468 ANDREW WILES where R is the universal deformation ring of po viewed as a representation of Gal(Qp/Qp)on O-algebras and pR is the ideal of R corresponding to pD (i.e.,its inverse image in R).Since a 0,associated to (a)is a quotient /()ofwhich isa free -module of rank one.We then obtain homomorphisn Pa:Gal(Qp/Qp)-GL2(R/(pR a)) induced from the universal deformation (we picka repres sentation in the uni versal class).This is associated to an O-module of rank 4 which tensored with K gives a K-vector space E(K)4 which is an extension (1.10) 0→u→E→u0 where K2 has the Galois representation p(viewed locally). In the first construction a (Q,if and only if the extension (1.9)is crystalline,as the extension given in(1)ssum of copies of themor e usua extension where p replaces K in (1.9).On the other hand (a)H(Qp,V)if and only if the second construction can be made through R,or equivalently if and only if Eis the representation associated to ap-divisible group.(A priori, the representation associated to pa only has the property that on all finite quotients it comes from a finite flat group scheme.However a theorem of Raynaud [Ray1]says that then Pa comes from a p-divisible group.For more details on Rf,the universal flat deformation ring of the local representatior po,see [Ram].)Now the extension E comes from a p-divisible group if and only if it is crystalline;cf.Fo,$6).So we have to show that(1.9)is crystalline if and only if (1.10)is One obtains (1.10)from (1.9)as follows.We view V as Homg(u,u)and let X=ker:{HomK(u,U)⑧→ where the map is the natural one fwf(w).(All tensor products in this proof will be as K-vector spaces.)Then as KDp]-modules E≈(E⑧0/X To check this,one calculates explicitly with the definition of the action on E (given above on e)and on E'(given in the proof of Proposition 1.1).It follows from standard properties of crystalline representations that if E is crystalline, so is E and also E'.Conversely we ecover E from E'as follows. Consider E⑧M≈(E⑧M⑧U)/(X⑧).Then there is a natural map p:E⑧(det)→E'⑧u induced by the direct sum decomposition u⑧tus (det)Sym2.Here det denotes a 1-dimensional vector space over K with Galois action via detpf.Now we claim that is injective on(det).For
468 ANDREW WILES where R is the universal deformation ring of po viewed as a representation of Gal(Qp/Qp) on 0-algebras and p~ is the ideal of R corresponding to p~ (i.e., its inverse image in R). Since a # 0, associated to (a)is a quotient p~/(pi,a) of p~/& which is a free 0-module of rank one. We then obtain a homomorphism pa: Gal(Qp/Qp) + GL2 (R/(P~,a)) induced from the universal deformation (we pick a representation in the universal class). This is associated to an 0-module of rank 4 which tensored with K gives a K-vector space El 21 (K)4 which is an extension where U II K2 has the Galois representation pf,~ (viewed locally). In the first construction a E H$(Q~, V) if and only if the extension (1.9) is crystalline, as the extension given in (1.9) is a sum of copies of the more usual extension where Qp replaces K in (1.9). On the other hand (a) Hf'(Qp, V) if and only if the second construction can be made through Rfl, or equivalently if and only if El is the representation associated to a pdivisible group. ( A priori, the representation associated to p, only has the property that on all finite quotients it comes from a finite flat group scheme. However a theorem of Raynaud [Rayl] says that then p, comes from a pdivisible group. For more details on Rfl, the universal flat deformation ring of the local representation po, see [Ram].) Now the extension E' comes from a pdivisible group if and only if it is crystalline; cf. [Fo, $61. So we have to show that (1.9) is crystalline if and only if (1.10) is crystalline. One obtains (1.10) from (1.9) as follows. We view V as HomK(U, U) and let X = ker : {HomK(U, U) 8U -+ U) where the map is the natural one f 8 w