MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 463 Oe]/(,2))which is an O-algebra deformation of po (see the proof of Propo sition 1.1 below).Let E=Onle)2 where the Galois action is via po.Then there is an exact sequence 0eE/Am→E/Am→(E/E)/Am→0 Uan Uam and hence an extension class in Ext(Uam,Uan).One checks now that (1.8) is a map of O-modules.We define H(Qp,V)to be the inverse image of ExtaU,U)under (1.8),i.e.,those extensions which are already extensions in the category of finite flat group schemes Zp.Observe that Exta(Uan,Uan)n ExtD (U,U)is an O-module,so H(Qp,V)is seen to be an O-sub- module of H(Qp,Va).We observe that our definition is equivalent to requir- ing that the classes in H(Qp,Vam)map under (1.8)to Exta(Uam,U)for all n.For if em is the extension class in Ext(Um,Un)then em en⊕Um as Galois-modules and we can apply results of Rayl]to see that em comes from a finite flat group scheme over Zp if en does. In the flat(non-ordinary)dete mined by Raynaud's results as mentioned at the beginning of the chapter.It follows in particular that,since is absolutely irreducible,V()H V)is divisible in this case (in fact V(Qp)K/O).Thus H(Qp,Van)H(Qp,V)am and hence we can define H(Qp,V=UH(Qp,m） and we claim that H(Qp,V)anH(Qp,Van).To see this we have to compare representations for m>n, Pn.m:Gal(Qp/Qp)-GL2(Onlel/Xm) Pm.n pm,m:Gal(@p/Qp)→GL2(Omle]/Am） where Pn,m and Pm,m are obtained from n∈H'(Qp,n）and im(an)∈ H1(Qp,Vam)and m.n:a+be-a+xm-nbe.By Ram,Prop 1.1 and Lemma 2.1]ifnm comes from a finite flat group scheme then so does Pm.n Conversely Pm,n is injec ctive and so Pn,m con afinite flat groupscheme if m docs cf.[Ray1].The definitions of H(Q/Q,Van)and H(Qs/Q,V)now extend to the flat case and we note that (1.7)is also valid in the flat case. Still in the fat()case we can again use the determination of pol to see that H(Qp,V)is divisible.For it is enough to check that H2(Qp,Va)=0 and this follows by duality from the fact that H(Q,V)=0
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 463 O[e]/(Xne, e2)) which is an 0-algebra deformation of po (see the proof of Proposition 1.1 below). Let E = On[el2 where the Galois action is via pa. Then there is an exact sequence and hence an extension class in Extl(~Am, UAn). One checks now that (1.8) is a map of 0-modules. We define Hfl(Qp, VAn) to be the inverse image of ~xti (uA~, UAn) under (1.8), i.e., those extensions which are already extensions in the category of finite flat group schemes Zp. Observe that Extfi(uAn, Uxn) n ~xtb~~,~ (uA~, UAn) is an (7-module, so Hfl(Qp, vAn) is seen to be an 0-submodule of H1(Qp, VAn). We observe that our definition is equivalent to requiring that the classes in Hfl(Qp, VAn) map under (1.8) to Exti(uAm, UAn) for all m 2 n. For if em is the extension class in Ext1(UAm, Uxn) then em ~f en$UAm as Galois-modules and we can apply results of [Rayl] to see that em comes from a finite flat group scheme over Zp if en does. In the flat (non-ordinary) case pol I, is determined by Raynaud's results as mentioned at the beginning of the chapter. It follows in particular that, since polD is absolutely irreducible, V(Qp) = HO(Qp, V) is divisible in this case P (in fact V(QP) N K/C3). Thus H1(Qp, VAn) 11 H1(Qp, V)An and hence we can define and we claim that Hfl(Qp, V) N Hfl(Qp, VAn ). To see this we have to compare representations for m 2 n, where p, and p, are obtained from an E H~(Q~, VAn) and im(an) E H1(Qp, VAm) and cp,: a +be +a +Xm-nbe. By [Ram, Prop 1.1 and Lemma 2.11 if p, comes from a finite flat group scheme then so does ,om,. Conversely cp, is injective and so ,on, comes from a finite flat group scheme if p, does; cf. [Rayl] . The definitions of Hh(Qc/Q, VAn) and HA (Qc/Q, V) now extend to the flat case and we note that (1.7) is also valid in the flat case. Still in the flat (non-ordinary) case we can again use the determination of po11, to see that H1(Qp, V) is divisible. For it is enough to check that H2(Qp, VA) = 0 and this follows by duality from the fact that HO (Q~,Vc) = 0
