ANDREW JOHN WILES with residue field k. The universal representation associated to po is defined over Rl and the universal property of R then defines a map R- R1. So we obtain a section to the map R(Po)R W(k') and the map is therefore W(k) an isomorphism.(I am grateful to Faltings for this observation. )We will also need to extend the consideration of O-algebras tp the restricted cases. In each case we can require a to be an O-algebra and again it is easy to see tha Bs⑧ O in each W(k) The second generalization concerns primes q+p which are ramified in Po We distinguish three special cases(types(A)and(C)need not be disjoint) (A)PolD=(l x2)for a suitable choice of basis, with xI and x2 unramified, x1x2=w and the fixed space of Ig of dimension 1, (B)Polls=(01),Xf 1, for a suitable choice of basis (C)H(Qq, W)=0 where Wa is as defined in(1.6) Then in each case we can define a suitable deformation theory by imposing additional restrictions on those we have already considered, namely: (A)PlDa =(i a)for a suitable choice of basis of A with v1 and v2 un ramified and 1v,2=E: (B)Plla =(0 1) for a suitable choice of basis(xg of order prime to p, so the same character as above) (C)det pll, det PolL,, i.e., of order prime to p Thus if M is a set of primes in 2 distinct from p and each satisfying one of (A),(B)or(C)for Po, we will impose the corresponding restriction at each prime in M unrestricted, we can associate a deformation theory to po provided ord. Thus to each set of data D=[, 2,0, M where is Se,str fat or (1.3) po:Gal(Qx/Q)→GL2(k) is itself of type D and O is the ring of integers of a totally ramified extension of W(k); po is ordinary if. is Se or ord, strict if. is strict and fat if. is fl (meaning fat); po is of type M, i.e., of type(A),(B)or(C)at each ramified primes q+p, q E M. We allow different types at different q's. We will refer to these as the standard deformation theories and write rp for the universal ring associated to D and Pp for the universal deformation(or even p if D is clear from the context) We note here that if d=(ord, 2,0, M) and D=(Se, 2,0, M)then there is a simple relation between RD and RD/. Indeed there is a natural map
458 ANDREW JOHN WILES with residue field k. The universal representation associated to ρ 0 is defined over R1 and the universal property of R then defines a map R → R1. So we obtain a section to the map R(ρ 0) → R ⊗ W(k) W(k ) and the map is therefore an isomorphism. (I am grateful to Faltings for this observation.) We will also need to extend the consideration of O-algebras tp the restricted cases. In each case we can require A to be an O-algebra and again it is easy to see that R· Σ,O R· Σ ⊗ W(k) O in each case. The second generalization concerns primes q = p which are ramified in ρ0. We distinguish three special cases (types (A) and (C) need not be disjoint): (A) ρ0|Dq = ( χ1 ∗ χ2 ) for a suitable choice of basis, with χ1 and χ2 unramified, χ1χ−1 2 = ω and the fixed space of Iq of dimension 1, (B) ρ0|Iq = ( χq 0 0 1 ), χq = 1, for a suitable choice of basis, (C) H1(Qq, Wλ) = 0 where Wλ is as defined in (1.6). Then in each case we can define a suitable deformation theory by imposing additional restrictions on those we have already considered, namely: (A) ρ|Dq = ( ψ1 ∗ ψ2 ) for a suitable choice of basis of A2 with ψ1 and ψ2 unramified and ψ1ψ−1 2 = ε; (B) ρ|Iq = ( χq 0 0 1 ) for a suitable choice of basis (χq of order prime to p, so the same character as above); (C) det ρ|Iq = det ρ0|Iq , i.e., of order prime to p. Thus if M is a set of primes in Σ distinct from p and each satisfying one of (A), (B) or (C) for ρ0, we will impose the corresponding restriction at each prime in M. Thus to each set of data D = {·, Σ, O,M} where · is Se, str, ord, flat or unrestricted, we can associate a deformation theory to ρ0 