod p^{n} reductions γ_{i,n} such that ϕ̃_{1}ϕ̃_{2} = ϕ̃_{1}ϕ̃_{2}γ_{1}γ_{2}. (The last hypothesis is to fix the determinant). If

ρ_{n} : G → GL_{2}(W (k)/p^{n}) is given by ( \begin{matrix}ϕ_{1,n}γ_{1,n} & ∗ \\ 0 & ϕ_{2,n}γ_{2,n} \end{matrix} ) with mod p reduction ρ̄, then there is a

deformation ρ_{n+1} : G → GL_{2}(W (k)/p^{n+1}) with diagonal characters ϕ_{1,n+1}γ_{1,n+1} and ϕ_{2,n+1}γ_{2,n+1}. In other words, the reducible deformation theory of ρ̄ with fixed diagonal characters is unobstructed.

Proof: See Theorems 6.1 and 6.2 of [R1] for related computations. One easily sees the obstruction to lifting lies in H^{2}(G, U^{0}) which is assumed to be trivial.

Lemma 2 Suppose H^{2}(G_{p}, U^{1}) = 0. If ρ_{n} : G → GL_{2}(W (k)/p^{n}) is an upper triangular representation whose mod p reduction is ρ̄ then there is an upper triangular ρ_{n+1} : G → GL_{2}(W (k)/p^{n+1}) whose mod p^{n} reduction is ρ_{n}. In other words the reducible deformation theory of ρ̄ is unobstructed.

Proof: The obstruction to lifting lies in H^{2}(G, U^{1}).

\underline{Facts from Galois cohomology} We recall, in our context, a few well known facts from Galois cohomology that we will need. We refer the reader to [H], [Mi] and [Se2] for details and proofs. Here M is finite p torsion (local or global) Galois module. Let G_{V} = Gal(Q_{V}/Q) where V is a (possibly infinite) set of primes including p and infinity and Q_{V} is the maximal extension of Q unramified outside V. Also, G_{v} = Gal(Q̄_{v}/Q_{v}). Finally, M^{∗} = Hom(M, μ_{p}) is the Cartier dual of M.

Define P_{V}^{1} (M) to be the restricted direct product □_{v∈V} H^{1}(G_{v}, M) with respect to the images of the inflation maps H^{1}(G_{v}/I_{v}, M) → H^{1}(G_{v}, M) for those v ∈ V at which M is unramified. (So M^{I_{v}} = M for such v.) We define P_{V}^{2} (M) to be the direct sum ⊕_{v∈V} H^{2}(G_{v}, M). For i = 1, 2 we denote the kernel of the map H^{i}(G_{V} , M) → P_{V}^{i} (M) by � _{V}^{i} (M).

Fact 1 (Local Duality) For i = 0, 1, 2 there is a perfect pairing H^{i}(G_{v}, M) × H^{2−i}(G_{v}, M^{∗}) → H^{2}(G_{v}, μ_{p}) ≃ Z/p.

Fact 2 (Local Euler-Poincaré characteristic) Recall #M is a power of p.

1) For v finite, not equal to p we have #H^{0}(G_{v}, M) · #H^{2}(G_{v}, M) = #H^{1}(G_{v}, M).

2) For v = p we have #H^{0}(G_{p}, M) · #H^{2}(G_{p}, M) · #M = #H^{1}(G_{p}, M).

Fact 3 (Global duality) 1) The groups � _{V}^{1} (M^{∗}) and � _{V}^{2} (M) are dual to one another.

2) The images of the restriction maps H^{1}(G_{V} , M) → P_{V}^{1} (M) and H^{1}(G_{V} , M^{∗}) → P_{V}^{1} (M^{∗}) are exact annihilators of one another.

Fact 4 (Global Euler-Poincaré characteristic) For V finite we have

\frac{#H^{0}(G_{V} , M) · #H^{2}(G_{V} , M)}{#H^{1}(G_{V} , M)} = \frac{#H^{0}(G_{∞}, M)}{#M}.

Fact 5 Let Ẑ be the completion of Z. Then if M is a Ẑ module we have #H^{0}(Ẑ, M) = #H^{1}(Ẑ, M).

Lemma 3 Let M be a finite dimensional k vector space with G_{p} action. Then

dim_{k}H^{1}(G_{p}/I_{p}, M^{I_{p}}) = dim_{k}H^{0}(G_{p}, M).

Proof: Since G_{p}/I_{p} ≃ Ẑ we see dim_{k}H^{1}(G_{p}/I_{p}, M^{I_{p}}) = dim_{k}H^{0}(G_{p}/I_{p}, M^{I_{p}}) = dim_{k}H^{0}(G_{p}, M).

\underline{Local at l ≠ p considerations} In this section we analyze, for l ≠ p, local representations ρ̄ : G_{l} → GL_{2}(k). We recall Diamond’s classification of two dimensional mod p representations of G_{l}. Diamond has four cases that he calls P, S, V and H. We recall these below under the hypotheses that Ad^{0}ρ̄ is ramified. We do not give all of Diamond’s equivalent formulations of each case and note that we interchange his roles of p and l. Let I_{l} be inerita at l and let Ĩ_{l} and G̃_{l} denote the images of I_{l} and G_{l} in the projective representation associated to ρ̄. Note Ĩ_{l} and G̃_{l} act faithfully on Ad^{0}ρ̄. Here we assume p ≥ 5.

• P : ρ̄ is twist equivalent to ( \begin{matrix}ϕ & 0 \\ 0 & 1 \end{matrix} ) for some ramified character ϕ of G_{l}.

• S : a) l ≡ 1 mod p and ρ̄ is twist equivalent to ( \begin{matrix}1 & ϕ \\ 0 & 1 \end{matrix} ) where ϕ is a ramified additive

character of G_{l} or

b) l ≢ 1 mod p and ρ̄ is twist equivalent to the indecomposable representation ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} )

where χ is the mod p cyclotomic character.

• V : ρ̄ is twist equivalent to Ind_{G_{E}}^{G_{l}}ξ where E is the unramified quadratic extension of Q_{l} and ξ is a character of G_{E} = Gal(Ē/E) that is not equal to its conjugate under the action of Gal(E/Q_{l}).

• H : a) l is odd and ρ̄ is twist equivalent to Ind_{G_{E}}^{G_{l}}ξ where E is a ramified quadratic extension of Q_{l} and ξ is a character of G_{E} = Gal(Ē/E) whose restriction to I_{E} is not equal to its conjugate under the action of Gal(E/Q_{l}).

b) l = 2, Ĩ_{l} (respectively G̃_{l}) is isomorphic to D_{4} (respectively A_{4}), A_{4} (respectively A_{4}), or A_{4} (respectively S_{4}).

We remark that here and throughout the rest of this paper we feel free to twist our mod p Galois representations, both local and global, by characters of finite order. Twisting does not affect the cohomology of Ad^{0}ρ̄ and hence does not affect the deformation theory. Because ultimately we are interested in deformations to W (k) and we can always “untwist” these deformations by the Teichmüller lift of these characters of finite order.

Recall the facts below from [R2]. The precise references in [R2] are included parenthetically.

Fact 6 H^{2}(G_{v}, Ad^{0}ρ̄) ≠ 0 if and only if Ad^{0}ρ̄ has a one dimensional quotient by a two dimensional G_{v} stable subspace on which G_{v} acts by χ, the cyclotomic character. This results holds for all finite v including v = p. ( Lemma 3 of [R2].)

Fact 7 If l ≢ ±1 mod p then H^{2}(G_{l}, Ad^{0}ρ̄) = 0. (Proposition 2 of [R2].)

Fact 8 Suppose ρ̄ : G → GL_{2}(k) where G is any profinite group. Let ρ_{n} : G → GL_{2}(W (k)/p^{n}) be a deformation of ρ̄ to W (k)/p^{n}. Suppose ρ_{n+1,1} and ρ_{n+1,2} are two deformations of ρ_{n} to W (k)/p^{n+1}. Then there is an element g ∈ H^{1}(G, Ad^{0}ρ̄) such that (1 + p^{n}g).ρ_{n+1,1} = ρ_{n+1,2}. (The remarks following the Fact in the Deformation Theory section of [R2].)

Lemma 4 Let E/Q_{l} be a quadratic extension. Suppose ρ̄ is twist equivalent to Ind_{G_{E}}^{G_{l}}ξ where ξ is a character of G_{E} that is not equal to its conjugate under the action of Gal(E/Q_{l}). Then Ad^{0}ρ̄ ≃ A_{1} ⊕ A_{2} where A_{i} is an absolutely irreducible k[G_{l}] module with dim_{k}A_{i} = i and H^{0}(G_{l}, Ad^{0}ρ̄) = 0. Furthermore H^{2}(G_{l}, Ad^{0}ρ̄) = 0 unless E/Q_{l} is unramified and l ≡ −1 mod p in which case H^{2}(G_{l}, Ad^{0}ρ̄) is one dimensional.

Proof: We see, that up to a twist, ρ̄ |_{G_{E}}= ( \begin{matrix}ξ & 0 \\ 0 & ξ^{σ} \end{matrix} ) where σ generates Gal(E/Q_{l}). Then as a

k[G_{E}] module we see Ad^{0}ρ̄ ≃ k ⊕ k(ξ/ξ^{σ}) ⊕ k(ξ^{σ}/ξ). Observe Gal(E/Q_{l}) acts nontrivially on k and permutes k(ξ/ξ^{σ}) and k(ξ^{σ}/ξ). We take A_{1} = k and A_{2} = k(ξ/ξ^{σ}) ⊕ k(ξ^{σ}/ξ) and the absolute irreducibility results are clear. That H^{0}(G_{l}, Ad^{0}ρ̄) = 0 is also clear. As for the H^{2} result, by Fact 6 we have H^{2}(G_{l}, Ad^{0}ρ̄) ≠ 0 if and only if Ad^{0}ρ̄ has a one dimensional quotient on which G_{l} acts via χ. But A_{1} is the only one dimensional quotient of Ad^{0}ρ̄ and G_{l} acts via Gal(E/Q_{l}). This is the action of χ precisely when E is unramified over Q_{l} and l ≡ −1 mod p. In this case, since A_{1} is the only one dimensional quotient of Ad^{0}ρ̄ we have H^{2}(G_{l}, Ad^{0}ρ̄) is one dimensional.

Proposition 1 There is a class C_{l} of deformations of ρ̄ to W (k) and a subspace N_{l} of H^{1}(G_{l}, Ad^{0}ρ̄) of codimension dim_{k}H^{2}(G_{l}, Ad^{0}ρ̄) such that if we fix any h ∈ H^{1}(G_{l}, Ad^{0}ρ̄), h ∉ N_{l} then ρ̄ can be successively deformed to an element of C_{l} by deforming from W (k)/p^{n} to W (k)/p^{n+1} with adjustments at each step made only by a multiple of h.

Proof: If H^{2}(G_{l}, Ad^{0}ρ̄) = 0 then all deformation problems are unobstructed and we take N_{l} = H^{1}(G_{l}, Ad^{0}ρ̄) and C_{l} to be all deformations of ρ̄ to W (k). We do this because after deforming from mod p^{n} to mod p^{n+1} there is no need to adjust the deformation by an element of H^{1}(G_{l}, Ad^{0}ρ̄) so that we can deform to W (k)/p^{n+2}. Henceforth we need only consider cases where H^{2}(G_{l}, Ad^{0}ρ̄) ≠ 0. Thus by Fact 7 we may assume l ≡ ±1 mod p.

We treat Diamond’s four cases separately. In each case we follow the outline below. First we must exhibit our class C_{l} of deformations of ρ̄ to W (k). If ρ_{n} is the mod p^{n} reduction of an element in this class we will abuse notation and write ρ_{n} ∈ C_{l}. Then we will find a one dimensional space of cohomology classes spanned by g ∈ H^{1}(G_{l}, Ad^{0}ρ̄) such that for all a ∈ k we have (1 + ap^{n−1}g).ρ_{n} ∈ C_{l}. Thus altering the mod p^{n} reduction of an element in C_{l} by a multiple of g gives the mod p^{n} reduction of another element in C_{l}. We say that N_{l} preserves C_{l}. We let N_{l} be the space spanned by g. (N_{l} will always be trivial or one dimensional in this section.) If N_{l} is codimension one in H^{1}(G_{l}, Ad^{0}ρ̄) (as it will be) we may choose h to be any element in H^{1}(G_{l}, Ad^{0}ρ̄) not in N_{l}. Clearly ρ_{n} has a deformation ρ_{n+1} to W (k)/p^{n+1} that is in C_{l}. By Fact 8 any deformation τ_{n+1} of ρ_{n} to W (k)/p^{n+1} differs from ρ_{n+1} by an element of H^{1}(G_{l}, Ad^{0}ρ̄).

Since h and g span H^{1}(G_{l}, Ad^{0}ρ̄) we see τ_{n+1} = (1 + (ah + bg)p^{n}).ρ_{n+1} for some a, b ∈ k. Thus (1 − ah).τ_{n+1} = (1 + bgp^{n}).ρ_{n+1} ∈ C_{l}. The crucial point in the argument is that N_{l} is codimension one.

We apply the outline to the four cases.