464 ANDREW WILES where V=Hom(VA,p)and p is the group of pth roots of unity.(Again this follows from the explicitform ofMuch subtler is the fact that H(Qp,V)is divisible.This result is essentially due to Ramakrishna.For using a local version of Proposition 1.1 below we have that Homo (pR/pR,K/O)H(Qp,V) whereRisthe universal oa flat defomation ring for and 0-algebras (This exists by Theorem 1.1 of [Ram]because po is absolutely irreducible.) Since RR where Ra is the corresponding ring for W(k)-algebras the main theoren of [Ram,Th.4.2]shows that R is a power series ring and the divisibility of H(Qp,V)then follows.We refer to Ram]for more details about Rf. Next we need an analogue of(1.5)for V.Again this is a variant of standard results in deformation theory and is given (at least for D=(ord,W(k), with some restriction on x1,x2 in i(a))in [MT,Prop 25]. PROPOSITION 1.2.Suppose that pf is a deformation of po of type D=（,∑，O,M)with an unramified extension of Of.A.Then as O-modules Homo(pD/pi,K/O)(Qs/Q,V). Remark.The isomorphism is functorial in an obvious way if one changes D to a larger D'. Proof.We will just describe the Selmer case with M=as the other cases use similar arguments.Suppose that a is a cocycle which represents a cohomology class in Hse(Q/Q,VA).Let Onle]denote the ring Ofel/(A"e,e2). We can associate to a a representation Pa:Gal(Qs/Q)-GL2(Onle]) as follows:set pa(g)=a(g)pfa(g)where pfa(g),a priori in GL2(O),is viewed in GL(via the natural Here a basis for 2 is chosen so that the representation Pfon the decomposition group D Gal(Qs/Q)has the upper triangular form of (i)(a),and then a(g)E Van is viewed in GL2(On[e])by identifying 1-te了 ={ker:GL2(On[e])-GL2()} Then = )}
464 ANDREW WILES where V: = Hom(VA,p,) and y is the group of pth roots of unity. (Again this follows from the explicit form of pol .) Much subtler is the fact that DP Hfl(Qp, V) is divisible. This result is essentially due to Ramakrishna. For, using a local version of Proposition 1.1 below we have that where R is the universal local flat deformation ring for polDp and 0-algebras. (This exists by Theorem 1.1 of [Ram] because pol Dp is absolutely irreducible.) Since R 21 Rfl 8 O where Rfl is the corresponding ring for W(lc)-algebras W(k) the main theorem of [Ram, Th. 4.21 shows that R is a power series ring and the divisibility of Hfl(Q, V) then follows. We refer to [Ram] for more details about RA. Next we need an analogue of (1.5) for V. Again this is a variant of standard results in deformation theory and is given (at least for V = (ord, C, W(k), 4) with some restriction on XI, ~2 in i(a)) in [MT, Prop 251. PROPOSITION 1.2.Suppose that pf,~ is a deformation of po of type V = (-,C,O,M)with 0 an unramified extension of Of,. Then as 0-modules Remark. The isomorphism is functorial in an obvious way if one changes V to a larger 23'. Proof. We will just describe the Selmer case with M = 4 as the other cases use similar arguments. Suppose that a is a cocycle which represents a cohomology class in H&(Q~/Q, Vxn). Let On[&] denote the ring O[&]/(X~&, E~). We can associate to a a representation as follows: set pa (g) = a(g)pf,~ (g) where pf ,A (g), a priori in GL2 (0), is viewed in GL2(0,[&]) via the natural mapping 0 + On[&]. Here a basis for O2 is chosen so that the representation pf,~ on the decomposition group Dp c Gal(Qc/Q) has the upper triangular form of (i)(a), and then a(g) E Vxn is viewed in GL2 (0,[el) by identifying Then
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 465 W以n= (+严)} 、1-ej and 以= {（1+e)} One checks readily that pa is a continuous homomorphism and that the defor- mation [po]is unchanged if we add a coboundary to a. We need to check that lpol is a Selmer deformation.Let Gal(Qp/Qnr)and g=Gal(/Qp).Consider the exact sequence of] modules 0→(/wR)H→(n/wR)→X→0 where X is a submodule of (V/V)4.Since the action of Dp on V/V is via a character which is nontrivial mod A (it equals x2x11 mod A and xx2), we see that X9=0 and H(g,X)=0.Then we have an exact diagram of O-modules 0 H(G,(vA/Wo))H(G,(Var/Wo.)) H(Qp:Van/Wo) H(Qpnr,Va/Wo.) By hypothesis the image of a is zero in H,V/Wo)9.Hence it is in the image of H((V/W)).Thus we can assume that it is rep resented in H(Qp,Van/Won)by a cocycle,which maps g to Vin/Won;i.e., f(D)CV/Wf()=0.The difference between f and the image of a is a coboundary fo-}for some uV.By subtracting the coboundary foou-u)from a globally we get a new a such that a f as cocycles 、Oce to Since [Pa]is a Selmer deformation there is a unique map of local algebras a:Ro→On[e]inducing it.(IfM≠中we must check the