provided (1.3) ρ0 : Gal(QΣ/Q) → GL2(k) is itself of type D and O is the ring of integers of a totally ramified extension of W(k); ρ0 is ordinary if · is Se or ord, strict if · is strict and flat if · is fl (meaning flat); ρ0 is of type M, i.e., of type (A), (B) or (C) at each ramified primes q = p, q ∈ M. We allow different types at different q’s. We will refer to these as the standard deformation theories and write RD for the universal ring associated to D and ρD for the universal deformation (or even ρ if D is clear from the context). We note here that if D = (ord, Σ, O,M) and D = (Se, Σ, O,M) then there is a simple relation between RD and RD . Indeed there is a natural map
MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM RD- RD by the universal property of RD, and its kernel is a principal ideal generated by T=E(r det pp(a-l where y E Gal(Qz/Q) is any element whose restriction to Gal(Qoo/Q)is a generator(where Qoo is the Zp-extension of Q) and whose restriction to Gal(Q(SN)/Q)is trivial for any N prime to p with N E Q>, SN being a primitive Nth root of 1 (1.4) Rn/T≈RD It turns out that under the hypothesis that po is strict, i.e. that polD is not associated to a finite fat group scheme, the deformation problems in (i)(a) and (i)(c) are the same; i. e, every Selmer deformation is already a strict deformation. This was observed by Diamond. the argument is local, so the decomposition group Dp could be replaced by Gal(Qp/Q) PROPOSITION 1.1(Diamond). Suppose that T: D, -GL2(A)is a con tinuous representation where A is an Artinian local ring with residue field k, a finite field of characteristic p. Suppose TN(1%) with xI and x2 unramifie and x1 +x2. Then the residual representation i is associated to a finite flo group scheme over Zp Proof(taken from Dia, Prop. 6.1). We may replace T by T x2 and we let 4=X1X2. Then T =(o 1) determines a cocycle t: Dp -M(1) where M is a free A-module of rank one on which Dp acts via Let u denote the cohomology class in H(Dp, M(1)) defined by t, and let uo denote its image in H(Dp, Mo(1)where Mo= M/mM. Let G=ker p and let F be the fixed field of G(so F is a finite unramified extension of Qp). Choose n so that p"A =0. Since H-(G, Hpr-H(G, Aps)is injective for r s, we see that the natural map of A(Dp/G-modules H(G, Hpn Oz, M)-++H(G, M(1)is an isomorphism By Kummer theory, we have H(G, M()EFX/(FX)POz. M as D-module Tow consider the commutative diagram H1(G,M(1)2(FX/(FX)P”8znM)—MDp H(G, Mo(1))-(FX/(FX)P)F, Mo Mo where the right-hand horizontal maps are induced by Up: FX-Z If +1 then Mp C mM, so that the element res uo of H(G, Mo(1) is in the image of(OF/(OF)P)oF, Mo. But this means that i is "peu ramifies"in the sense of Se] and therefore T comes from a finite flat group scheme.( See El,(8.20.) Remark. Diamond also observes that essentially the same proof shows that if T: Gal(Q/Q)-GL2(A), where A is a complete local Noetherian
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 459 RD → RD by the universal property of RD, and its kernel is a principal ideal generated by T = ε−1(γ) det ρD(γ) − 1 where γ ∈ Gal(QΣ/Q) is any element whose restriction to Gal(Q∞/Q) is a generator (where Q∞ is the Zp-extension of Q) and whose restriction to Gal(Q(ζNp )/Q) is trivial for any N prime to p with ζN ∈ QΣ, ζN being a primitive Nth root of 1: (1.4) RD/T R D. It turns out that under the hypothesis that ρ0 is strict, i.e. that ρ0|Dp is not associated to a finite flat group scheme, the deformation problems in (i)(a) and (i)(c) are the same; i.e., every Selmer deformation is already a strict deformation. This was observed by Diamond. the argument is local, so the decomposition group Dp could be replaced by Gal(Q¯ p/Q). Proposition 1.1 (Diamond). Suppose that π : Dp → GL2(A) is a continuous representation where A is an Artinian local ring