In case P, we see Ad^{0}ρ̄ ≃ k(ϕ) ⊕ k ⊕ k(ϕ^{−1}), where ϕ is ramified. Since l ≠ p we see, since the cyclotomic character χ is unramified, that Ad^{0}ρ̄ has a quotient as in Fact 6 if and only if l ≡ 1 mod p. In this case using local duality we have dim_{k}H^{2}(G_{l}, Ad^{0}ρ̄) = 1. It is easy to see dim_{k}H^{0}(G_{l}, Ad^{0}ρ̄) = 1. Using the local Euler-Poincaré characteristic (Fact 2) we have dim_{k}H^{1}(G_{l}, Ad^{0}ρ̄) = 2. Let ϕ̃ be the Teichmüller lift of ϕ. Let C_{l} be the set of deformations ρ of ρ̄

to W (k) that are given by ( \begin{matrix}ϕ̃γ & 0 \\ 0 & γ^{−1} \end{matrix} ) where γ : G_{p} → W (k)^{∗} is any unramified character congru-

ent to 1 mod p. (Recall we are fixing the determinant of all deformations we consider.) For σ ∈ G_{l} let n_{σ} be its image under the composition G_{l} → G_{l}/I_{l} ≃ Ẑ → Z/p. Consider g ∈ H^{1}(G_{l}, Ad^{0}ρ̄)

given by g(σ) = ( \begin{matrix}n_{σ} & 0 \\ 0 & −n_{σ} \end{matrix} ). For ρ ∈ C_{l} we study ρ_{n}. For a ∈ k we see that

(1 + ap^{n−1}g).ρ_{n}(σ) = ( \begin{matrix}ϕ̃(σ)γ(σ)(1 + an_{σ}p^{n−1}) & 0 \\ 0 & γ^{−1}(σ)(1 − an_{σ}p^{n−1}) \end{matrix} ) .

One easily sees this is the reduction mod p^{n} of an element of C_{l} with the character γ′ chosen to be any unramified lift to W (k)^{∗} of the unramified mod p^{n} character γ(1 + ap^{n−1}g). (It is easy to see such lifts exist.) Thus in case P we choose N_{l} to be the space spanned by g. If we choose h ∈ H^{1}(G_{l}, Ad^{0}ρ̄) so that h ∉ N_{l} the proposition is proved in this case.

In both parts of case S observe ρ̄ cuts out a tamely ramified extension of Q_{l} so any deformation to W (k) factors through the Galois group of the maximal tamely ramified extension of Q_{l}. This quotient of Gal(Q̄_{l}/Q_{l}) is well understood. It is topologically generated by σ_{l} and τ_{l} where τ_{l} topologically generates inertia and the relation σ_{l}τ_{l}σ_{l}^{−1} = τ_{l}^{l} holds. In both cases the class C_{l} is

defined to be deformations of ρ̄ to W (k) given by τ_{l} ↦ ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) and σ_{l} ↦ ( \begin{matrix}l & x \\ 0 & 1 \end{matrix} ) for any

suitable x, y ∈ W (k). Note any mod p^{n} deformation of ρ̄ given as above deforms to W (k). One need only lift x and y from W (k)/p^{n} to W (k).

In case S, part a) one easily sees, using that l ≡ 1 mod p and local duality (Fact 1), that dim_{k}H^{2}(G_{l}, Ad^{0}ρ̄) = 1 and dim_{k}H^{0}(G_{l}, Ad^{0}ρ̄) = 1. By the local Euler-Poincaré characteristic we

see dim_{k}H^{1}(G_{l}, Ad^{0}ρ̄) = 2. Consider the element g ∈ H^{1}(G_{l}, Ad^{0}ρ̄) given by g(τ_{l}) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} )

and g(σ_{l}) = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ). We see that if ρ_{n} is the mod p^{n} reduction of an element of C_{l} and α ∈ k we

have that

(1 + ap^{n−1}g).ρ_{n}(τ_{l}) = ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) , (1 + ap^{n−1}g).ρ_{n}(σ_{l}) = ( \begin{matrix}l & x + ap^{n−1} \\ 0 & 1 \end{matrix} ) .

Thus (1 + ap^{n−1}g).ρ_{n} ∈ C_{l} with a different value for x. In case S part a) we let N_{l} be the space of codimension one in H^{1}(G_{l}, Ad^{0}ρ̄) spanned by g and we choose h to be any element of H^{1}(G_{l}, Ad^{0}ρ̄) not in N_{l}.

In case S, part b) we have H^{0}(G_{l}, Ad^{0}ρ̄) = 0. One sees that Ad^{0}ρ̄ has a unique one dimensional quotient on which G_{l} acts via χ^{−1}. It is a simple calculation using Facts 1 and 2 to show H^{2}(G_{l}, Ad^{0}ρ̄) and H^{1}(G_{l}, Ad^{0}ρ̄) are both trivial unless l ≡ −1 mod p. In this case both H^{1}(G_{l}, Ad^{0}ρ̄) and H^{2}(G_{l}, Ad^{0}ρ̄) are one dimensional and we choose N_{l} = {0} and h any nonzero element of H^{1}(G_{l}, Ad^{0}ρ̄). We have now proved the Proposition for case S.

In case V, by Lemma 4 we have that H^{0}(G_{l}, Ad^{0}ρ̄) = 0 and that if H^{2}(G_{l}, Ad^{0}ρ̄) ≠ 0 its dimension is one. By the local Euler-Poincaré characteristic we see dim_{k}H^{1}(G_{l}, Ad^{0}ρ̄) = 1. Thus if H^{2}(G_{l}, Ad^{0}ρ̄) ≠ 0 we take N_{l} = 0 and C_{l} to be the Teichmüller deformation of ρ̄. (The order of the image of ρ̄ is prime to p.)

In case H, if l is odd we have, since E/Q_{l} is ramified, that H^{2}(G_{l}, Ad^{0}ρ̄) = 0 by Lemma 4. If l = 2 , then keeping in mind p ≥ 5 so the order of the image of ρ̄ is prime to p, we see Ad^{0}ρ̄ is semisimple as a k[G_{l}] module. In all the possibilities G̃_{l} contains A_{4} as a subgroup that acts faithfully on Ad^{0}ρ̄. But from the representation theory of A_{4} in characteristics other than 2 or 3 it is easy to see that Ad^{0}ρ̄ must be absolutely irreducible and thus Ad^{0}ρ̄ has no one dimensional quotients by G_{l} stable k subspaces. Thus by Fact 6 we see H^{2}(G_{l}, Ad^{0}ρ̄) = 0 for l = 2 as well.

Remark: In all cases we find a smooth quotient of the local deformation ring in dimH^{1}-dimH^{2} variables. In each case N_{l} is the subspace of H^{1}(G_{l}, Ad^{0}ρ̄) one gets from the surjective map from the deformation ring to this smooth quotient by dualizing the map on tangent spaces.

\underline{Local at p considerations} Recall we assume p ≥ 5. In this section we carry out an analysis of the possible ρ̄: G_{p} → GL_{2}(k) and their deformations to W (k). Note that here we do not assume that ρ̄ and Ad^{0}ρ̄ are ramified at p. We determine, in most cases two kinds of N_{p} and C_{p}, one for the odd case and one for the even case. We will insist that

dim_{k}H^{1}(G_{p}, Ad^{0}ρ̄) − dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄) = dim_{k}N_{p} + δ

where δ = 0 or 2 as our global situation is even or odd. (Actually, in the even case we will require H^{2}(G_{p}, Ad^{0}ρ̄) = 0. This assumption is simply a shortcoming of our method.) Our purpose, as in the previous section, is to show ρ̄ can be deformed to an element of C_{p} by deforming from W (k)/p^{m} to W (k)/p^{m+1} with adjustments at each step made only by an element of a subspace of H^{1}(G_{p}, Ad^{0}ρ̄) complementary to N_{p}.

We refer the reader to the end of this section for an explanation of the reasons why some cases provide difficulties. We advise the reader, during a first reading, to accept the values in the tables of this section and that N_{p} preserves C_{p} and proceed to the Recollections section. Finally, we note that the results here are due to Böckle and can be found in [Bö2]. We have recast his results in a language more suitable for our application.

We treat the cases where inertia acts via fundamental characters of level one and two separately. First suppose that inertia acts through ρ̄ via fundamental characters of level two. Following

section 2.2 of [Se1], we may assume (after possibly extending scalars) that ρ̄ |_{I_{p}}= ( \begin{matrix}ψ^{u+pv} & 0 \\ 0 & ψ^{v+pu} \end{matrix} )

where ψ is a fundamental character of level two. After twisting by a suitable power of χ = ψ^{p+1}

we may assume ρ̄ |_{I_{p}}= ( \begin{matrix}ψ^{a} & 0 \\ 0 & ψ^{pa} \end{matrix} ) where 1 ≤ a ≤ p − 1. We suppose det(ρ̄) = ϕχ^{a} where ϕ is an

unramified character of G_{p}. We only consider deformations with determinant ϕ̃χ^{a} where ϕ̃ is the Teichmüller lift of ϕ.

It is shown in Lemma 5 of [R2] that H^{2}(G_{p}, Ad^{0}ρ̄) = 0 in the above case. Thus all deformation problems are unobstructed when inertia at p acts via fundamental characters of level two. It is easy to see the projective image of ρ̄ is nonabelian so H^{0}(G_{p}, Ad^{0}ρ̄) = 0 and dim_{k}H^{1}(G_{p}, Ad^{0}ρ̄) = 3 by the local Euler-Poincaré characteristic. The universal deformation ring is a power series ring over W (k) in 3 variables. By Theorem B2 of [FM] we have that ρ̄ is crystalline and the universal crystalline deformation ring (since the determinant is fixed) is a power series ring in one variable over W (k). (Actually, the result of [FM] is for ρ̄ : G_{p} → GL_{2}(F_{p}) and deformations to Z_{p}. It is easy to see that their results extend to k and W (k). Alternatively, the arguments of Theorem 3.1 and Lemma 4.4 of [R1] are easily generalized to this situation. Also, [FM] computes the universal crystalline deformation ring with arbitrary determinant which is smooth in two variables. It is easy to see the determinant restriction eliminates one of these variables.) In the odd case it is this family of crystalline deformations to W (k) with det = ϕ̃χ^{a} that we take to be C_{p}. In the even case we let C_{p} be the family of all deformations of ρ̄ to W (k). In the odd case we take N_{p} to be H_{cr}^{1}(G_{p}, Ad^{0}ρ̄), the one dimensional subspace of H^{1}(G_{p}, Ad^{0}ρ̄) that is the tangent space of the crystalline ring. If ρ̄ is even we choose N_{p} to be all of H^{1}(G_{p}, Ad^{0}ρ̄). In tables below h^{i} = dim_{k}H^{i}(G_{p}, Ad^{0}ρ̄).

T able 1, ρ̄ |_{I_{p}}= ( \begin{matrix}ψ^{a} & 0 \\ 0 & ψ^{pa} \end{matrix} )

C_{p}, odd  N_{p}, odd, dim C_{p}, even N_{p}, even h^{0} h^{1} h^{2} ; Crystalline defs  H_{cr}^{1}, 1  All  All  0 3 0

The word “All” refers to C_{p} being all deformations to W (k) and N_{p} being all of H^{1}(G_{p}, Ad^{0}ρ̄). Recall the statement “N_{p} preserves C_{p}” means that if ρ ∈ C_{p} with mod p^{n} reduction ρ_{n} then (1 + p^{n−1}g).ρ_{n} is the mod p^{n} reduction of an element of C_{p} for all g ∈ N_{p}. That N_{p} preserves C_{p} in the odd case follows from the existence and smoothness of the crystalline deformation ring. That N_{p} preserves C_{p} in the even case follows from the existence and smoothness of the unrestricted deformation ring. We have thus proved the following Proposition.

Proposition 2 Suppose inertia acts through ρ̄ via fundamental characters of level 2. Then the data in Table 1 above is correct, and in both the odd and even cases N_{p} preserves C_{p}.

It remains to consider the case when inertia acts on the semisimplification of ρ̄ via fundamental characters of level one. After twisting by a character of order prime to p we may assume that

ρ̄ = ( \begin{matrix}ϕχ^{a} & 0 \\ 0 & 1 \end{matrix} ) or ρ̄ = ( \begin{matrix}ϕχ^{a} & ∗ \\ 0 & 1 \end{matrix} ) and is indecomposable. Here 1 ≤ a ≤ p − 1, and ϕ : G_{p} → k^{∗}

is an unramified character.

We fix the determinant of all deformations we consider to be ϕ̃χ^{a} where ϕ̃ is the Teichmüller lift of ϕ. Each of the two cases above breaks into several subcases. In most subcases, we find, for ρ̄ both even and odd, classes C_{p} of deformations of ρ̄ to W (k) and subspaces N_{p} of H^{1}(G_{p}, Ad^{0}ρ̄) such that the action of an element of N_{p} preserves C_{p}.

