MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 461 In the Selmer case we make an analogous definition for H(Q,Wa)by replacing V by similarly in the strict case.In the fat the fact that there is a natural isomorphism of k-vector spaces '(Qp,)一ExtD(Ux,U) where the extensions are omputed in the category of k-vector spaces with local Galois action.Then H(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 flat commutative group schemes ove Zp killed by p,G being the (unique)finite flat group scheme over Zp associated to Ux.By [Rayl]all such extensions in the inverse image even correspond to k-vector space schemes.For more details and calculations see [Ram). For g different from p and ge M we have three cases (A),(B),(C).In case (A)there is a filtration by Do entirely analogous to the one for p.We write this 0cWgcW9C Wa and we set (ker H(Qa,VA) →H'(Qg,w/Ww)⊕H'(Qgnr,k)in case(A) Hb(Qa,VA)= ker:H(Qg,VA) →H(Qgnr,) in case (B)or (C). Again we make an analogous definition for (Wa)by replacingV by Wa and deleting the last term in case (A).We now define the k-vector space Hb(Q/Q,Va)as Hb(Qs/Q,)={a∈H(Qz/Q,):ag∈Hb,(Qg,)for all g∈M, ap∈H(Qp,)} where*is Se,str,ord,fl or unrestricted according to the type of D.A similar definition applies to H(Qs/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 E,and that there is a prime of such thatmod.Here 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 O./A 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 po as being of type D then D is defined using O-algebras where DOfx is an unramified extension whose residue field isk.(Although this condition is unnecessary,it is convenient to useas the uniformizer for .Finally we assume thatMODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 461 In the Selmer case we make an analogous definition for Hi,(Q, Wx) by replacing VA by Wx, and similarly in the strict case. In the flat case we use the fact that there is a natural isomorphism of k-vector spaces where the extensions are computed in the category of k-vector spaces with local Galois action. Then Hfl(Qp, VA) is defined as the k-subspace of H1 (Q, VA) which is the inverse image of EX~;(G,G), the group of extensions in the cate￾gory of finite flat commutative group schemes over Zp killed by p, G being the (unique) finite flat 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 (A) there is a filtration by D, entirely analogous to the one for p. We write this 0 c wX~, c wtqc WX and we set I - H1(Q, WA/w:") @ H1(Qinr,k) in case (A) HA,(Qq,VA)= ker : H1(Q, V,) ( + xl(qyr,VA) in case (B) or (C). Again we make an analogous definition for H~,(Q,WA) by replacing VA by Wx and deleting the last term in case (A). We now define the k-vector space H&(QclQ, Vx) as H&(QE/Q,VA)= {a E H'(QE/Q, VA): a, E H~,(Q,VA) for all q E M, where c is Se, str, ord, fl or unrestricted according to the type of 27. A similar definition applies to H&(Qc/Q, Wx)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 C, and that there is a prime A of Of such that po = pf,~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 > Of,x/A 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 po as being of type 27 then D is defined using 0-algebras where 0 > is an unramified extension whose residue field is k. (Although this condition is unnecessary, it is convenient to use X as the uniformizer for 0.) Finally we assume that pf