with residue field k, a finite field of characteristic p. Suppose π ≈ ( χ1ε 0 ∗ χ2 ) with χ1 and χ2 unramified and χ1 = χ2. Then the residual representation π¯ is associated to a finite flat group scheme over Zp. Proof (taken from [Dia, Prop. 6.1]). We may replace π by π ⊗ χ−1 2 and we let ϕ = χ1χ−1 2 . Then π ∼= ( ϕε 0 t 1 ) determines a cocycle t : Dp → M(1) where M is a free A-module of rank one on which Dp acts via ϕ. Let u denote the cohomology class in H1(Dp, M(1)) defined by t, and let u0 denote its image in H1(Dp, M0(1)) where M0 = M/mM. Let G = ker ϕ and let F be the fixed field of G (so F is a finite unramified extension of Qp). Choose n so that pnA = 0. Since H2(G, µpr → H2(G, µps ) is injective for r ≤ s, we see that the natural map of A[Dp/G]-modules H1(G, µpn ⊗Zp M) → H1(G, M(1)) is an isomorphism. By Kummer theory, we have H1(G, M(1)) ∼= F ×/(F ×)pn ⊗Zp M as Dp-modules. Now consider the commutative diagram H1(G, M(1))Dp ∼ −−−−→((F ×/(F ×)pn ⊗Zp M)Dp−−−−→MDp , H1(G, M0(1)) ∼ −−−−→ (F ×/(F ×)p) ⊗Fp M0 −−−−→ M0 where the right-hand horizontal maps are induced by vp : F × → Z. If ϕ = 1, then MDp ⊂ mM, so that the element res u0 of H1(G, M0(1)) is in the image of (O× F /(O× F )p) ⊗Fp M0. But this means that ¯π is “peu ramifi´e” in the sense of [Se] and therefore ¯π comes from a finite flat group scheme. (See [E1, (8.20].) Remark. Diamond also observes that essentially the same proof shows that if π : Gal(Q¯ q/Qq) → GL2(A), where A is a complete local Noetherian
ANDREW JOHN WILES ring with residue field k, has the form lIs e(01) with i ramified then T is of type(A) Globally, Proposition 1.1 says that if Po is strict and if D=( Se, 2,O, M) and D=(str, 2, O, M)then the natural map Rp- RD, is an isomorphism In each case the tangent space of Rp may be computed as in Mal]. Let a be a uniformizer for O and let UA k be the representation space for po (The motivation for the subscript A will become apparent later. Let va be the representation space of Gal( Qz/Q)on Adpo= Homk (UA, U)c M2(k). Then there is an isomorphism of k-vector spaces(cf. the proof of Prop. 1.2 below) (15) Homk(mp/(mD, A),k)cHD(Qz/Q,V) where HD(Qz/Q, VA)is a subspace of H(Q=/Q, VA) which we now describe and mD is the maximal ideal of RcalD. It consists of the cohomology classes which satisfy certain local restrictions at p and at the primes in M. We call mp/mp, A)the reduced cotangent space of RD We begin with p. First we may write(since p# 2), as k gal(Qx/Q)]- module (1.6) VA=WA ek, where WA=f E Homk UA, UA): tracef=0J (Sym2@ det-)po and k is the one-dimensional subspace of scalar multiplications. Then if Po is ordinary the action of D, on Ua induces a filtration of Ua and also on WA and VA. Suppose we write these 0CURC UA,OCWoCWC WAand OCVC V C VA. Thus UA is defined by the requirement that Dp act on it via the character X1 (cf.(1.))and on UA/UA via X2. For W the filtrations are defined by ={f∈Wx:f(UQ)cU}, ={f∈Wk:f=0onU} and the filtrations for VA are obtained by replacing w by V. We note that these filtrations are often characterized by the action of D,. Thus the action of Dp on wo is via x1/x2; on w/wo it is trivial and on Qx/wA it is via X2/x1. These determine the filtration if either x1/x2 is not quadratic or polD is not semisimple. We define the k-vector spaces voa=fev: f=0 in Hom(UA/UA,U/UV) HSe(Qp, va)= ker(H(Qp, V)-H(Qp , VA/w), Hord(Qp, V)=ker(h(QI Hstr(Qp, Vi)=ker(h(Qp, V)h(Qp, W/W)e H(Qo,k))