In Tables 2 and 3 below, for the split and indecomposable cases, we give the relevant data. We again take h^{i} to be dim_{k}H^{i}(G_{p}, Ad^{0}ρ̄). Note that by the local Euler-Poincaré characteristic (Fact 2) we always have h^{1} = 3 + h^{0} + h^{2}. Let us now explain (with proof) all the information in the Tables 2 and 3 except the values of N_{p} and C_{p} in the odd case. The sets of dashes indicate the two (even) cases that we cannot produce a (global) deformation of ρ̄ to W (k). By Fact 6 we know h^{2} = 0 unless Ad^{0}ρ̄ has a one dimensional quotient (by a two dimensional G_{p} stable subspace) on which G_{p} acts via χ. In the split case one easily see this happens only when ϕχ^{a} = χ or χ^{p−2}. But these cases are twists of each other and are treated together. Recall U^{1} ⊂ Ad^{0}ρ̄ is the set of all 2×2

trace zero matrices over k of the form ( \begin{matrix}x & y \\ 0 & −x \end{matrix} ). Then for ρ̄ indecomposable U^{1} is the unique two dimensional G_{p} stable subspace of Ad^{0}ρ̄. It is easy to see G_{p} acts on the quotient Ad^{0}ρ̄/U^{1} via ϕ^{−1}χ^{−a}. Thus h^{2} ≠ 0 exactly when ϕχ^{a} = χ^{−1} = χ^{p−2}. In either the split or indecomposable case we see h^{2} = 1. Note that these situations are excluded in Theorem 1, part a). The h^{0} computations are easy and left to the reader. The h^{1} results follow immediately. When h^{2} = 0, for ρ̄ even, we choose C_{p} to be all deformations of ρ̄ to W (k) and N_{p} to be H^{1}(G_{p}, Ad^{0}ρ̄). Since H^{2}(G_{p}, Ad^{0}ρ̄) = 0 the deformation ring is smooth and it is clear that N_{p} preserves C_{p}. As remarked earlier, when H^{2}(G_{p}, Ad^{0}ρ̄) ≠ 0 we do not know how to proceed in the even case.

The dimension of N_{p} and the fact that N_{p} preserves C_{p} in the odd case with inertia acting via fundamental characters of level one are justified in the rest of this section. We also need to prove that the various C_{p} are nonempty! Roughly speaking we are showing the relevant (restricted) deformation ring is smooth. This will complete our verification of the tables.

Suppose that ρ̄ is flat, that is the Galois action of ρ̄ on the two dimensional k vector space is the generic fiber of a finite flat group scheme over Z_{p}. This notion corresponds to ρ̄ being weight 2 of level prime to p in the (global) modular case. A flat deformation ring exists. See [R1]. If ρ̄ is indecomposable then it is established in [R1] that the deformation ring is universal and smooth in one variable. (In [R1] arbitrary determinants are allowed, so the ring there is smooth in two variables.) We take the W (k) valued points of this deformation ring to be C_{p} and the tangent space of this ring, H_{flat}^{1}, to be N_{p}. (Note C_{p} is nonempty!) The existence and smoothness of this deformation ring implies N_{p} preserves C_{p} in this case. If ρ̄ is flat and split, i.e.

ρ̄ = ( \begin{matrix}χ & 0 \\ 0 & 1 \end{matrix} ), then the flat deformation is only versal. One easily computes the tangent space to

be 2 dimensional (keeping in mind we are fixing the determinant) and one sees that N_{p} preserves C_{p} by the following argument. If ρ ∈ C_{p} with mod p^{n} reduction ρ_{n} consider, for g ∈ N_{p}, (1+p^{n−1}g).ρ_{n}. The underlying Galois module is the generic fiber of a finite flat group scheme over Z_{p} and one sees the corresponding Fontaine-Laffaille module (or Honda system if one prefers) lifts to a p-dvisible group. Thus N_{p} preserves C_{p} in the case where ρ̄ is flat and split. This argument justifies all the values in tables 2 and 3 for ρ̄ flat in the odd case.

In most of the remaining cases we take N_{p} to be H_{ord}^{1}(G_{p}, Ad^{0}ρ̄), the ordinary cohomology classes in H^{1}(G_{p}, Ad^{0}ρ̄). Recall U^{0} is the subgroup of Ad^{0}ρ̄ consisting of matrices of the form

( \begin{matrix}0 & ∗ \\ 0 & 0 \end{matrix} ). Then we define H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) to be the kernel of the map

H^{1}(G_{p}, Ad^{0}ρ̄) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}}.

We first compute the dimensions of those N_{p} that are not taken to be H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) before proceeding to the ordinary cases. Afterwards we will establish that N_{p} preserves the nonempty set C_{p}. We remark here that the values in Table 4 are established by routine computations in Galois cohomology so no proofs are given.

Proposition 3 Suppose ρ̄ = ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ) is indecomposable and not flat and we are in the odd case.

Then N_{p} is one dimensional.

Proof: In this case we define N_{p} to be the kernel of the map H^{1}(G_{p}, Ad^{0}ρ̄) → H^{1}(G_{p}, Ad^{0}ρ̄/U^{0}). We use the symbol H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) for this N_{p}. The result follows from taking G_{p} cohomology of the short exact sequence

0 → U^{0} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{0} → 0

of G_{p} modules. We easily see (U^{0})^{G_{p}} and (Ad^{0}ρ̄)^{G_{p}} are trivial so we have the the sequence

0 → (Ad^{0}ρ̄/U^{0})^{G_{p}} → H^{1}(G_{p}, U^{0}) → H^{1}(G_{p}, Ad^{0}ρ̄) → H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0}).

One easily sees the first three terms have dimension 1, 2 and 3. The result follows.

In the tables below we give the relevant data for the local at p deformation problems in the even

and odd cases. The entries ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) mean all upper triangular deformations of ρ̄ to W (k). The

term \overline{H^{1}(G_{p}, U^{1})} means the image of H^{1}(G_{p}, U^{1}) under the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄). In the tables γ : G_{p} → W (k)^{∗} is any unramified character congruent to 1 mod p. Recall ϕ̃ : G_{p} → W (k)^{∗} is the Teichmüller lift of the unramified character ϕ : G_{p} → k^{∗}.

T able 2, ρ̄ = ( \begin{matrix}ϕχ^{a} & 0 \\ 0 & 1 \end{matrix} )

ϕχ^{a}  C_{p}, odd  N_{p}, odd, dim C_{p}, even N_{p}, even h^{0} h^{1} h^{2}
ϕχ^{a} = Id  ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )  \overline{H^{1}(G_{p}, U^{1})}, 4  All  All  3 6 0
ϕχ^{a} = χ, χ^{p−2}  Flat defs  H_{flat}^{1}, 2  —  —  1 5 1
ϕχ^{a} ≠ Id, χ, χ^{p−2} ρ = ( \begin{matrix}χ^{a}γ & ∗ \\ 0 & γ^{−1} \end{matrix} )  H_{ord}^{1}, 2  All  All  1 4 0

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T able 3, ρ̄ = ( \begin{matrix}ϕχ^{a} & ∗ \\ 0 & 1 \end{matrix} )

ϕχ^{a}  C_{p}, odd  N_{p}, odd, dim C_{p}, even N_{p}, even h^{0} h^{1} h^{2}
ϕχ^{a} = Id, unramified  ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )  \overline{H^{1}(G_{p}, U^{1})}, 2  All  All  1 4 0
ϕχ^{a} = Id, ramified ( \begin{matrix}χ^{p−1}γ & ∗ \\ 0 & γ^{−1} \end{matrix} )  H_{ord}^{1}, 2  All  All  1 4 0
ϕχ^{a} = χ, Flat  Flat Defs  H_{flat}^{1}, 1  All  All  0 3 0
ϕχ^{a} = χ, Not Flat  ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} )  H_{nf}^{1}, 1  All  All  0 3 0
ϕχ^{a} = χ^{p−2}  ( \begin{matrix}χ^{p−2}γ & ∗ \\ 0 & γ^{−1} \end{matrix} )  H_{ord}^{1}, 1  —  —  0 4 1
ϕχ^{a} ≠ Id, χ, χ^{p−2}  ( \begin{matrix}χ^{a} & ∗ \\ 0 & γ^{−1} \end{matrix} )  H_{ord}^{1}, 1  All  All  0 3 0

We compute H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) via some exact sequences. Associated to the short exact sequence

0 → U^{0} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{0} → 0

of G_{p} modules we have the long exact sequence of group cohomology. The part we are interested in is

... → H^{1}(G_{p}, U^{0}) → H^{1}(G_{p}, Ad^{0}ρ̄) → H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0})) → H^{2}(G_{p}, U^{0}).

We also have the inflation restriction sequence

0 → H^{1}(G_{p}/I_{p}, (Ad^{0}ρ̄/U^{0})^{I_{p}}) → H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0})) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}} → 0.

The last term in this sequence is in fact H^{2}(G_{p}/I_{p}, (Ad^{0}ρ̄/U^{0})^{I_{p}}) which is trivial as G_{p}/I_{p} ≃ Ẑ, a group of cohomological dimension 1. Gluing these sequences together as below we have

[figure: H^{1}(G_{p}, Ad^{0}ρ̄); H^{1}(G_{p}, Ad^{0}ρ̄/U^{0}); H^{2}(G_{p}, U^{0}); 0; H^{1}(G_{p}/I_{p}, (Ad^{0}ρ̄/U^{0})^{I_{p}}); H^{1}(I_{p}, Ad^{0}ρ̄/U^{0})^{G_{p}/I_{p}}; 0]

Lemma 5 H^{2}(G_{p}, U^{0}) ≠ 0 exactly when ϕ is trivial and a = 1 (or a = p−2 in the split case which is twist equivalent to the case when a = 1). If H^{2}(G_{p}, U^{0}) = 0 then

dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = 1 + h^{0} + h^{2} − dim_{k}H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0})).

Proof: Recall h^{i} = dim_{k}H^{i}(G_{p}, Ad^{0}ρ̄) and G_{p} acts on U^{0} via ϕχ^{a}. But H^{2}(G_{p}, U^{0}) ≠ 0 if and only if this action equals that of χ. If H^{2}(G_{p}, U^{0}) = 0 then a simple diagram chase shows

H^{1}(G_{p}, Ad^{0}ρ̄) → H^{1}(I_{p}, Ad^{0}ρ̄/U^{0})^{G_{p}/I_{p}}

is surjective. Since H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) is the kernel of this map we have

dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = h^{1} − dim_{k}H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0})) + dim_{k}H^{1}(G_{p}/I_{p}, (Ad^{0}ρ̄/U^{0})^{G_{p}/I_{p}}).

By Lemma 3

dim_{k}H^{1}(G_{p}/I_{p}, (Ad^{0}ρ̄/U^{0})^{I_{p}}) = dim_{k}H^{0}(G_{p}, (Ad^{0}ρ̄/U^{0})).

The result finally follows by using the local Euler-Poincaré characteristic to reduce all H^{1}’s to H^{0}’s and H^{2}’s.

We compute the dimensions of the N_{p} for ρ̄ odd and I_{p} acting via fundamental characters of level one. Suppose ρ̄ is split.

• Suppose ϕχ^{a} = Id. Since ρ̄ is trivial the sequence

0 → U^{1} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{1} → 0

of k[G_{p}] modules splits. Taking G_{p} cohomology we see the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) is injective. One easily sees dim_{k}H^{1}(G_{p}, U^{1}) = 4 so dim_{k}\overline{H^{1}(G_{p}, U^{1})} = 4 and we are done.

• Suppose ϕχ^{a} = χ or χ^{p−2}. In these cases ρ̄ is twist equivalent to ( \begin{matrix}χ & 0 \\ 0 & 1 \end{matrix} ). Since ρ̄ is flat

we see from the remarks prior to Proposition 3 that the flat tangent space is 2 dimensional.

• ϕχ^{a} ≠ Id, χ, χ^{p−2}. By Lemma 5 we see H^{2}(G_{p}, U^{0}) = 0. We easily see h^{0} = 1 and h^{2} = 0. By the latter part of Lemma 5 we have H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0})) = 0 so dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = 2.

Suppose ρ̄ is indecomposable. This case also splits into several parts. Before addressing these we include Table 4 below which will be useful for calculations throughout the rest of this section. The numbers represent the k dimension of the cohomology group when ρ̄ is indecomposable as described. Note that ρ̄ flat is omitted from Table 4 since we have already addressed this case. The values in the table below are easily verified and therefore considered proved. The H^{2} values are found via minor variant of Fact 6, the H^{0} values are easy to compute, and the H^{1}’s are computed by the local Euler-Poincaré characteristic (Fact 2). We use Table 4 to verify the values in Tables 2 and 3 above for ρ̄ indecomposable and to show (in the rest of this section) that N_{p} preserves C_{p}.

T able 4, ρ̄ = ( \begin{matrix}ϕχ^{a} & ∗ \\ 0 & 1 \end{matrix} )

ϕχ^{a}  H^{1}(G_{p}, U^{1}) H^{1}(G_{p}, (U^{1}/U^{0})) H^{2}(G_{p}, U^{0}) H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0}))
ϕχ^{a} = Id, unramified  3  2  0  0
ϕχ^{a} = Id, ramified  3  2  0  0
ϕχ^{a} = χ, Not Flat  2  2  1  0
ϕχ^{a} = χ^{p−2}  2  2  0  1
ϕχ^{a} ≠ Id, χ, χ^{p−2}  2  2  0  0

Let us now turn to verifying the dimensions of N_{p} in the odd indecomposable cases that remain.

