444 ANDREW WILES Our approach to the study of elliptic curves is via their associated Galois representations.Suppose that Pp is the representation of Gal(Q/Q)on the pdivision points of an elliptic urve over Q,and suppose for the moment that p3 is irreducible.The choice of 3 is critical because a crucial theorem of Lang- lands and Tunnell shows that if ps is irreducible then it is also modular.We then proceed by showing that under the hypothesis at3. together with some milder restrictions on the ramification of p3 at the other primes,every suitable lifting of ps is modular.To do this we link the problem, via some novel arguments from commutative algebra,to a class number prob lem of a well-known type.This we then solve with the help of the paper [TW] This suffices to prove the modularity of E as it is known that E is modular if and only if the associated 3-adic representation is modular. The key development in the proof is a new and surprising link between two strong but distinct traditions in number theory,the relationship between Galois representations and modular forms on the one hand and the interpretation of special values of L-functions on the other. The former tradition is of course more recent. Following the original results of Eichler and Shimura in the 1950's and 1960's the other main theorems were proved by Deligne,Serre and Langlands in the period up to 1980.This included the construction of Galois representations associated to modular forms,the refinements of Langlands and Deligne (later completed by Caravol).and the crucial application by langlands of base change methods to give se results in weight one.However with the exception of the rather special weight one case,including the extension by Tunnell of Langlands'original theorem,there was no progress in the direction of associating modular forms to Galois representations.From the mid 1980's the main impetus to the field was given by the conjectures of Serre which elaborated on the s-conjecture alluded to before.Besides the work of Ribet and others on this problem we draw on some of the more specialized developments of the 1980's,notably those of Hida and Mazur. The second tradition goes back to the famous analytic class number for mula of Dirichlet,but owes its modern revival to the conjecture of Birch and Swinnerton-Dyer.In practice however,it is the ideas of Iw sawa in this field on which we attempt to draw,and which to a large extent we have to replace.The principles of Galois cohomology,and in particular the fundamental theorems of Poitou and Tate,also playan importantrole here The restriction that p3 be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic curves which share a common P5.Using this,we complete the proof that all semistable elliptic curves are modular.In particular,this finally yields a proof of Fermat's Last Theorem.In addition,this method seems well suited to establishing that all elliptic curves over Q are modular and to generalization to other totally real number fields. Now we present our meth ods and results in more detail 444 ANDREW WILES Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that p, is the representation of Gal(Q/Q) on the pdivision points of an elliptic curve over Q, and suppose for the moment that p3 is irreducible. The choice of 3 is critical because a crucial theorem of Lang￾lands and Tunnell shows that if p3 is irreducible then it is also modular. We then proceed by showing that under the hypothesis that p3 is semistable at 3, together with some milder restrictions on the ramification of p3 at the other primes, every suitable lifting of p3 is modular. To do this we link the problem, via some novel arguments from commutative algebra, to a class number prob￾lem of a well-known type. This we then solve with the help of the paper [TW]. This suffices to prove the modularity of E as it is known that E is modular if and only if the associated 3-adic representation is modular. The key development in the proof is a new and surprising link between two strong but distinct traditions in number theory, the relationship between Galois representations and modular forms on the one hand and the interpretation of special values of L-functions on the other. The former tradition is of course more recent. Following the original results of Eichler and Shimura in the 1950's and 1960's the other main theorems were proved by Deligne, Serre and Langlands in the period up to 1980. This included the construction of Galois representations associated to modular forms, the refinements of Langlands and Deligne (later completed by Carayol), and the crucial application by Langlands of base change methods to give converse results in weight one. However with the exception of the rather special weight one case, including the extension by Tunnell of Langlands' original theorem, there was no progress in the direction of associating modular forms to Galois representations. From the mid 1980's the main impetus to the field was given by the conjectures of Serre which elaborated on the &-conjecture alluded to before. Besides the work of Ribet and others on this problem we draw on some of the more specialized developments of the 1980's, notably those of Hida and Mazur. The second tradition goes back to the famous analytic class number for￾mula of Dirichlet, but owes its modern revival to the conjecture of Birch and Swinnerton-Dyer. In practice however, it is the ideas of Iwasawa in this field on which we attempt to draw, and which to a large extent we have to replace. The principles of Galois cohomology, and in particular the fundamental theorems of Poitou and Tate, also play an important role here. The restriction that p3 be irreducible at 3 is bypassed by means of an " intriguing argument with families of elliptic curves which share a common p5. Using this, we complete the proof that all semistable elliptic curves are modular. In particular, this finally yields a proof of Fermat's Last Theorem. In addition, this method seems well suited to establishing that all elliptic curves over Q are modular and to generalization to other totally real number fields. Now we present our methods and results in more detail