460 ANDREW JOHN WILES ring with residue field k, has the form π|Iq ∼= ( 1 0 ∗ 1 ) with ¯π ramified then π is of type (A). Globally, Proposition 1.1 says that if ρ0 is strict and if D = (Se, Σ, O,M) and D = (str, Σ, O,M) then the natural map RD → RD is an isomorphism. In each case the tangent space of RD may be computed as in [Ma1]. Let λ be a uniformizer for O and let Uλ k2 be the representation space for ρ0. (The motivation for the subscript λ will become apparent later.) Let Vλ be the representation space of Gal(QΣ/Q) on Adρ0 = Homk(Uλ, Uλ) M2(k). Then there is an isomorphism of k-vector spaces (cf. the proof of Prop. 1.2 below) (1.5) Homk(mD/(m2 D, λ), k) H1 D(QΣ/Q, Vλ) where H1 D(QΣ/Q, Vλ) is a subspace of H1(QΣ/Q, Vλ) which we now describe and mD is the maximal ideal of RC alD. It consists of the cohomology classes which satisfy certain local restrictions at p and at the primes in M. We call mD/(m2 D, λ) the reduced cotangent space of RD. We begin with p. First we may write (since p = 2), as k[Gal(QΣ/Q)]- modules, (1.6) Vλ = Wλ ⊕ k, where Wλ = {f ∈ Homk(Uλ, Uλ) : tracef = 0} (Sym2 ⊗ det−1)ρ0 and k is the one-dimensional subspace of scalar multiplications. Then if ρ0 is ordinary the action of Dp on Uλ induces a filtration of Uλ and also on Wλ and Vλ. Suppose we write these 0 ⊂ U0 λ ⊂ Uλ, 0 ⊂ W0 λ ⊂ W1 λ ⊂ Wλ and 0 ⊂ V 0 λ ⊂ V 1 λ ⊂ Vλ. Thus U0 λ is defined by the requirement that Dp act on it via the character χ1 (cf. (1.2)) and on Uλ/U0 λ via χ2. For Wλ the filtrations are defined by W1 λ = {f ∈ Wλ : f(U0 λ) ⊂ U0 λ}, W0 λ = {f ∈ W1 λ : f = 0 on U0 λ}, and the filtrations for Vλ are obtained by replacing W by V . We note that these filtrations are often characterized by the action of Dp. Thus the action of Dp on W0 λ is via χ1/χ2; on W1 λ/W0 λ it is trivial and on Qλ/W1 λ it is via χ2/χ1. These determine the filtration if either χ1/χ2 is not quadratic or ρ0|Dp is not semisimple. We define the k-vector spaces V ord λ = {f ∈ V 1 λ : f = 0 in Hom(Uλ/U0 λ, Uλ/U0 λ)}, H1 Se(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qunr p , Vλ/W0 λ)}, H1 ord(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qunr p , Vλ/V ord λ )}, H1 str(Qp, Vλ) = ker{H1(Qp, Vλ) → H1(Qp, Wλ/W0 λ) ⊕ H1(Qunr p , k)}.
MODULAR ELLIPTIC CURVES AND FERMATS LAST THEOREM In the Selmer case we make an analogous definition for HSe( Qp, WA) placing va by WA, and similarly in the strict case. In the flat case we use the fact that there is a natural isomorphism of k-vector spaces H(Qp,VA)- ExtAD,(UA, UX where the extensions are computed in the category of k-vector spaces with local Galois action. Then HF(Qp, VA) is defined as the k-subspace of H(Qp, VA) which is the inverse image of ExtA(G, G), the group of extensions in the cate- gory of finite fat commutative group schemes over Zp killed by p, G being the (unique) finite fat group scheme over Zp associated to UA. By[Rayl] all such extensions in the inverse image even correspond to k-vector space schemes. For more details and calculations see [Ram For q different from p and q E M we have three cases(A),(B),(C). In case()there is a filtration by Dg entirely analogous to the one for p. We write this 0C w0,qcw, q cW and we set H(Qq, WA/Wx,)H(Qunr, k) in case(A) Hb、(Qq,V) Qa, v) H(Qq,VA) Again we make an analogous definition for HD Qa, W) by replacing VA by Wa and deleting the last term in case(A). We now define the k-vector HD(Qx/Q, VA) H(Q/Q,V)={a∈H1(Q/Q,):aq∈Hn(Q9,V) for all g∈M ∈H(Qp,V)} where is Se, str, ord, fl or unrestricted according to the type of D. A similar definition applies to HD(Qz/ Q, Wa) if. is Selmer or strict Now and for the rest of the section we are going to assume that po arises from the reduction of the A-adic representation associated to an eigenform More precisely we assume that there is a normalized eigenform f of weight 2 and level N, divisible only by the primes in 2, and that there ia a prime a of Of such that po =Pf, A mod A. Here Of is the ring of integers of the field generated by the Fourier coefficients of f so the fields of definition of the two representations need not be the same. However we assume that k 2 Ofx/A and we fix such an embedding so the comparison can be made over k. It will that if b type D then D is defined using O-algebras where 02 Ofx is an unramified extension whose residue field is k.(Although this condition is unnecessary, it is convenient to use A as the uniformizer for O. )Finally we assume that pf, a