• ϕχ^{a} = Id and ρ̄ is unramified. Consider the short exact sequence

0 → U^{1} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{1} → 0

and take G_{p} cohomology. We get the long exact sequence

0 → (U^{1})^{G_{p}} → (Ad^{0}ρ̄)^{G_{p}} → (Ad^{0}ρ̄/U^{1})^{G_{p}} → H^{1}(G_{p}, U^{1}) →^{β} H^{1}(G_{p}, Ad^{0}ρ̄) → ...

whose first three terms are easily seen to be one dimensional. Consulting Table 4 the next term is three dimensional. Thus the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) has two dimensional image so dim_{k}\overline{H^{1}(G_{p}, U^{1})} = 2 and we are done.

• ϕχ^{a} = Id and ρ̄ is ramified. As H^{2}(G_{p}, U^{0}) = 0, Lemma 5 applies. Since h^{0} = 1, h^{2} = 0 and dim_{k}H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0})) = 0 we see dim_{k}N_{p} = dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = 2.

• ϕχ^{a} = χ and ρ̄ is flat. This case has been addressed. N_{p} = H_{f}^{1} is one dimensional.

• ϕχ^{a} = χ and ρ̄ is not flat. This case has been addressed by Proposition 3.

• ϕχ^{a} = χ^{p−2}. Since H^{2}(G_{p}, U^{0}) = 0 and h^{0} = 0, h^{2} = 1, and dim_{k}H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0})) = 1 we see by Lemma 5 dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = 1.

• ϕχ^{a} ≠ Id, χ, χ^{p−2}. Since H^{2}(G_{p}, U^{0}) = 0, h^{0} = 0, h^{2} = 0 and dim_{k}H^{2}(G_{p}, (Ad^{0}ρ̄/U^{0})) = 0 we see dim_{k}H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) = 1 by Lemma 5.

We have established the correctness of the values in Tables 2 and 3. We now turn to the questions of showing, when ρ̄ is odd and inertia acts via fundamental characters of level one, the various C_{p}’s are nonempty and preserved by their respective N_{p}’s. Note that we have already established this for ρ̄ flat and odd. We need not worry about the even cases as there we have taken C_{p} to be all deformations to the smooth deformation ring and it is clear N_{p} preserves C_{p}.

First we show the C_{p} are nonempty.

Lemma 6 Suppose we are in the odd case and ρ̄ is not an indecomposable representation ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ).

Then there is a deformation ρ of ρ̄ to W (k) of the form ( \begin{matrix}ϕ̃χ^{a}γ & ∗ \\ 0 & γ^{−1} \end{matrix} ) where γ : G_{p} → W (k)^{∗}

is an unramified character congruent to 1 mod p. Thus C_{p} is nonempty.

Proof: We show ρ̄ deforms to a characteristic zero representation of the form ρ = ( \begin{matrix}ϕ̃χ^{a}γ & ∗ \\ 0 & γ^{−1} \end{matrix} )

regardless of whether N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) or \overline{H^{1}(G_{p}, U^{1})}. Observe in either case ρ ∈ C_{p}.

Note ρ̄ may or may not be decomposable. By hypothesis ϕχ^{a} ≠ χ. From Table 4 we have H^{2}(G_{p}, U^{0}) = 0. Let γ : G_{p} → W (k)^{∗} be any unramified character congruent to 1 mod p. By Lemma 1 the deformation theory for ρ̄ where we only consider reducible deformations with diagonal characters ϕ̃χ^{a}γ and γ^{−1} is unobstructed. Thus ρ ∈ C_{p} as desired.

Lemma 7 Suppose ρ̄ = ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ) and is indecomposable. C_{p} is nonempty in this case.

Proof: It suffices to show ρ̄ lifts to a characteristic zero representation of the form ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ). Note

the kernel of ρ̄ fixes the field Q_{p}(μ_{p}, a^{1/p}) where μ_{p} are the pth roots of unity and a ∈ Q_{p}(μ_{p}). One easily finds a deformation ρ_{2} of ρ̄ to W (k)/p^{2} that factors through Gal(Q_{p}(μ_{p^{2}}, a^{1/p^{2}}) by Kummer Theory.

We have now shown C_{p} is nonempty for all ρ̄ in the odd case. It remains to prove N_{p} preserves C_{p} in the odd cases where inertia acts via fundamental characters of level one.

Proposition 4 Suppose ρ̄ is odd and split. Then N_{p} preserves C_{p}.

Proof: If ϕχ^{a} = χ then ρ̄ is flat and we have already addressed this case.

If ϕχ^{a} = Id then C_{p} is all upper triangular deformations and N_{p} is \overline{H^{1}(G_{p}, U^{1})}. As H^{2}(G_{p}, U^{1}) = 0 in this case the versal upper triangular deformation ring is smooth in this case so N_{p} preserves C_{p}.

In the remaining cases we will exhibit a basis for H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) that clearly preserves C_{p}. This will prove the proposition. Observe Ad^{0}ρ̄ ≃ k ⊕ k(ϕχ^{a}) ⊕ k(ϕ^{−1}χ^{−a}) where k(φ) is the one dimensional k vector space with G_{p} action via the character φ : G_{p} → k^{∗}. Recalling that cohomology commutes with direct sums we see H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) is spanned by a nonzero element of the one dimensional space H^{1}(G_{p}, k(ϕχ^{a})) and nonzero element of the unramified one dimensional subspace in the two dimensional space H^{1}(G_{p}, k). Suppose ρ_{n} ∈ C_{p}. Then ρ_{n} : G_{p} → GL_{2}(W (k)/p^{n}) is

given by ( \begin{matrix}ϕ̃χ^{a}γ_{n} & ∗ \\ 0 & γ_{n}^{−1} \end{matrix} ) is a deformation of ρ̄ to W (k)/p^{n}. For any g ∈ H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) we see

(1 + p^{n−1}g).ρ_{n} = ( \begin{matrix}ϕ̃χ^{a}γn^{′} & ∗′ \\ 0 & γ′ _{n}^{−1} \end{matrix} ) where γn^{′} is a (possibly) different unramified character of G_{p}

congruent to 1 mod p.

If we can show that (1 + p^{n−1}g).ρ_{n} ∈ C_{p} we will have that N_{p} preserves C_{p}. Since ϕχ^{a} ≠ χ we see H^{2}(G_{p}, Ad^{0}ρ̄) = 0 by Lemma 5. By Lemma 1 the reducible deformation theory of ρ̄ with fixed diagonal characters is unobstructed. Choose the diagonal characters to be ϕ̃χ^{a}γ′ and γ′ where γ′ is any characteristic zero lift of γ′_{n}. The proof is complete.

We now prove an analog of Proposition 4 in the indecomposable case. We first deal with the cases where N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄). The primary difficulty is to make sure that the elements of N_{p} are “upper triangular”, that is their action on upper triangular deformations gives upper triangular deformations. Recall U^{1} is the subgroup of Ad^{0}ρ̄ of upper triangular matrices. The short exact sequence

0 → U^{1} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{1} → 0

gives the long exact G_{p} cohomology sequence. We are interested in the map

β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄).

In the cases where we have taken N_{p} to be H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) we will see N_{p} is contained in the image of β.

Lemma 8 Suppose ρ̄ is odd, indecomposable, and that we are in a case where N_{p} is taken to be H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) or H_{nf}^{1}(G_{p}, Ad^{0}ρ̄). If the image of the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) contains N_{p} then N_{p} preserves C_{p}.

Proof: Let ρ_{n} ∈ C_{p} with diagonal characters ϕ̃χ^{a}γ_{n} and γ_{n}^{−1} where γ_{n} is unramified. Let g ∈ H_{ord}^{1}(G_{p}, Ad^{0}ρ̄). By hypothesis there exists a g̃ ∈ H^{1}(G_{p}, U^{1}) be such that β(g̃) = g. Since ρ_{n} ∈ C_{p} is reducible we see (1 + p^{n}g̃).ρ_{n} is reducible. Thus (1 + p^{n}g).ρ_{n} is also reducible, and since g ∈ N_{p} we see the diagonal characters of (1 + p^{n}g).ρ_{n}, are ϕ̃χ^{a}γ′_{n} and γ^{′−1}_{n} or χ and 1 as N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) or H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) where γ′_{n} is unramified. When N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) we see by the argument in the proof of Lemma 6 that (1 + p^{n−1}g).ρ_{n} deforms to a characteristic zero element of C_{p}. When N_{p} = H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) the argument in the proof of Lemma 7 applies. Thus in either case N_{p} preserves C_{p}.

It now suffices to show the hypothesis of Lemma 8 is satisfied in each of the nonflat indecomposable cases.

Proposition 5 Suppose ρ̄ = ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) is ramified and we are in the odd case. The image of

β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) contains N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄). Thus N_{p} preserves C_{p} in this case.

Proof: Consider the commutative diagram below.

[figure: β; H^{1}(G_{p}, U^{1}); H^{1}(G_{p}, Ad^{0}ρ̄); H^{1}(G_{p}, U^{1}/U^{0}); H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0})); H^{1}(I_{p}, U^{1}/U^{0})^{G_{p}/I_{p}}; H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}}]

We have the exact sequence

0 → H^{1}(G_{p}/I_{p}, (U^{1}/U^{0})^{I_{p}}) → H^{1}(G_{p}, (U^{1}/U^{0})) → H^{1}(I_{p}, (U^{1}/U^{0}))^{G_{p}/I_{p}} → 0.

By Lemma 3

dim_{k}H^{1}(G_{p}/I_{p}, (U^{1}/U^{0})^{I_{p}}) = dim_{k}H^{0}(G_{p}, (U^{1}/U^{0})) = 1.

Since from Table 4 we have dim_{k}H^{1}(G_{p}, (U^{1}/U^{0})) = 2 we see dim_{k}H^{1}(I_{p}, (U^{1}/U^{0}))^{G_{p}/I_{p}} = 1. We will show the bottom horizontal map is trivial. Thus the map H^{1}(G_{p}, U^{1}) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}} by going down twice and then to the right is trivial. Since the diagram commutes we see the map H^{1}(G_{p}, U^{1}) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}} by going to the right and then down twice is trivial. Thus we will have shown that the image of H^{1}(G_{p}, U^{1}) under the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) lies in H_{ord}^{1}(G_{p}, Ad^{0}ρ̄), the kernel of the composite of the two right vertical maps.

We consider the I_{p} cohomology sequence of the short exact sequence

0 → U^{1}/U^{0} → Ad^{0}ρ̄/U^{0} → Ad^{0}ρ̄/U^{1} → 0.

We get the long exact sequence

0 → (U^{1}/U^{0})^{I_{p}} → (Ad^{0}ρ̄/U^{0})^{I_{p}} → (Ad^{0}ρ̄/U^{1})^{I_{p}} → H^{1}(I_{p}, (U^{1}/U^{0})) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{1})) → ...

whose first three terms are all easily seen to be one dimensional. (N.B. That (Ad^{0}ρ̄/U^{0})^{I_{p}} is one dimensional follows from the fact that ρ̄ is ramified. If ρ̄ had been indecomposable and unramified (Ad^{0}ρ̄/U^{0})^{I_{p}} would have been two dimensional. This one dimensionality is the crucial step in establishing the triviality of the bottom horizontal map in our diagram.) Thus the map H^{1}(I_{p}, (U^{1}/U^{0})) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0})) has one dimensional kernel. We have the left exact sequence

0 → (Ad^{0}ρ̄/U^{1})^{I_{p}} → H^{1}(I_{p}, (U^{1}/U^{0})) → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0})).

Since taking G_{p}/I_{p} invariants is left exact we get the the exact sequence

0 → ((Ad^{0}ρ̄/U^{1})^{I_{p}})^{G_{p}/I_{p}} → H^{1}(I_{p}, (U^{1}/U^{0}))^{G_{p}/I_{p}} → H^{1}(I_{p}, (Ad^{0}ρ̄/U^{0}))^{G_{p}/I_{p}}.

But the first term is simply (Ad^{0}ρ̄/U^{1})^{G_{p}} which is easily seen to be one dimensional. Since dim_{k}H^{1}(I_{p}, (U^{1}/U^{0}))^{G_{p}/I_{p}} = 1 the bottom horizontal map in our diagram is trivial.

Proposition 6 Suppose ρ̄ = ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} )is not flat. Then N_{p} = H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) preserves C_{p} in the

odd case.

Proof: As in Lemma 8, it suffices to show that the image of the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) contains N_{p} = H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) which is the kernel of the right vertical map in the diagram below.

[figure: β; H^{1}(G_{p}, U^{1}); H^{1}(G_{p}, Ad^{0}ρ̄); H^{1}(G_{p}, (U^{1}/U^{0})); H^{1}(G_{p}, (Ad^{0}ρ̄/U^{0}))]

We know dim_{k}H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) = 1. We will show the horizontal maps are injective and the left vertical map has one dimensional kernel. The commutativity of the diagram then shows β(H^{1}(G_{p}, U^{1})) ⊃ H_{nf}^{1}(G_{p}, Ad^{0}ρ̄) and we are done.