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 465 and One checks readily that p, is a continuous homomorphism and that the deformation [pa] is unchanged if we add a coboundary to a. We need to check that [p,] is a Selmer deformation. Let 'FI = G~~(Q,/Q,u"')and G = Gal(Qr/Qp). Consider the exact sequence of CJ[G]- modules 0 -t (V;n/wxn)0 H + (vxn/~Xn)~+ X+O where X is a submodule of (v~~/v:~)~. Since the action of D, on vAn/V$ is via a character which is nontrivial mod X (it equals X2X11 mod X and $ x2), we see that xG= 0 and H1(G,X) = 0. Then we have an exact diagram of 0-modules 0 By hypothesis the image of a is zero in H'(QY. G~,/W:~)~. Hence it is in the image of H1(G,(V~~IWX~)~). Thus we can assume that it is represented in H1(Qp, VXn/wXn) by a cocycle, which maps G to v~~/w&; i.e., f (Dp) C VinIwXO, f (Ip)= 0. The difference between f and the image of a is a coboundary {a ++ azl -u) for some u E Vxn. By subtracting the coboundary {a H au -u) from cr globally we get a new a such that cr = f as cocycles mapping G to V;n/~;n. Thus a(Dp) c Vin, a(Ip)c Wfn and it is now easy to check that [pa] is a Selmer deformation of po. Since [p,] is a Selmer deformation there is a unique map of local 0- algebras cp, : Rv + On[&]inducing it. (If M # 4 we must check the
466 ANDREW WILES other conditions also.)Since pa =pfa mod e we see that restricting to pp gives a homomorphism of O-modules pa:pp→e.O/An such that (p)=0.Thus we have defined a map:a Hse(Qx/Q,Van)Homo(pp/pi,O/x"). It is straightforward to check that this is a map of O-modules.To check the injectivity of p suppose that a(pp)=0.Then a factors through RD/pD and being an -algebra homon orphism this determines .Thus [pf]=[Pal. If A-pAthen Amode is seen to be central by Schur's lemma and so may be taken to be I.A simple calculation now shows that a is a coboundary. To see that is surjective choose Ψ∈Homo(pp/p2,O/A"). Then pu:Gal(Qs/Q)GL2(Rp/(p,ker )is induced by a representative of the universal deformation(cho equal Pf.when reduced mod pD)and we define a map aw Gal(Qs/Q)-Van by 1+pp/(p%,kerΨ)pp/(p2,kerΨ) av(g)=pv(g)psx(g)-1E ∈Vn pp/(p畅，ker)1+pp/(p2,kerΨ) where pfa(g)is viewed in GL2(RD/(p),ker )via the structural map Rp (Rp being an O-algebra and the structural map being local because of the existence of a section).The right-hand inclusion comes from pD/p2,ker)兰O/An÷(O/An)·e E. Then aw is readily seen to be a continuous cocycle whose cohomology class lies in Hse(Q/Q,Vam).Finally (aw)=V.Moreover,the constructions are compatible ith change of n,i.e.,for and O//+1. We now relate the local cohomology groups we have defined to the theory of Fontaine and in particular to the groups of Bloch-Kato BK We will dis tinguish these by writing H for the cohomology groups of Bloch-Kato.None of the results described in the rest of this section are used in the rest of the paper.They serve only to relate the Selmer groups we have defined (and later compute)to the more standard versions.Using the lattice associated to pf we obtain also a lattice T4 with Galois action via Adpf.Let V=TzQp be the associated vector space and identify V with V/T.Let pr:be
466 ANDREW WILES other conditions also.) Since p, - pf,~mod E we see that restricting cp, to p~ gives a homomorphism of 0-modules, such that cpa(p&) =0. Thus we have defined a map cp : cr + cp, It is straightforward to check that this is a map of 0-modules. To check the injectivity of cp suppose that cp,(pD) = 0. Then cp, factors through RD/pD -v 0 and being an 0-algebra homomorphism this determines pa. Thus [pflx]= [pa]. If A-lp,A = pf,x then Amod E is seen to be central by Schur's lemma and so may be taken to be I. A simple calculation now shows that a is a coboundary. To see that cp is surjective choose Then pq: Gal(Qc/Q) + GL~(R=/(~&, ker Q)) is induced by a representative of the universal deformation (chosen to equal pf,~when reduced mod pz>) and we define a map crq : Gal(Qc/Q) + Vxn by where pf,x(g) is viewed in GL~(R~/(~&, ker Q)) via the structural map O + RD (RD being an 0-algebra and the structural map being local because of the existence of a section). The right-hand inclusion comes from Then a* is readily seen to be a continuous cocycle whose cohomology class lies