H f (w). (All tensor products in this proof will be as K-vector spaces.) Then as K[Dp]-modules To check this, one calculates explicitly with the definition of the action on E (given above on e) and on E' (given in the proof of Proposition 1.1). It follows from standard properties of crystalline representations that if E is crystalline, so is E 8 U and also E'. Conversely, we can recover E from E' as follows. Consider El @ U E (E 8 U 8 U)/(X 8 U). Then there is a natural map cp : E 8 (det) + El 8 U induced by the direct sum decomposition U 8 U - (det) @ Sym2U. Here det denotes a 1-dimensional vector space over K with Galois action via det pf,~. Now we claim that cp is injective on V 8 (det). For
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 469 iff∈then(f)=f⑧(w1⑧w2-w2⑧w）where w1,w2 are a basis for u for which w1Aw2=1 in det≈K.So if o(f)eX⑧then f(w)⑧w2-f(2)⑧w1=0inu⑧4. But this is false unless f(w)=f(w2)=0 whence f=0.So o is injective on y det and if o itself were not injective then E would split contradicting .we have exhibitedE(det)as asubrep of E which is crystalline.We deduce that E is crystalline if E'is.This completes the proof of(i). To prove (ii)we check first that Hs(Qp,Vam)=jn(Hs(Qp,V)(this was already used in(1.7)).We next have to show that H(Qp,V)Hs(Qp,V) where the latter is defined by Hse(Qp:V)=ker:H(Qp:V)H(Qunr,v/vo) with )o the subspace of y on which I acts via &But this follows from the computations in Corollary 3.8.4 of [BK].Finally we observe that pr(Hse(Qp,V)Hs(Qp,V) although the inclusion may be strict,and pr(H(Qp,V)）=H(Qp,V) by definition.This completes the proof. These groups have the property that for s>r, (1.11) H(Qp:Var)nH(Qp,VA))=H(Qp,Var) where jr.s:Vr-Va is the natural injection.The same holds for Vr and V.in place of Var and Va.where Vir is defined by Var Hom(Var,Hp) and similarly for V.Both results are immediate from the definition (and indeed were part of the motivation for the definition). We also give a finite level version of a result of Bloch-Kato which is easily deduced from the vector space version.As before let TcV be a Galois stable lattice so that T4.Define H(Qp,T)=i1(H(Qp,y)】 under the natural inclusion i:T,and likewise for the dual lattice T= Homz (V,(Qp/Zp)(1))in V.(Here V*=Hom(V,Qp(1));throughout this paper we use M to denote a dual of M with a Cartier twist.)Also write
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 469 if f E V then cp(f) = f 8 (wl @ w2 -w2 8 wl) where wl, w2 are a basis for U for which wlr\w2 = 1 in det - K. So if p(f) E X 8 U then But this is false unless f (wl) = f (w2) = 0 whence f = 0. So cp is injective on V 8 det and if cp itself were not injective then E would split contradicting a # 0. So cp is injective and we have exhibited E @ (det) as a subrepresentation of El 8 U which is crystalline. We deduce that E is crystalline if El is. This completes the proof of (i). To prove (ii) we check first that HAe(Qp, Vh") = f1 (HAe(QP, v)) (this was already used in (1.7)). We next have to show that H$ (Q, V) & Hie(Q, V) where the latter is defined by Hke(QP,V) = ker : H1(Qp, V) -+ H1(Q~, v/vO) with VO the subspace of V on which I, acts via E. But this follows from the computations in Corollary 3.8.4 of [BK]. Finally we observe that although the inclusion may be strict, and by definition. This completes the proof. These groups have the property that for s 2 r, where j,: VAT+ Vxs is the natural injection. The same holds for V,*, and VxS in place of V,T and VAs where Vxr is defined by and similarly for Vx*,. Both results are immediate from the definition (and indeed were part of the motivation for the definition). We also give a finite level version of a result of Bloch-Kato which is easily deduced from the vector space version. As before let T c V be a Galois stable lattice so that T - 04.Define under the natural inclusion i : T ~fV, and likewise for the dual lattice T* = HomZp(V, (Qp/Zp)(l)) in V*. (Here V* = Hom(V, Qp(l)); throughout this paper we use M* to denote a dual of M with a Cartier twist.) Also write