Consider the exact sequence

0 → U^{1} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{1} → 0

and take G_{p} cohomology. Since (Ad^{0}ρ̄/U^{1})^{G_{p}} = 0 we have

... → 0 → H^{1}(G_{p}, U^{1}) →^{β} H^{1}(G_{p}, Ad^{0}ρ̄) → ...

so the top horizontal map is injective. The same argument applied to the exact sequence

0 → U^{1}/U^{0} → Ad^{0}ρ̄/U^{0} → Ad^{0}ρ̄/U^{1} → 0

gives the injectivity of the bottom horizontal map.

Finally, consider the short exact sequence

0 → U^{0} → U^{1} → U^{1}/U^{0} → 0

and take G_{p} cohomology. We have

0 → (U^{0})^{G_{p}} → (U^{1})^{G_{p}} → (U^{1}/U^{0})^{G_{p}} → H^{1}(G_{p}, U^{0}) → H^{1}(G_{p}, U^{1})→H^{1}(G_{p}, (U^{1}/U^{0})).

We easily see the first two terms are trivial, the third is one dimensional, and the last three terms are two dimensional. Thus the map H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, (U^{1}/U^{0})) has one dimensional kernel so the the image of β contains H_{nf}^{1}(G_{p}, Ad^{0}ρ̄). By Lemma 8 the proposition is proved.

Proposition 7 Suppose ρ̄ = ( \begin{matrix}ϕχ^{a} & ∗ \\ 0 & 1 \end{matrix} ) is indecomposable where ϕχ^{a} ≠ χ or the identity char-

acter. Then N_{p} = H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) preserves C_{p} in the odd case.

Proof: Again we must show the image of the map β : H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄) contains the one dimensional subspace H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) of H^{1}(G_{p}, Ad^{0}ρ̄). We refer the reader to the diagram of Proposition 5. Taking the G_{p} cohomology of the short exact sequence

0 → U^{1} → Ad^{0}ρ̄ → Ad^{0}ρ̄/U^{1} → 0

we see (using p ≥ 5 when ϕχ^{a} = χ^{p−2}) that (Ad^{0}ρ̄/U^{1})^{G_{p}} = 0 so we have

... → 0 → H^{1}(G_{p}, U^{1}) →^{β} H^{1}(G_{p}, Ad^{0}ρ̄) → ...

so β is injective. To finish the proof, using the commutativity of the diagram, it suffices to show that the composed map of going down twice and then to the right has nontrivial kernel. For then the composition of going to the right and then down twice has nontrivial kernel. Since the kernel of the composition of the two right vertical maps is the one dimensional space H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) and β is injective we see the image of β would then contain H_{ord}^{1}(G_{p}, Ad^{0}ρ̄).

Considering the short exact sequence

0 → U^{0} → U^{1} → U^{1}/U^{0} → 0

and taking G_{p} cohomology we get

... → H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, (U^{1}/U^{0})) → H^{2}(G_{p}, U^{0}).

Since ϕχ^{a} ≠ χ we have H^{2}(G_{p}, U^{0}) = 0 by Lemma 5. the top left vertical map in our diagram is then surjective. The kernel of the bottom left vertical map is the image under inflation of H^{1}(G_{p}/I_{p}, (U^{1}/U^{0})^{I_{p}}). By Lemma 3 the dimension of this cohomology group over k is simply dim_{k}H^{0}(G_{p}, (U^{1}/U^{0})) = 1. Thus the composition of the two left vertical maps has nontrivial kernel and the proposition is proved.

Proposition 8 If we are in either of the two cases where (C_{p}, odd) = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) then N_{p} preserves

C_{p}.

Proof: In both cases we easily see H^{2}(G_{p}, U^{1}) = 0 so the upper triangular deformation theory of ρ̄ is unobstructed. Thus N_{p} preserves C_{p} in these cases.

We have now verified all the results in the tables and shown N_{p} preserves C_{p} in all cases.

Proposition 9 With the exception of the two cases where C_{p} = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} )and N_{p} = \overline{H^{1}(G_{p}, U^{1})},

all deformations to W (k) in (C_{p}, odd) above are semistable.

Proof: When inertia acts via fundamental characters of level two the elements of (C_{p}, odd) are all crystalline and known to besemistable. See [Fo].

When inertia acts via fundamental characters of level one then when (C_{p}, odd) ≠ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) they

are semistable by the Proposition of section 3.1 of [PR].

We remark that we have felt free to twist ρ̄ by a character of finite order and replace ρ̄ with the twisted version. This affects Proposition 9 only up to the character of finite order. In particular, if one does not twists our characteristic zero representations are only potentially semistable or crystalline.

What is the “difference” between the cases we can handle and those we cannot? In the even case, if H^{2}(G_{p}, Ad^{0}ρ̄) = 0 there are no difficulties. Otherwise dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄) = 1. Let d = dim_{k}H^{1}(G_{p}, Ad^{0}ρ̄). Our requirements for N_{p} and C_{p} amount to insisting the deformation ring associated to ρ̄ has W (k)[[T_{1}, ...T_{d−1}]] as a quotient and insisting that C_{p} consist exactly of the W (k) valued points of this quotient. G. Böckle has kindly pointed out to us that no such quotients exist when H^{2}(G_{p}, Ad^{0}ρ̄) ≠ 0.

In the odd case, after twisting, the deformation ring must have a smooth quotient in r − 2 − dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄) variables, all of whose W (k) valued points are semistable. We do not know

how to show such a ring exists in the two cases where (C_{p}, odd) = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ). In these cases

H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) is of the appropriate codimension in H^{1}(G_{p}, Ad^{0}ρ̄), but H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) is not contained in the image of the map H^{1}(G_{p}, U^{1}) → H^{1}(G_{p}, Ad^{0}ρ̄), that is the elements of H_{ord}^{1}(G_{p}, Ad^{0}ρ̄) are not upper triangular. It might perhaps be that we are limited in that we are trying to only construct ordinary representations. A more refined semistable theory could settle these cases.

\underline{Recollections} In this section we recall a number of facts that we will need from [R2].

We have ρ̄ : G_{S} → GL_{2}(k) and Ad^{0}ρ̄ is the additive group of two by two trace zero matrices over k with G_{S} action through ρ̄ and by conjugation. We assume S contains p and infinity even if ρ̄ is unramified at these places. Let Q(ρ̄) be the field fixed by the kernel of ρ̄. Let (Ad^{0}ρ̄)^{∗} be the Cartier dual of Ad^{0}ρ̄. We assume that ρ̄ has minimal conductor among it twists. In particular we may assume Q(ρ̄)/Q and Q(Ad^{0}ρ̄)/Q are ramified at exactly the same finite primes. Let N be the kernel of the action of G_{S} on Ad^{0}ρ̄ and let N^{d} be the kernel of the action of G_{S} on (Ad^{0}ρ̄)^{∗}. Let Q(Ad^{0}ρ̄) be the field fixed by N and let D = Q(Ad^{0}ρ̄) ∩ Q(μ_{p}) and K = Q(Ad^{0}ρ̄)Q(μ_{p}). It is easy to see N^{d} ⊇ Gal(Q_{S}/Q).

[figure: Q; D; Q(μ_{p}); Q(Ad^{0}ρ̄); K; Q(ρ̄)]

Our hypotheses on ρ̄ are as follows:

• ρ̄ and Ad^{0}ρ̄ are absolutely irreducible representations of G_{S}.

• H^{1}(G_{S}/N, Ad^{0}ρ̄) = 0.

• H^{1}(G_{S}/N^{d}, (Ad^{0}ρ̄)^{∗}) = 0.

• There is an element

c = a × b ∈ Gal(Q(Ad^{0}ρ̄)/D) × Gal(Q(μ_{p})/D) ≃ Gal(K/D) ⊆ Gal(K/Q)

such that under the map Gal(Q(μ_{p})/Q) → (Z/p)^{∗} we have that b maps to t ≠ ±1 and a lifts to an element of Gal(Q(ρ̄)/Q) with necessarily distinct eigenvalues of ratio t. (Note the hypothesis on t immediately implies p ≥ 5.

It is easy to see the absolute irreducibility of Ad^{0}ρ̄ implies that of ρ̄.

Lemma 9 Let r = dim_{k}� _{S}^{1}((Ad^{0}ρ̄)^{∗}), s be the number of v ∈ S such that H^{2}(G_{v}, Ad^{0}ρ̄) ≠ 0, and δ = 0 or 2 as ρ̄ is even or odd. Then dim_{k}H^{1}(G_{S}, Ad^{0}ρ̄) = r + s + δ.

Proof: By the proof of Proposition 1 and the tables in the Local at p considerations section we know that dim_{k}H^{2}(G_{v}, Ad^{0}ρ̄) = 0 or 1 for all v ∈ S. We then have, using Theorem 4.10 of Chapter I of [Mi] the exact sequence

0 → � _{S}^{2}(Ad^{0}ρ̄) → H^{2}(G_{S}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{2}(G_{v}, Ad^{0}ρ̄) → H^{0}(G_{S}, (Ad^{0}ρ̄)^{∗})^{∗}.

The last term is trivial as Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗} are assumed absolutely irreducible. By Fact 3 we have � _{S}^{2}(Ad^{0}ρ̄) and � _{S}^{1}((Ad^{0}ρ̄)^{∗}) are dual so dim_{k}H^{2}(G_{S}, Ad^{0}ρ̄) = r + s. The result follows using the global Euler-Poincaré characteristic (Fact 4) keeping in mind dim_{k}H^{0}(G_{∞}, Ad^{0}ρ̄) = 1 + δ.

Lemma 10 If p ≥ 7 and the projective image of ρ̄: G_{S} → GL_{2}(k) equals a conjugate of P SL_{2}(k′) or P GL_{2}(k′) over the quadratic extension of k for some subfield k′ of k, then the four hypotheses on ρ̄ above are satisfied.

Proof: The proofs of Lemmas 17 through 21 of [R2] guarantee the first three conditions hold. The existence of the element c follows from the proof of Theorem 2 of [R2]. All of this follows as we only need consider the projective image of ρ̄ to understand Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗}. The results of [R2] only require the projective image of ρ̄ contain P SL_{2}(k′) for some field k′ of characteristic greater than or equal to 7.

It is the Lemmas of [R2] that require p ≥ 7 and this is why we have this hypothesis here and in Theorem 1.

The following facts are proved or cited in sections 6 and 7 of [R2].

Fact 9 A finite dimensional absolutely irreducible mod p representation of a profinite group has a minimal field of definition. If the image of ρ̄ contains SL_{2}(k) the minimal field of definition of ρ̄ and Ad^{0}ρ̄ is k. (The fact of the Descents section of [R2] and Lemma 17 of that paper.)

Remark: The minimal fields of definition of ρ̄ and Ad^{0}ρ̄ need not be the same. In this case we denote the minimal field of definition of Ad^{0}ρ̄ by k̃ and refer to the descended representation space by Ãd^{0}ρ̄. If k̃ is the minimal field of definition of Ad^{0}ρ̄ it is easy to see k̃ is the minimal field of definition of (Ad^{0}ρ̄)^{∗}. We refer to its descent by (Ãd^{0}ρ̄)^{∗}. The existence of these descents is used in this paper only insofar as we recall various facts from [R2] that depend on the descents. Ultimately we need the descents to use Chebotarev’s Theorem to provide us with primes satisfying certain properties such as the set Q in Fact 16 of this section and the various sets T, T_{i}, T̃_{i} of the next section. Since our primary interest is in the applications of Chebotarev’s Theorem we only mention Ãd^{0}ρ̄ and (Ãd^{0}ρ̄)^{∗} in a few key places.

[figure: Q; D; Q(μ_{p}); Q(Ad^{0}ρ̄); K; M̃; L_{i}; L; LM̃]

Fact 10 Let {f_{1}, .., f_{r+s+δ}} be a basis of H^{1}(G_{S}, Ãd^{0}ρ̄). Let g ∈ H^{1}(G_{S}, (Ãd^{0}ρ̄)^{∗}) be nonzero.

1) There correspond to the f_{i} extensions L_{i}, Galois over Q, with Gal(L_{i}/K) ≃ Ãd^{0}ρ̄ as Gal(K/Q) modules where Gal(L_{i}/K) inherits a k̃ structure from f_{i}. The exact sequence

1 → Gal(L_{i}/K) → Gal(L_{i}/Q) → Gal(K/Q) → 1

splits. Each L_{i} is linearly disjoint over K with the composite of the L_{j}, j ≠ i. Let L denote the composite of the L_{i}, 1 ≤ i ≤ r + s + δ. Then Gal(L/K) ≃ (Ãd^{0}ρ̄)^{r+s+δ} (with inherited k̃ structure) and the exact sequence

1 → Gal(L/K) → Gal(L/Q) → Gal(K/Q) → 1

splits.

2) There corresponds to g an extension M̃, Galois over Q, of K with Gal(M̃/K) ≃ (Ãd^{0}ρ̄)^{∗} as Gal(K/Q) modules where the k̃ structure is inherited from g. The exact sequence

1 → Gal(M̃/K) → Gal(M̃/Q) → Gal(K/Q) → 1

splits.