in Hie(Qc/Q, VAn).Finally cp(crq) = Q. Moreover, the constructions are compatible with change of n, i.e., for Vxn - VAn+land X : O/Xn c-+0/Xn+'. We now relate the local cohomology groups we have defined to the theory of Fontaine and in particular to the groups of Bloch-Kato [BK].We will distinguish these by writing H; for the cohomology groups of Bloch-Kato. None of the results described in the rest of this section are used in the rest of the paper. They serve only to relate the Selmer groups we have defined (and later compute) to the more standard versions. Using the lattice associated to pf,x we obtain also a lattice T 2 O4with Galois action via Ad pf,x. Let V = T @zpQp be the associated vector space and identify V with V/T. Let pr : V + V be
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 467 the natural projection and define cohomology modules by (Qp,V)=ker HQp V)HQp.VBery). (Qp,V)=pr (H(Qp,V)H(Qp,V), H(Qp,Vn）= Gn-1((Q,V)）cH'(Q, where jn:Vam-V is the natural map and the two groups in the definition of H(Q,V)are defined using continuous cochains.Similar definitions apply to y Hom (,(1))and indeed to any finite-dimens onal continuous p-adic representation space.The reader is cautioned that the definition of H(Q,Vn)is dependent on the lattice T(or equivalently on V).Under conditio ns Bloch and Kato show,using the theory of Fontaine and Lafaille,that this is independent of the lattice (see BK,Lemmas 4.4 and 4.5]).In any case we will consider in what follows a fixed lattice associated to Adp,etc.Henceforth we will only use the notation)when the underlying vector space is crystalline. PROPOSITION 1.3.(i)If po is flat but not ordinary and pf.is associated to a p-divisible group then for alln H(Qp:Van)=H(Qp:Van). (间fpfAsotinary,detpyAl,=eamdpfAisasociatedtoap-dinistblc group,then for all n H(Qp:Van)C Hse(Qp:Van). Proof.Beginning with (i),we define H(Qp,V)={a E H(Qp,V) K(a/An)H(Qp,V)for all n}where K:H(Qp,V)H(Qp,V).Then we see that in case (i),H(Qp,V)is divisible.So it is enough to show that (Qp:V)=H(Qp:V). We have to compare two constructions associated to a nonzero element a of H(Qp,V).The first is to associate an extension (1.9) 0+y→E点K→0 of K-vector spaces with commuting continuou Galois action.Ifwe fixane with 6(e)=1 the action on e is defined by ge =e+a(o)with aa cocycle representing a.The second construction begins with the image of the subspace (a)in).By the analogue of Proposition 1.2 in the local case,there is an O-module isomorphism H(Qp,V)Homo(pR/p K/O)
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 467 the natural projection and define cohomology modules by where j,: Vxn -t V is the natural map and the two groups in the definition of H&(Qp, V) are defined using continuous cochains. Similar definitions apply to V* = HornQp(V, Qp(l)) and indeed to any finite-dimensional continuous padic representation space. The reader is cautioned that the definition of H&(Qp, VAn) is dependent on the lattice T (or equivalently on V). Under certain conditions Bloch and Kato show, using the theory of Fontaine and Lafaille, that this is independent of the lattice (see [BK, Lemmas 4.4 and 4.51). In any case we will consider in what follows a fixed lattice associated to p = pf,x, Ad p, etc. Henceforth we will only use the notation H&(QP, -) when the underlying vector space is crystalline. PROPOSITION 1.3. (i) If p0 is flat but not ordinary and pf,x is associated to a p-divisible group then for all n (ii) Ifpf,x is ordinary, det pf,x I = E and pf,x is associated to a p-divisible I, group, then for all n, H&(QP,vA~) C_ ~S1,(Qp,~xn). Proof. Beginning with (i), we define Hfl(Q,V) = {a E H1(Qp,V) : K(~/X,) E Hfl(~,V) for all n) where K : H1(Qp,V) -+ H1 (Qp, V). Then we see that in case (i), Hfl(Qp, V) is divisible. So it is enough to show that H&(Q,V) = H;(Q, v). We have to compare two constructions associated to a nonzero element a of H1(QP, V). The first is to associate an extension of K-vector spaces with commuting continuous Galois action. If we fix an e with 6(e) = 1 the action on e is defined by ae = e + &(a)with 6 a cocycle representing a. The second construction begins with the image of the subspace (a)in H1(Qp, V). By the analogue of Proposition 1.2 in the local case, there is an 0-module isomorphism