470 ANDREW WILES pr:TT for the natural projection map,and for the mapping it induces on cohomology. PROPOSITION 1.4.If py is associated to a p-divisible group (the ordi- nary case is allowed)then (i)prn (H(Qp,T)=H(Qp,T/X")and similarly for T,T*/". (ii)H(Qp,Van)is the orthogonal complement of H(Qp,V)under Tate local duality between H(Qp,V)and H(Qp,V)and similarly for Wx and Win replacing Van and Vn. More generally these results hold for any crystalline representation in place of y and X'a uniformizer in K'where K'is any finite ertension of Qp with K'C EndV. Proof.We first observe that pr (H(Qp,T))CH(Qp,T/X").Now from the construction we may identify T/An with Va.A result of Bloch- Kato ([BK,Prop.3.8])says that H(Qp,V)and H(Qp,are orthogonal compleme under Tate local duality.It follows formally that H(Qp,V and prn ((Qp,T))are orthogonal complements,so to prove the proposition it is enough to show that (1.12) #(Qp:Van)#(Qp,Van)=#(Qp:Van). Now if r dimk H(Qp,V)and s dimk H(Qp,V)then (1.13) r+s=dimk H(Qp:V)+dimk H(Qp:V*)+dimk V. From the definition, (1.14)#(Qp,Vam)=#(O/X")".#ker{(Qp,Van)H (Qp,V)}. The second factor isq)V().When we write V()iv for the maximal divisible subgroup of V(Qp)this is the same as #(V(Qp)/V(Qp)dI)/X"=#(V(Qp)/V(Qp)diM)x =V(Qp)x/#(V(Qp)div)xn. Combining this with (1.14)gives (1.15)#H(Qp,n)=#(O/An)r #H,Vn)/#(/")dimxv). This,together with an analogous formula for H(Qp,Vn)and(1.13),gives #(Qp,Van)#(Qp,VA)=#(O/").(Qp:Van)#(Qp:V)
470 ANDREW WILES pr, : T -+ T/Xn for the natural projection map, and for the mapping it induces on cohomology. PROPOSITION1.4. If pf,~is associated to a p-divisible group (the ordinary case is allowed) then (i) pr, (H; (Qp,T)) = H; (Qp, T/Xn) and similarly for T*, T*/Xn (ii) H~(Q, V,n) is the orthogonal complement of Hk(QP, Vxn) under Tate local duality between H' (Q, Vxn) and H1(QP, Vx*,) and similarly for Wxn and Win replacing Vxn and Vxn. More generally these results hold for any crystalline representation V' in place of V and A' a uniformizer in K' where K' is any finite extension of Qp with K' c End,a,(Gp,Qp) V'. Proof. We first observe that pr, (H&(QP, T)) c Hk(QP, T/Xn). Now from the construction we may identify T/Xn with VAn. A result of BlochKato ([BK, Prop. 3.81) says that H&(QP, V) and Hk(QP, V*) are orthogonal complements under Tate local duality. It follows formally that Hk(Qp, Vxn) and pr, (Hk(QP, T)) are orthogonal complements, so to prove the proposition it is enough to show that Now if r = dimK H~(Q, V) and s = dimK Hk(QP, V*) then From the definition, The second factor is equal to # {V (Qp)/Xn V (Qp)). When we write V for the maximal divisible subgroup of V (Qp) this is the same as Combining this with (1.14) gives This, together with an analogous formula for # Hk (QP, Vxn) and (1.13), gives
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 471 As#(Qp,Vin)=#H2(Qp,Vam)the assertion of (1.12)now follows from the formula for the Euler characteristic of V. The proof for W,or indeed more generally for any crystalline represen- tation,is the same. We also give a characterization of the orthogonal complements of Hs(Qp,Wan)and Hs(Qp,Vn),under Tate's local duality.We write these duals as Hs(Qp,Wi)and Hs(Qp,VA)respectively.Let w:H(Qp,Win)H(Qp,Win/(Win)) be the natural map where (W)i is the orthogonal complement of W in Wi,and let n be defined as the image under the composite map Xn,t=im:Z/(Z)P"⑧0/A”.一'(Qp,pn⑧O/A") →H(Qp,Win/(Wi)°) where in the middle term p/n is to be identified with (W)1/(W). Similarly if we replace Wn by Vin we let Yni be the image of Z/(Z)P (O/A)2 in H(Qp,V/(WA)),and we replace by the analogous map PROPOSITION 1.5. He(Qp,Wn）=Po(Xn,i), He(Qp,Vn）=p(Yni Proof.This can be checked by dualizing the sequence 0→H5(Qp,Wn)一Hse(Qp,Wn) →ker:{H'(Qp,Wn/(Wn))→H(Qr,Wn/(Wn), whereW)=ker :HQ,Wa)pW/(W)).The first term is orthogonal toker:H W)H W/(W)).By the naturality of the cup product pairing with respect to quotients and subgroups the claim then reduces to the well known fact that under the cup product pairing H'(Qp,4pn）×H'(Qp,Z/p）→Z/p the orthogonal complement of the unramified homomorphisms is the image of the units Z(Z)Hp,p).The proof for Var is essentially the same
MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 471 As #HO (Qp, V,)= # H2 (Qp,VAn)the assertion of (1.12) now follows from the formula for the Euler characteristic of Vxn. The proof for Wxn,or indeed more generally for any crystalline representation, is the same. We also give a characterization of the orthogonal complements of Hie(Qp,WX~) and Hie(Qp,VAn),under Tate's local duality. We write these duals as Hi, (Qp, Win)and Hi, (Qp,VTn) respectively. Let be the natural map where (Win)iis the orthogonal complement of w;,' in Win,and let Xn,+be defined as the image under the composite map where in the middle term ppn @ O/Xn is to be identified with (Win)l/(W;I")O. Similarly if we replace Winby V& we let Yn,i be the image of Z,X/(Z,X)pn@ (O/Xn)2in H 1(Qp,vX*,/(win)O), and we replace cpw by the analogous map cp,. Proof. This can be checked by dualizing the sequence 0 + Hitr(QP,Wxn)-+H~,(Q, Wx.1 - ker : {H1(Qp,WA~/(WX~)O) +H~(QY, WX~/(WA~)O}, where Hitr(Qp,WAn)= ker : H1(Qp,Wxn) + H1(Qp,WX~/(WX~)O). The first term is orthogonal to ker : H1(Qp,Win)-'H 1(Qp,Win/(Win)').By the naturality of the cup product pairing with respect to quotients and subgroups the claim then reduces to the well known fact that under the cup product pairing H' (Qp, pPn) x H1(Qp,ZIP") +Zlpn the orthogonal complement of the unramified homomorphisms is the image of the units ZpX /(Z,X)pn -,H 1(Qp,ppn) The proof for Vxn is essentially the same
472 ANDREW WILES 2.Some computations of cohomology groups We now make some comparisons of orders of cohomology groups using the theorems of Poitou and Tate.We retain the notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general context.Suppose that L=ΠL,sΠH'(Qg,X) ,whereXis a finite module for Gal)of p-power order. We define L*to be the orthogonal complement of L under the perfect pairing (local Tate duality) Q×盟'QXy）-Q,/2 where X*=Hom(X,ps).Let λx:H(Q/Q,X)一ΠH(Qg,X) be the localization map and similarly x.for X.Then we set HL(Q/Q.x)=Ax(L),HL-(Qs/Q,X*)=x(L). The following result was suggested by a result of Greenberg(cf.[Grel])and is a simple consequence of the theorems of Poitou and Tate.Recall that p is always assumed odd and that p. PROPOSITION 1.6. #H吐(Qz/Q,X)/#H.(Qz/QX)=h∞Πh where ∫hg=#H(QgX*)/八H(Qg,X):Lgl hoo=(R,X)(Q,X)/#(Q,X*). Proof.Adapting the exact sequence of Poitou and Tate(cf.[Mi2,Th.0) we get a seven term exact sequence 0→ (Q/Q,XN）一H(Q/Q,X）一H(Q,X/Lg 及H(Q,X)）-r(Qz/Qx)-l.(Q=/Q,xy LH(Qz/Q,X)一0
ANDREW WILES 2. Some computations of cohomology groups We now make some comparisons of orders of cohomology groups using the theorems of Poitou and Tate. We retain the notation and conventions of Section 1 though it will be convenient to state the first two propositions in a more general context. Suppose that is a subgroup, where X is a finite module for Gal(Qc/Q) of ppower order. We define L* to be the orthogonal complement of L under the perfect pairing (local Tate duality) nff1(Qq,x) x n~'(Qq,x*) + Qplzp qEC qEC where X* = Hom(X, ppM). Let AX : H~(QEIQ, X) + n WQ,X) qEC be the localization map and similarly Ax* for X*. Then we set The following result was suggested by a result of Greenberg (cf. [Grel]) and is a simple consequence of the theorems of Poitou and Tate. Recall that p is always assumed odd and that p E C. - where Proof. Adapting the exact sequence of Poitou and Tate (cf. [Mi2, Th. 4.201) we get a seven term exact sequence + HZ(QcIQ, X) -+ H1(QcIQ, X) +II H1(Qq, x)ILq '2EC I 0