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)=0MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 463 O[e]/(Xne, e2)) which is an 0-algebra deformation of po (see the proof of Propo￾sition 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-sub￾module of H1(Qp, VAn). We observe that our definition is equivalent to requir￾ing 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