3) M̃ is linearly disjoint from L over K and Gal(LM̃/K) ≃ Gal(L/K) × Gal(M̃/K). (Lemmas 8, 11, and 12 of [R2].)

Fact 11 The composite LM̃ is Galois over Q and the exact sequence

1 → Gal(LM̃/K) → Gal(LM̃/Q) → Gal(K/Q) → 1

splits. Thus Gal(LM̃/Q) ≃ Gal(LM̃/K)× Gal(K/Q). (Lemma 13 of [R2].)

Consider the element

c = a × b ∈ Gal(Q(Ad^{0}ρ̄)/D) × Gal(Q(μ_{p})/D) ≃ Gal(K/D) ⊆ Gal(K/Q)

as in the hypotheses on ρ̄ and its action on Ãd^{0}ρ̄ and (Ãd^{0}ρ̄)^{∗}. In [R2] it was observed that Ãd^{0}ρ̄ decomposes under the action of c into F_{p} eigenspaces with distinct eigenvalues t, 1 and 1/t. Recall t ≠ ±1 mod p. By duality (Ãd^{0}ρ̄)^{∗}, keeping in mind the b component of c acts on (Ad^{0}ρ̄)^{∗} by multiplication by t, decomposes into F_{p} eigenspaces with eigenvalues 1, t, and t^{2}.

Fact 12 Gal(L_{i}/K) contains a nontrivial F_{p} subspace on which c acts trivially. Gal(M̃/K) contains an F_{p} subspace on which c acts trivially. (Lemma 14 of [R2].)

Let α_{i} ∈ Gal(L/K) ≃ (Ãd^{0}ρ̄)^{r+s+δ} be an element that is zero at all entries except the ith at which it is an element of Ãd^{0}ρ̄ on which c acts trivially. Let ζ be a nonzero element of Gal(M̃/K) on which c acts trivially. Consider η_{i} = (α_{i} × ζ)× c in

(Gal(L/K) × Gal(M̃/K)) × Gal(K/D) ⊆ Gal(LM̃/K)× Gal(K/Q) ≃ Gal(LM̃/Q).

We need a few local facts. (These will apply to primes with Frobenius in the conjugacy class of η_{i} ∈ Gal(LM̃/Q).)

Fact 13 Let t ≠ ±1 ∈ F_{p}^{∗}. Let q ≡ t be a prime and ρ̄: G_{q} → GL_{2}(k) be unramified. Suppose that ρ̄(F rob_{q}) has distinct eigenvalues with ratio t. Then dim_{k}H^{0}(G_{q}, Ad^{0}ρ̄) = 1, dim_{k}H^{1}(G_{q}, Ad^{0}ρ̄) = 2, and dim_{k}H^{2}(G_{q}, Ad^{0}ρ̄) = 1. (Lemma 1 of [R2].)

Consider ρ̄ : G_{q} → GL_{2}(k) with q as above. Observe ρ̄ is unramified so any deformation to W (k) is tamely ramified and thus factors through the Galois group over Q_{q} of the maximal tamely ramified extension. Let σ_{q} and τ_{q} be topological generators of the tame Galois group where τ_{q}

generates inertia. Then, up to a twist, ρ̄ is given by σ_{q} ↦ ( \begin{matrix}q & 0 \\ 0 & 1 \end{matrix} ) and τ_{q} ↦ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ). Define

C_{q} to be all deformations given by σ_{q} ↦ ( \begin{matrix}q & px \\ 0 & 1 \end{matrix} ) and τ_{q} ↦ ( \begin{matrix}1 & py \\ 0 & 1 \end{matrix} ). Define N_{q} to be space

spanned by g ∈ H^{1}(G_{q}, Ad^{0}ρ̄) where g(τ_{q}) = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) and g(σ_{q}) = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ).

Fact 14 Let ρ̄ : G_{q} → GL_{2}(k) be unramified with ρ̄(F rob_{q}) having distinct eigenvalues with ratio t ∈ F_{p}, t ≠ ±1 and q ≡ t mod p. Let N_{q} and C_{q} be as above and let h ∈ H^{1}(G_{q}, Ad^{0}ρ̄) be such that h ∉ N_{q}. Then ρ̄: G_{q} → GL_{2}(k) can be deformed to an element of C_{q} one step at a time with adjustments at each step made only by a multiple of h. Thus N_{q} preserves C_{q}. (Proposition 1 of [R2].)

Fact 15 η_{i} = (α_{i} × ζ)× c has order pd in Gal(LM̃/Q). Let u_{i} be a prime of Q unramified in LM̃ with Frobenius in the conjugacy class of η_{i}. Note u_{i} is a prime as in Fact 13. Then f_{i} |_{G_{u_{i}}} ∉ N_{u_{i}}. For i ≠ j we have f_{i} |_{G_{u_{j}}} = 0. For g ∈ H^{1}(G_{S}, (Ad^{0}ρ̄)^{∗}) corresponding to M̃ we have g |_{G_{u_{i}}} ≠ 0. (Corollaries 1 and 2 of [R2].)

Fact 16 There is a set Q = {q_{1}, q_{2}, .., q_{r}} of primes, q_{i} ∉ S such that

1) q_{i} is unramified in K/Q and F rob_{q_{i}} is in the conjugacy class of c in Gal(K/Q). Also, ρ̄(F rob_{q_{i}}) has distinct eigenvalues with ratio t ≠ ±1 in F_{p} and q_{i} ≡ t mod p.

2) � _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}) = 0 and � _{S∪Q}^{2}(Ad^{0}ρ̄) = 0.

3) f_{i} |_{G_{q_{j}}} = 0 for i ≠ j, 1 ≤ i ≤ r + s + δ and 1 ≤ j ≤ r.

4) f_{j} |_{G_{q_{j}}} ∉ N_{q_{j}} for 1 ≤ j ≤ r.

5) The inflation map H^{1}(G_{S}, Ad^{0}ρ̄) → H^{1}(G_{S∪Q}, Ad^{0}ρ̄) is an isomorphism. (Proposition 4 of [R2] and its proof.)

Proof: Parts 1) through 4), proved in [R2], use the existence of the element c ∈ Gal(K/Q) in our hypotheses on ρ̄. Only part 5 is not proved in [R2]. Recall r = dim_{k}� _{S}^{2}(Ad^{0}ρ̄) = dim_{k}� _{S}^{1}((Ad^{0}ρ̄)^{∗}). Proposition 1.6 of [W] becomes, after taking L_{v} = H^{1}(G_{v}, Ad^{0}ρ̄) for all v ∈ S ∪ Q,

\frac{#H^{1}(G_{S∪Q}, Ad^{0}ρ̄)}{#� _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗})} = h_{∞} ∏_{v∈S∪Q,v≠∞} h_{v}

where

h_{∞} = \frac{#H^{0}(G_{∞}, (Ad^{0}ρ̄)^{∗}) · #H^{0}(G_{S}, Ad^{0}ρ̄)}{#H^{0}(G_{S}, (Ad^{0}ρ̄)^{∗})}

and

h_{v} = \frac{#H^{2}(G_{v}, Ad^{0}ρ̄)}{#[H^{1}(G_{v}, Ad^{0}ρ̄) : L_{v}]}.

Since dim_{k}H^{2}(G_{v}, Ad^{0}ρ̄) = 1 for the r primes in Q and dim_{k}H^{2}(G_{v}, Ad^{0}ρ̄) = 1 for s primes of S and dim_{k}H^{2}(G_{v}, Ad^{0}ρ̄) = 0 for the remaining primes of S we see ∏_{v∈S∪Q,v≠∞} h_{v} = (#k)^{r+s}. As Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗} are assumed to be absolutely irreducible we see H^{0}(G_{S}, Ad^{0}ρ̄) and H^{0}(G_{S}, (Ad^{0}ρ̄)^{∗}) are trivial. Finally we see dim_{k}H^{0}(G_{∞}, (Ad^{0}ρ̄)^{∗}) = δ where we recall δ = 0 or 2 as ρ̄ is even or odd so h_{∞} = (#k)^{δ} as ρ̄ is even or odd. Assembling all this and keeping in mind that we already know � _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}) is trivial wesee #H^{1}(G_{S∪Q}, Ad^{0}ρ̄) = (#k)^{r+s+δ}. Thus the (necessarily injective) inflation map H^{1}(G_{S}, Ad^{0}ρ̄) → H^{1}(G_{S∪Q}, Ad^{0}ρ̄) is an isomorphism.

\underline{Global considerations} We have ρ̄: G_{S} → GL_{2}(k) where k is a finite field of characteristic p ≥ 7. For the time being we will assume the projective image of ρ̄ is conjugate over the quadratic extension of k to P SL_{2}(k′) or P GL_{2}(k′) where k′ is a subfield of the quadratic extension of k. Thus Lemma 10 applies and the four hypotheses at the beginning of the Recollections section hold.

Recall that for l ∈ S, l ≠ p we have defined N_{l} ⊆ H^{1}(G_{l}, Ad^{0}ρ̄) where dim_{k} (H^{1}(G_{l}, Ad^{0}ρ̄)) /N_{l} = dim_{k}H^{2}(G_{l}, Ad^{0}ρ̄) ≤ 1. We have also defined a family C_{l} of deformations of ρ̄ |_{G_{l}} to W (k). The key property is that N_{l} preserves C_{l}, that is if a ∈ k, g ∈ N_{l} and ρ_{n} is the mod p^{n} reduction of an element of C_{p} then (1 + ap^{n−1}g).ρ_{n} is also the mod p^{n} reduction of an element of C_{l}.We have similarly defined C_{q} and N_{q} for unramified primes q as in Facts 13 and 14. Also, recall for the prime p and most ρ̄ that we have a subspace N_{p} ⊆ H^{1}(G_{p}, Ad^{0}ρ̄) with

dim_{k} (H^{1}(G_{p}, Ad^{0}ρ̄)/N_{p}) = δ + dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄)

where δ = 0 or 2 as ρ̄ is even or odd. The ρ̄ for which we do not have such an N_{p} are those we cannot deform to W (k). If ρ̄ : G_{S} → GL_{2}(k) is even and ρ̄ |_{G_{p}} is an excluded case we do not know how to proceed. Otherwise take N_{p} and C_{p} as in the tables for ρ̄ is even or odd and recall N_{p} preserves C_{p}.

Our aim is to find two finite sets of primes Q and T as in Facts 13 and 14. The set Q will be, as in Fact 16, such that the map H^{2}(G_{S∪Q}, Ad^{0}ρ̄) → ⊕_{v∈S∪Q}H^{2}(G_{v}, Ad^{0}ρ̄) is injective, that is � _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}), and by duality � _{S∪Q}^{2}(Ad^{0}ρ̄) are trivial. The set T will be such that the map

H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄) → ⊕_{v∈S} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v})

is surjective. We will eventually have that the map

H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄) → ⊕_{v∈S∪Q∪T} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v})

is surjective and that � _{S∪Q∪T}^{1} ((Ad^{0}ρ̄)^{∗}) and its dual � _{S∪Q∪T}^{2} (Ad^{0}ρ̄) are trivial. Thus all obstructions to deformation problems (unramified outside S ∪ Q ∪ T) will be local and we will have enough global 1 cohomology classes to remove these local obstructions.

We may think of Q as annihilating a dual Selmer group and T as annihilating the cokernel of a Selmer group map.

Consider ⊕_{v∈S}N_{v} ⊆ ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄).

Lemma 11 Dim_{k} (⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) = s + δ.

Proof: For l ≠ p we have dim_{k}(H^{1}(G_{l}, Ad^{0}ρ̄)/N_{l}) = dim_{k}H^{2}(G_{l}, Ad^{0}ρ̄) which is 0 or 1. For p we see dim_{k}(H^{1}(G_{p}, Ad^{0}ρ̄)/N_{p}) = δ + dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄) where dim_{k}H^{2}(G_{p}, Ad^{0}ρ̄) = 0 or 1. Recalling that ⊕_{v∈S}H^{2}(G_{v}, Ad^{0}ρ̄) has dimension s (essentially by definition) the codimension result follows.

Lemma 12 Recall r = dim_{k}� _{S}^{1}((Ad^{0}ρ̄)^{∗}) = r. Consider the restriction map

Res : H^{1}(G_{S}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄).

Let d = dim_{k}Res^{−1}(⊕_{v∈S}N_{v}). Then d ≥ r.

Proof: Res induces an injection R̃es : H^{1}(G_{S}, Ad^{0}ρ̄)/Res^{−1}(⊕_{v∈S}N_{v}) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}. Since (⊕_{v∈S}N_{v}) is of codimension s+δ in ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄) by Lemma 11 and dim_{k}H^{1}(G_{S}, Ad^{0}ρ̄) = r + s + δ by Lemma 9 we have r + s + δ − d ≤ s + δ so d ≥ r.

Let {f_{1}, f_{2}, .., f_{d}} be a basis of Res^{−1}(⊕_{v∈S}N_{v}). We augment this to a basis {f_{1}, f_{2}, .., f_{r+s+δ}} of H^{1}(G_{S}, Ad^{0}ρ̄).