MODULAR ELLIPTIC CURVES AND FERMAT’S LAST THEOREM 461 In the Selmer case we make an analogous definition for H1 Se(Qp, Wλ) by replacing Vλ by Wλ, and similarly in the strict case. In the flat case we use the fact that there is a natural isomorphism of k-vector spaces H1(Qp, Vλ) → Ext1 k[Dp](Uλ, Uλ) where the extensions are computed in the category of k-vector spaces with local Galois action. Then H1 f (Qp, Vλ) is defined as the k-subspace of H1(Qp, Vλ) which is the inverse image of Ext1 fl(G, G), the group of extensions in the category of finite flat commutative group schemes over Zp killed by p, G being the (unique) finite flat group scheme over Zp associated to Uλ. By [Ray1] all such extensions in the inverse image even correspond to k-vector space schemes. For more details and calculations see [Ram]. For q different from p and q ∈ M we have three cases (A), (B), (C). In case (A) there is a filtration by Dq entirely analogous to the one for p. We write this 0 ⊂ W0,q λ ⊂ W1,q λ ⊂ Wλ and we set H1 Dq (Qq, Vλ) = ker : H1(Qq, Vλ → H1(Qq, Wλ/W0,q λ ) ⊕ H1(Qunr q , k) in case (A) ker : H1(Qq, Vλ) → H1(Qunr q , Vλ) in case (B) or (C). Again we make an analogous definition for H1 Dq (Qq, Wλ) by replacing Vλ by Wλ and deleting the last term in case (A). We now define the k-vector space H1 D(QΣ/Q, Vλ) as H1 D(QΣ/Q, Vλ) = {α ∈ H1(QΣ/Q, Vλ) : αq ∈ H1 Dq (Qq, Vλ) for all q ∈ M, αq ∈ H1 ∗ (Qp, Vλ)} where ∗ is Se, str, ord, fl or unrestricted according to the type of D. A similar definition applies to H1 D(QΣ/Q, Wλ) if · is Selmer or strict. Now and for the rest of the section we are going to assume that ρ0 arises from the reduction of the λ-adic representation associated to an eigenform. More precisely we assume that there is a normalized eigenform f of weight 2 and level N, divisible only by the primes in Σ, and that there ia a prime λ of Of such that ρ0 = ρf,λ mod λ. Here Of is the ring of integers of the field generated by the Fourier coefficients of f so the fields of definition of the two representations need not be the same. However we assume that k ⊇ Of,λ/λ and we fix such an embedding so the comparison can be made over k. It will be convenient moreover to assume that if we are considering ρ0 as being of type D then D is defined using O-algebras where O⊇Of,λ is an unramified extension whose residue field is k. (Although this condition is unnecessary, it is convenient to use λ as the uniformizer for O.) Finally we assume that ρf,λ
ANDREW JOHN WILES itself is of type D. Again this is a slight abuse of terminology as we are really considering the extension of scalars pf. x 8 O and not pf, x itself, but we will do this without further mention if the context makes it clear.(The analysis of this section actually applies to any characteristic zero lifting of po but in all our applications we will be in the more restrictive context we have described here With these hypotheses there is a unique local homomorphism RD -O of O-algebras which takes the universal deformation to(the class of)Pf, A. Let pD=ker: RD++0. Let K be the field of fractions of O and let Uf=(K/0)2 with the Galois action taken from Pf, A. Similarly, let Vf= Adpf, 80 K/O2 (K/0)4 with the adjoint representation so that V≈Wf⊕K/C where W has Galois action via Sym pf, x det pr and the action on the second factor is trivial. Then if po is ordinary the filtration of Uf under the Adp action of Dp induces one on Wf which we write0cwicwfCWf Often to simplify the notation we will drop the index f from wf, Vf etc. There is also a filtration on WAn=( ker