MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 455 Chapter 1 This chapter is devoted to the study of certain Galois representations. In the first section we introduc and study Mazur's deformation theory and discuss various refinements of it.These refinements will be needed later to make precise the correspondence between the universal deformation rings and the Hecke rings in Chapter 2.The main results needed are Proposition 1.2 which is used to interpret various generalized cotangent spaces as Selmer groups and(1.7)which later will be used to study them.At the end of the section we relate these Selmer groups to ones used in the Bloch-Kato conjecture,but this connection is not needed for the proofs of our main results. In the second section we extract from the results of Poitou and Tate on Galois cohomology certain general relations between Selmer groups as varies, as well as between Selmer groups and their duals.The most important obser vation of the third section is Lemma 1.10(i)which guarantees the existence of the special primes used in Chapter 3 and [TW]. 1.Deformations of Galois representations Let p be an odd prime.Let be a finite set of primes including p and let Qs be the maximal extension of Q unramified outside this set and oo. Throughout we fix an embedding of Q,and so also of Q,in C.We will also fix a choice of decomposition group D for all primes g in Z.Suppose that k is a finite field of characteristic p and that (1.1) po:Gal(Qs/Q)→GL2(k) is an irreducible representation.In contrast to the introduction we will assume in the rest of the paper with its field of definitionk Suppos further that det po is odd.In particular this implies that the smallest field of definition for po is given by the field ko generated by the traces but we will not assume that k=ko.It also implies that po is absolutely irreducible.We con- sider the deformations [p]to GL2(A)of po in the sense of Mazur [Mal].Thus if W(k)is the ring of Witt vectors of k,A is to be a complete Noetherian local W(k)-algebra with residue field k and maximal ideal m,and a deformation [pl is just a strict equivalence class of homomorphisms p:Gal(Q/Q) →GL2(A) such that pmod m po,two such homomorphisms being called strictly equiv- alent if one can be brought to the other by conjugation by an element of ker GL2(A)GL2(k).We often simply write p instead of [p]for the equivalence classMODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 455 Chapter 1 This chapter is devoted to the study of certain Galois representations. In the first section we introduce and study Mazur's deformation theory and discuss various refinements of it. These refinements will be needed later to make precise the correspondence between the universal deformation rings and the Hecke rings in Chapter 2. The main results needed are Proposition 1.2 which is used to interpret various generalized cotangent spaces as Selmer groups and (1.7) which later will be used to study them. At the end of the section we relate these Selmer groups to ones used in the Bloch-Kato conjecture, but this connection is not needed for the proofs of our main results. In the second section we extract from the results of Poitou and Tate on Galois cohomology certain general relations between Selmer groups as C varies, as well as between Selmer groups and their duals. The most important obser￾vation of the third section is Lemma l.lO(i) which guarantees the existence of the special primes used in Chapter 3 and [TW]. 1. Deformations of Galois representations Let p be an odd prime. Let C be a finite set of primes including p and let Qc be the maximal extension of Q unramified outside this set and w. Throughout we fix an embedding of Q,and so also of Qc, in C. We will also fix a choice of decomposition group D, for all primes q in Z. Suppose that k is a finite field of characteristic p and that is an irreducible representation. In contrast to the introduction we will assume in the rest of the paper that po comes with its field of definition k. Suppose further that det po is odd. In particular this implies that the smallest field of definition for po is given by the field ko generated by the traces but we will not assume that k = ko. It also implies that po is absolutely irreducible. We con￾sider the deformations [p] to GL2(A) of po in the sense of Mazur [Mall. Thus if W(k) is the ring of Witt vectors of k, A is to be a complete Noetherian local W(k)-algebra with residue field k and maximal ideal m, and a deformation [p] is just a strict equivalence class of homomorphisms p: Gal(Qc/Q) -t GL2(A) such that p mod m = po, two such homomorphisms being called strictly equiv￾alent if one can be brought to the other by conjugation by an element of ker : GL2(A) -t GL2(lc). We often simply write p instead of [p] for the equivalence class