Recall Q(Ad^{0}ρ̄) is the field fixed by N, the maximal subgroup of G_{S} that acts trivially on Ad^{0}ρ̄, that D = Q(Ad^{0}ρ̄) ∩ Q(μ_{p}) and K = Q(Ad^{0}ρ̄)Q(μ_{p}), that N^{d} is the maximal subgroup of G_{S} that acts trivially on (Ad^{0}ρ̄)^{∗}, and that N^{d} ⊇ Gal(Q_{S}/K). Also recall the four hypotheses on ρ̄ at the beginning of the Recollections section. Since we are assuming the projective image of ρ̄ contains P SL_{2}(k′) for a suitable subfield k′ of k there is, by Lemma 10 an element

c = a × b ∈ Gal(Q(Ad^{0}ρ̄)/D) × Gal(Q(μ_{p})/D) ≃ Gal(K/D)

with properties given at the beginning of the Recollections section.

For i = 1, 2, .., r + s + δ recall L_{i} is the extension of K arising from f_{i} as in Fact 10, and L is the composite of all the L_{i}. Recall also each L_{i} is disjoint over K from the composite of the L_{j} for j ≠ i. Then Gal(L/K) ≃ (Ãd^{0}ρ̄)^{r+s+δ} as k̃[G_{S}] modules. (Recall k̃ is the minimal field of definition of Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗} and Ãd^{0}ρ̄ and (Ãd^{0}ρ̄)^{∗} are descents of Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗} to k̃.) Let Q be the set of r primes whose existence and properties are described in Fact 16.

For r + 1 ≤ i ≤ d take α_{i} to be an element of Gal(L/K) ≃ (Ãd^{0}ρ̄)^{r+s+δ} all of whose entries are 0 except the ith which is a nonzero element on which c acts trivially. Such an element exists by Fact 12. Let T_{i} be those primes not in S ∪ Q with Frobenius in the conjugacy class of

α_{i}× c ∈ Gal(L/K)× Gal(K/D) ⊆ Gal(L/K)× Gal(K/Q) ≃ Gal(L/Q).

The following proposition is the key technical ingredient of this paper that we were not aware of in [R2]. It is motivated by Proposition 2 of section 8.4 of [H] which is due to Neukirch.

Proposition 10 For i = r + 1, .., d the restriction map

θ : H^{1}(G_{S∪Q∪T_{i}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)

is surjective.

Proof: Recall that for a finite Galois module M and a (possibly infinite) set W we define P_{W}^{1} (M) = □_{v∈W} H^{1}(G_{v}, M) to be the restricted direct product with respect to the images H_{nr}^{1}(G_{v}, M) of the of the inflation maps H^{1}(Gal(Q_{v}^{nr}/Q_{v}), M) → H^{1}(G_{v}, M) at those v for which M is unramified. By local duality P_{W}^{1} (M) is dual to P_{W}^{1} (M^{∗}) where M^{∗} is the Cartier dual of M. Here W need not contain p.

Consider the restriction map

θ̃: H^{1}(G_{S∪Q∪T_{i}}, Ad^{0}ρ̄) → P_{S∪Q∪T_{i}}^{1}(Ad^{0}ρ̄).

By global Tate-Poitou duality (Fact 3) the image of θ̃ is the exact annihilator of the image of the restriction map

ϑ̃: H^{1}(G_{S∪Q∪T_{i}}, (Ad^{0}ρ̄)^{∗}) → P_{S∪Q∪T_{i}}^{1}((Ad^{0}ρ̄)^{∗}).

The annihilator in P_{S∪Q∪T_{i}}^{1}((Ad^{0}ρ̄)^{∗}) of (P_{Q∪T_{i}}^{1}(Ad^{0}ρ̄) + Image(θ̃)) is the intersection of the annihilators of these two summands. (All spaces are closed). This consists of the set {ϑ̃(x)}for x ∈ H^{1}(G_{S∪Q∪T_{i}}, (Ad^{0}ρ̄)^{∗}) such that x |_{G_{v}}= 0 for all v ∈ Q ∪ T_{i}.

We show all such x are trivial. Suppose there exists a nonzero such x. As in part 2) of Fact 10, we see that x gives rise to an extension M̃/K with Gal(M̃/K) ≃ (Ãd^{0}ρ̄)^{∗} as k̃[G_{S}] modules. (The k̃ structure is inherited from (Ãd^{0}ρ̄)^{∗} as in Fact 10 and we are using the hypothesis on ρ̄ that H^{1}(G_{S}/N^{d}, (Ad^{0}ρ̄)^{∗}) = 0.) By Fact 10 we have M̃ ∩ L = K and the exact sequence

1 → Gal(M̃/K) → Gal(M̃/Q) → Gal(K/Q) → 1

splits so

Gal(LM̃/Q) ≃ (Gal(L/K) × Gal(M̃/K)) × Gal(K/Q).

[figure: Q; K; M̃; L; LM̃]

A nonzero ζ ∈ Gal(M̃/K) on which c ∈ Gal(K/Q) acts trivially exists by Fact 12. Let T̃_{i} be those primes not in S ∪ Q with Frobenius in the conjugacy class of (α_{i} × ζ)× c in

(Gal(L/K) × Gal(M̃/K)) × Gal(K/Q) ≃ Gal(LM̃/K)× Gal(K/Q) ≃ Gal(LM̃/Q).

Note (α_{i} × ζ)× c projects to α_{i}× c ∈ Gal(L/K) so T̃_{i} is a nonempty (infinite) subset of T_{i} and that for v ∈ T̃_{i} we have x |_{G_{v}}≠ 0 by Fact 15. But x ∈ H^{1}(G_{S∪Q∪T_{i}}, (Ad^{0}ρ̄)^{∗}) was chosen so x |_{G_{v}}= 0 for all v ∈ Q ∪ T_{i}. This contradiction shows x = 0 so the annihilator in P_{S∪Q∪T_{i}}^{1}((Ad^{0}ρ̄)^{∗}) of (P_{Q∪T_{i}}^{1}(Ad^{0}ρ̄) ⊕ Image(θ̃)) is trivial. Thus we have

(P_{Q∪T_{i}}^{1}(Ad^{0}ρ̄) + Image(θ̃)) = P_{S∪Q∪T_{i}}^{1}(Ad^{0}ρ̄).

Since the cokernel of θ is the cokernel of the composite map

H^{1}(G_{S∪Q∪T_{i}}, Ad^{0}ρ̄) →^{θ̃} P_{S∪Q∪T_{i}}^{1}(Ad^{0}ρ̄) → \frac{P_{S∪Q∪T_{i}}^{1}(Ad^{0}ρ̄)}{P_{Q∪T_{i}}^{1}(Ad^{0}ρ̄)} = ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)

the cokernel of θ is

\frac{P_{S∪Q∪T_{i}}^{1}(Ad^{0}ρ̄)}{P_{Q∪T_{i}}^{1}(Ad^{0}ρ̄) + Image(θ̃)} = 0.

We see θ is surjective.

Lemma 13 For i = r + 1, .., d let ℘_{i} ∈ T_{i}. Then for r + 1 ≤ m ≤ d

dim_{k}H^{1}(G_{S∪Q∪{℘_{r+1},..,℘_{m}}}, Ad^{0}ρ̄) = m − r + r + s + δ = m + s + δ.

Proof: We mimic the proof of Fact 16, part 5). The roles there of S and S ∪ Q are played here by S ∪ Q and S ∪ Q ∪ {℘_{r+1}, .., ℘_{m}} respectively. Since � _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}) is trivial it follows that � _{S∪Q∪{℘_{1},..,℘_{m}}}^{1}((Ad^{0}ρ̄)^{∗}) = 0. The rest of the proof is identical to that of Fact 16, part 5). We use that ℘_{i} is a prime satisfying the hypotheses of Fact 13 (see Fact 16) so #H^{2}(G_{℘_{i}}, Ad^{0}ρ̄) = #k.

Lemma 14 For i = r + 1, .., d there exist primes ℘_{i} ∈ T_{i} such that the map

H^{1}(G_{S∪Q∪{℘_{r+1},..,℘_{d}}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is onto.

Proof: Throughout we assume d > r. If d = r then the statements are vacuously true. Recall the inflation map H^{1}(G_{S}, Ad^{0}ρ̄) → H^{1}(G_{S∪Q}, Ad^{0}ρ̄) is an isomorphism by Fact 16, part 5) and the these cohomology groups have dimension r + s + δ. Consider the restriction map

H^{1}(G_{S}, Ad^{0}ρ̄) = H^{1}(G_{S∪Q}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}.

By definition the kernel is dimension d so the image has dimension r + s + δ − d. The target space has dimension s + δ by Lemma 11.

By Proposition 10 the map

H^{1}(G_{S∪Q∪T_{r+1}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is onto. Since (⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) is finite dimensional, there is a finite subset D_{r+1} of T_{r+1} such that the map

H^{1}(G_{S∪Q∪D_{r+1}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is onto. There is a (nonunique) prime, call it ℘_{r+1}, in D_{r+1} such that the image of

H^{1}(G_{S∪Q∪{℘_{r+1}}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is of dimension r + s + δ − d + 1. That the increase in the dimension of the image is exactly one follows from Lemma 13. If no such prime existed then the image of H^{1}(G_{S∪Q∪D_{r+1}}, Ad^{0}ρ̄) in ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v} would be r + s + δ − d dimensional. This contradicts the surjectivity.

Now consider the map H^{1}(G_{S∪Q∪{℘_{r+1}}∪T_{r+2}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}. It is also surjective by Proposition 10. As above we can find a prime ℘_{r+2} ∈ T_{r+2} such that the dimension of the image of H^{1}(G_{S∪Q∪{℘_{r+1},℘_{r+2}}}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v} is r + s + δ − d + 2.

Continuing in this fashion the Lemma is proved.

Denote by T the set {℘_{r+1}, .., ℘_{d}} of Lemma 14. Augment the basis {f_{1}, .., f_{r+s+δ}} of

H^{1}(G_{S}, Ad^{0}ρ̄) = H^{1}(G_{S∪Q}, Ad^{0}ρ̄)

to a basis {f_{1}, .., f_{d+s+δ}} of H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄).

Lemma 15 Recall from Lemma 12 that d ≥ r and {f_{1}, .., f_{d}} is a basis of Res^{−1}(⊕_{v∈S}N_{v}).

1) For 1 ≤ i ≤ r we have f_{i} |_{G_{v}}= 0 for all v ∈ Q ∪ T except v = q_{i} ∈ Q. Also f_{i} |_{G_{q_{i}}} ∉ N_{q_{i}}.

2) For r + 1 ≤ i ≤ d we have f_{i} |_{G_{v}}= 0 for all v ∈ Q ∪ T except v = ℘_{i} ∈ T. Also f_{i} |_{G_{℘_{i}}} ∉ N_{℘_{i}}.

3) Let X be the k vector space spanned by {f_{d+1}, .., f_{d+s+δ}}. Then the restriction map

X → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is an isomorphism.

Proof: The first two parts follow from Fact 15. Recall in the comments following Lemma 12 the kernel of the map

H^{1}(G_{S}, Ad^{0}ρ̄) = H^{1}(G_{S∪Q}, Ad^{0}ρ̄) → ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}

is the space spanned by {f_{1}, .., f_{d}}. Part 3) now follows by Lemma 14 and counting dimensions.

Lemma 16 The map H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄) → ⊕_{v∈S∪Q∪T} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) is an isomorphism.

Proof: Thus we may say the cokernel of the ‘Selmer group map’ is trivial. Recall {f_{1}, .., f_{d+s+δ}} is a basis of H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄). We separate this basis into three groups, some of which may be empty.

• Group I - {f_{1}, .., f_{r}}

• Group II - {f_{r+1}, .., f_{d}}

• Group III - {f_{d+1}, .., f_{d+s+δ}}

By Lemma 15, part 3) the span of group III elements under restriction projects isomorphically to ⊕_{v∈S}H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}.

By Lemma 15, part 2) the span of group II elements maps under restriction isomorphically to ⊕_{v∈T} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}). This is because the v ∈ T are as in Fact 14. Thus dim_{k}(H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) = 1, and for all i we have f_{i} |_{G_{v}}= 0 for all v ∈ Q ∪ T except v = ℘_{i}. Furthermore for r + 1 ≤ i ≤ d we have f_{i} |_{G_{℘_{i}}} ∉ N_{℘_{i}}.

Similarly, the span of group I elements maps isomorphically to ⊕_{v∈Q} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) by Lemma 15, part 1). Thus the map in question has trivial kernel. By counting dimensions we are done.

Recall that N and N^{d} are the maximal subgroups of G_{S} that act trivially on Ad^{0}ρ̄ and (Ad^{0}ρ̄)^{∗} respectively. Thus N fixes Q(Ad^{0}ρ̄) and N^{d} ⊇ Gal(Q_{S}/K). Recall that if

dim_{k}� _{S}^{2}(Ad^{0}ρ̄) = r = dim_{k}� _{S}^{1}((Ad^{0}ρ̄)^{∗})

then there is a set Q of primes {q_{1}, .., q_{r}} such that dim_{k}� _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}) = 0 with the q_{i} as in Facts 13 and 16. We have the field diagram below.