An:Wf-wf) given by Win=wa nwu (compatible with our previous description for n= 1). Likewise we write VAn for{ ker a:V→V We now explain how to extend the definition of HD to give meaning to HD(Qz/Q, VAn)and HD(Qx/Q, V)and these are O/An and O-modules, re- spectively. In the case where po is ordinary the definitions are the same with VAn or V replacing VA and O/An or K/O replacing k. One checks easily tha as O-modules HD(Qx/Q, VAn)c HD(Qz/Q, V)An where as usual the subscript an denotes the kernel of multiplication by An This just uses the divisibility of H(Qx/Q, V)and H(Qp, w/w) in the trict case. In the selmer case one checks that for m >n the kernel of H(Qun, VAn/won)H(Qnr, VAm/wAm) has only the zero element fixed under Gal( Qun/Qp) and the ord case is similar Checking conditions at q E M is dome with similar arguments. In the Selmer and strict cases we make analogous definitions with WAn in place of VAn and W in place of V and the analogue of (1.7) still holds fat(bi first that there is a natural map of O-modules H(Qp, VAn)-( UAm, UAn) for each m n where the extensions are of O-modules with local galois action. To describe this suppose that a E H(Qp, VAn). Then we can asso- ciate to a a representation Pa: Gal(Qp/Qp)- gl2(OnE)(where One=
462 ANDREW JOHN WILES itself is of type D. Again this is a slight abuse of terminology as we are really considering the extension of scalars ρf,λ ⊗ Of,λ O and not ρf,λ itself, but we will do this without further mention if the context makes it clear. (The analysis of this section actually applies to any characteristic zero lifting of ρ0 but in all our applications we will be in the more restrictive context we have described here.) With these hypotheses there is a unique local homomorphism RD → O of O-algebras which takes the universal deformation to (the class of) ρf,λ. Let pD = ker : RD → O. Let K be the field of fractions of O and let Uf = (K/O)2 with the Galois action taken from ρf,λ. Similarly, let Vf = Adρf,λ ⊗O K/O (K/O)4 with the adjoint representation so that Vf Wf ⊕ K/O where Wf has Galois action via Sym2ρf,λ ⊗ det ρ−1 f,λ and the action on the second factor is trivial. Then if ρ0 is ordinary the filtration of Uf under the Adρ action of Dp induces one on Wf which we write 0 ⊂ W0 f ⊂ W1 f ⊂ Wf . Often to simplify the notation we will drop the index f from W1 f , Vf etc. There is also a filtration on Wλn = {ker λn : Wf → Wf } given by Wi λn = Wλn ∩ Wi (compatible with our previous description for n = 1). Likewise we write Vλn for {ker λn : Vf → Vf }. We now explain how to extend the definition of H1 D to give meaning to H1 D(QΣ/Q, Vλn ) and H1 D(QΣ/Q, V ) and these are O/λn and O-modules, respectively. In the case where ρ0 is ordinary the definitions are the same with Vλn or V replacing Vλ and O/λn or K/O replacing k. One checks easily that as O-modules (1.7) H1 D(QΣ/Q, Vλn ) H1 D(QΣ/Q, V )λn , where as usual the subscript λn denotes the kernel of multiplication by λn. This just uses the divisibility of H0(QΣ/Q, V ) and H0(Qp, W/W0) in the strict case. In the Selmer case one checks that for m>n the kernel of H1(Qunr p , Vλn /W0 λn ) → H1(Qunr p , Vλm/W0 λm) has only the zero element fixed under Gal(Qunr p /Qp) and the ord case is similar. Checking conditions at q ∈ M is dome with similar arguments. In the Selmer and strict cases we make analogous definitions with Wλn in place of Vλn and W in place of V and the analogue of (1.7) still holds. We now consider the case where ρ0 is flat (but not ordinary). We claim first that there is a natural map of O-modules (1.8) H1(Qp, Vλn ) → Ext1 O[Dp](Uλm, Uλn ) for each m ≥ n where the extensions are of O-modules with local Galois action. To describe this suppose that α ∈ H1(Qp, Vλn ). Then we can associate to α a representation ρα : Gal(Q¯ p/Qp) → GL2(On[ε]) (where On[ε] =