[figure: Q; D; Q(μ_{p}); Q(Ad^{0}ρ̄); K; Q(ρ̄)]

Theorem 1 Let k be a finite field of characteristic p ≥ 7and ρ̄ : G_{Q} → GL_{2}(k). Let χ be the cyclotomic character.

a) If ρ̄ is even and ρ̄ |_{G_{p}} is not twist equivalent to( \begin{matrix}χ & 0 \\ 0 & 1 \end{matrix} ) or twist equivalent to the indecompos-

ablerepresentation ( \begin{matrix}χ^{p−2} & ∗ \\ 0 & 1 \end{matrix} ) then there is a finite set R of primes containing p and infinity and

a deformation ρ : G_{R} → GL_{2}(W (k)) of ρ̄.

b) If ρ̄ is odd then there is a finite set R of primes containing p and infinity and a deformation ρ : G_{R} → GL_{2}(W (k)) of ρ̄. Suppose further that ρ̄ |_{G_{p}} is not twist equivalent to the trivial repre-

sentation or the indecomposable unramified representation given by ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ). If ρ̄ |_{G_{p}} is a twist of

an ordinary representation then, up to a twist by a character of finite order, ρ can be taken to be ordinary. If inertia at p acts via fundamental characters of level 2 then, up to a twist by a character of finite order, ρ can be taken to be potentially crystalline.

Proof: If the order of the image of ρ̄ is prime to p we may take ρ to be the Teichmüller lift of ρ̄. This is independent of the parity of ρ̄ or whether weare in an excluded case. In this case ρ is ramified at exactly the same primes as ρ̄ so we may take R = S.

In our local computations we have taken the liberty of twisting ρ̄ |_{G_{v}} by a character of finite order. This only affects us insofar as the ρ we construct (in the odd cases where C_{p} consists of ordinary at p or crystalline at p deformations of ρ̄ to W (k)) is ordinary or crystalline only after twisting by a character of finite order. Thus in these cases ρ will be potentially ordinary or crystalline. In the even cases that we can handle ρ is simply some deformation of ρ̄ to W (k).

If the image of ρ̄ contains an element of order p, then since p ≥ 7 we see from section 260 of [D] that the image of ρ̄ is either contained in a Borel subgroup of GL_{2}(k) or the projective image of ρ̄ is conjugate over the quadratic extension of k to either P GL_{2}(k′) or P SL_{2}(k′) for some subfield k′ of the quadratic extension of k. In the Borel case Theorem 2 of [Kh1] provides us with a reducible potentially semistable deformation of ρ̄ to W (k) independent of the parity of ρ̄ or whether we are in an excluded case. This ρ is unramified at all primes outside of R, a set of primes containing S.

In the remaining cases Lemma 10 applies and the four conditions at the beginning of the Recollections section hold. We are in a position to use the results we have developed in this section. Here we will take R = S ∪ Q ∪ T where Q and T are as in Lemmas 15 and 16.

Note that since � _{S∪Q}^{1}((Ad^{0}ρ̄)^{∗}) = 0 we have � _{S∪Q∪T}^{1} ((Ad^{0}ρ̄)^{∗}) = 0 so � _{S∪Q∪T}^{2} (Ad^{0}ρ̄) = 0 by Tate-Poitou duality. (The set Q of primes was chosen to annihilate a ‘dual Selmer group’.)

By Chapter I, theorem 4.10 of [Mi], using that H^{0}(G_{S}, (Ad^{0}ρ̄)^{∗}) = 0 by absolute irreducibility of (Ad^{0}ρ̄)^{∗}, we see the map H^{2}(G_{S∪Q∪T} , Ad^{0}ρ̄) → ⊕_{v∈S∪Q∪T} H^{2}(G_{v}, Ad^{0}ρ̄) is an isomorphism.

We prove the Theorem by induction as in Theorem 1 of [R2]. Recall, for ρ̄ odd or even, we have classes C_{v} of deformations of ρ̄ to W (k) for primes v ∈ S. We also have classes C_{v} of deformations to W (k) for primes v ∈ Q ∪ T. See the discussion following Fact 13. Note that ρ̄ is (trivially) the mod p reduction of an element of C_{v} for all primes v ∈ S ∪ Q ∪ T. This is the base case of our induction.

Suppose then we have a deformation ρ_{n} : G_{S∪Q∪T} → GL_{2}(W (k)/p^{n}) of ρ̄ such that ρ_{n} |_{G_{v}} is the mod p^{n} reduction of an element of C_{v} for all v ∈ S ∪ Q ∪ T, that is ρ_{n} |_{G_{v}}∈ C_{v} for all v ∈ S ∪ Q ∪ T. To complete the induction we must find a deformation ρ_{n+1} : G_{S∪Q∪T} → GL_{2}(W (k)/p^{n+1}) of ρ_{n} such that ρ_{n+1} |_{G_{v}}∈ C_{v} for all v ∈ S ∪ Q ∪ T. Since ρ_{n} is the mod p^{n} reduction of an element of C_{v} for all v ∈ S ∪ Q ∪ T we see the local obstructions to deforming to W (k)/p^{n+1} all vanish. As � _{S∪Q∪T}^{2} (Ad^{0}ρ̄) = 0 the global obstruction vanishes and we see there exists a deformation ρ̃: G_{S∪Q∪T} → GL_{2}(W (k)/p^{n+1}) ofρ_{n} to W (k)/p^{n+1}. It remains to adjust ρ̃ by a suitable element of H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄) so that this adjustment is the mod p^{n+1} reduction of an element of C_{v} for all v ∈ S ∪ Q ∪ T.

For any v ∈ S∪Q∪T we know there exist f_{v} ∈ H^{1}(G_{v}, Ad^{0}ρ̄) such that (1+p^{n}f_{v}).ρ̃ |_{G_{v}}∈ C_{v} where f_{v} is defined up to an element of N_{v}. By Lemma 16 there is an f ∈ H^{1}(G_{S∪Q∪T} , Ad^{0}ρ̄) such that f and ⊕_{v∈S∪Q∪T} f_{v} have the same image in ⊕_{v∈S∪Q∪T} (H^{1}(G_{v}, Ad^{0}ρ̄/N_{v}). Thus (1 + p^{n}f).ρ̃ |_{G_{v}}∈ C_{v} for all v ∈ S ∪ Q ∪ T. The induction is complete.

We have thus constructed ρ : G_{R} → GL_{2}(W (k)), a deformation of ρ̄. If ρ̄ is odd and is not an excluded case as in part b) of the theorem, then ρ |_{G_{p}}∈ C_{p} is, up to a finite twist, ordinary or crystalline as described. In all these nonexcluded cases ρ is potentially semistable at p by Proposition 9.

In the introduction we remarked that the ρ of Theorem 1 is unique, that is when we insist

that, up to a twist, ρ |_{G_{q}}= ( \begin{matrix}χ & ∗ \\ 0 & 1 \end{matrix} ) for q ∈ R − S then ρ is unique. (Note that ρ |_{G_{q}} may be

decomposable). This uniqueness follows immediately from the two isomorphisms

H^{2}(G_{R}, Ad^{0}ρ̄) → ⊕_{v∈R}H^{2}(G_{v}, Ad^{0}ρ̄)

and

H^{1}(G_{R}, Ad^{0}ρ̄) → ⊕_{v∈R} (H^{1}(G_{v}, Ad^{0}ρ̄)/N_{v}) .

Indeed, the kernel of the latter map is the (reduced) tangent space for the deformation functor where we insist our deformation of ρ̄ be unramified outside R, minimally ramified in the sense of [Di1] at primes l ∈ S, l ≠ p, and of the form described at the beginning of this paragraph for q ∈ R − S. Finally, for p we insist our deformation be of type C_{p}. (Note this is no restriction at all if ρ̄ is even). Since the kernel of this map is trivial our restricted deformation ring is simply W (k).

References

[Bö1] Böckle, G. A local-to-global principle for deformations of Galois representations, Journal für die reine und angewandte Mathematik, Band \underline{509}, 1999 p. 199-236.

[Bö2] Böckle, G. Demŭskin groups with group actions and applications to deformations of local Galois representations, preprint.

[B1] Boston, N. Deformation Theory of Galois Representations, Harvard Ph.D Thesis, 1987.

[B2] Boston, N. Explicit deformation of Galois Representations, Inventiones Mathematicae \underline{103}, 1991 p. 181-196.

[B3] Boston, N. Families of Galois Representation- Increasing the Ramification, Duke Math Journal \underline{103}, 1990 p. 357-367.

[BM] Boston, N., Mazur, B. Explicit Universal Deformations of Galois Representations, Advanced Studies in Pure Mathematics 1989.

[BCDT] Breuil, C., Conrad, B., Diamond, F., Taylor, R., On the Modularity of Elliptic Curves over Q, or Wild 3-adic exercises, preprint.

[CDT] Conrad, B., Diamond, F., Taylor, R., Modularity of certain potentially Barsotti-Tate Galois representations, Journal of the AMS \underline{12} no. 2, 1999 p. 521-567.

[DDT] Darmon, H., Diamond, F., Taylor, R. Fermat’s Last Theorem, in Current Developments in Mathematics, 1995, Bott, R. etal. ed.

[deS] de Shalit, E. Hecke Rings and Universal Deformation Rings, in Modular Forms and Fermat’s Last Theorem. Cornell, G., Silverman, J. and Stevens, G. ed.

[Di1] Diamond, F. An Extension of Wiles’ Results in Modular Forms and Fermat’s Last Theorem, Cornell, G., Silverman, J. and Stevens, G. ed.

[Di2] Diamond, F. The refined conjecture of Serre, in Elliptic Curves, Modular Forms, and Fermat’s Last Theorem, Coates, J. and Yau, S.T. ed.

[D] Dickson, L. E. Linear Groups, B. G. Teubner 1901.

[Fl] Flach, M. A finiteness theorem for the symmetric square of an elliptic curve, Inventiones Mathematicae \underline{109}, 1992 p. 307-327.

[Fo] Fontaine, J-M., Représentations p-adiques semi-stables, Astérisque \underline{223}, 1994 p. 113-184.

[FL] Fontaine, J.-M., Laffaille, G. Construction de représentations p-adiques, Ann. Sci. École Norm. Sup. (4) \underline{15} 1982 p. 547-608. of

[FM] Fontaine, J-M., Mazur, B. Geometric Galois Representations, in Elliptic Curves, Modular Forms, and Fermat’s Last Theorem, Coates, J. and Yau, S.T. ed.

[H] Haberland, K. Galois Cohomology of Algebraic Number Fields VEB Deutscher Verlag der Wissenschaften 1978.

[Kh1] Khare, C. Base Change, Lifting and Serre’s Conjecture, Journal of Number Theory \underline{63} 1997 p. 387-395.

[Kh2] Khare, C. Mod p Descent for Hilbert Modular Forms, preprint.

[M1] Mazur, B. Deforming Galois Representations, In: Proceedings of the March 1987 Workshop on “Galois Groups over Q” held at MSRI, Berkeley, California, Y. Ihara, K. Ribet, J.-P. Serre, eds.

[M2] Mazur, B. An Introduction to the Deformation Theory of Galois Representations in Modular Forms and Fermat’s Last Theorem, Cornell, G., Silverman, J. and Stevens, G. ed.

[Me] Mestre, J.-F. Construction d’extensions régulières de Q(T) à groupe de Galois SL_{2}(F_{7}) et M̃_{12}, C.R. Acad. Sci. Paris \underline{319} 1994 p. 781-782.

[Mi] Milne, J. Arithmetic Duality Theorems, Academic Press, Inc. 1986.

[PR] Perrin-Riou, B. Représentations p-adiques ordinaires, In Périodes p-adiques, Astérisque \underline{223}, 1994 p. 185-207.

[R1] Ramakrishna, R. On a variation of Mazur’s deformation functor, Compositio Mathematica, \underline{87}. 1993 p. 269-286.

[R2] Ramakrishna, R. Lifting Galois representations, Inventiones Mathematicae, \underline{138}. 1999 p. 537-562.

[Ri] Ribet, K. on modular representations of Gal(Q̄/Q) arising from modular forms, Inventiones Mathematicae \underline{100}, 1990 p. 431-476.

[Se1] Serre, J.-P. Sur les Représentations Modulaires de Degré 2 de Gal(Q̄/Q), Duke Mathematical Journal \underline{54}, 1987 p. 179-230.

[Se2] Serre, J.-P. Cohomologie Galoisienne, Lecture Notes in Mathematics No. 5 Cinquième édition, révisée et complétée, Springer-Verlag, 1994.

[Se3] Serre, J.-P., Abelian l-adic representations and elliptic curves, Benjamin, 1968.

[TW] Taylor, R. and Wiles, A. Ring-theoretic properties of certain Hecke algebras, Annals of Mathematics (2) \underline{141}, 1995 no. 3, p. 553-572.

[W] Wiles, A. Modular Elliptic Curves and Fermat’s Last Theorem, Annals of Mathematics (2) \underline{141}, 1995 no. 3, p. 443-551.

[ZM] Zeh-Marschke, A. SL_{2}(Z/7) als Galoisgruppe über Q, unpublished note.