《数学教学论》课程教学资源（学习资料）费马大定理证明论文原文 Modular elliptic curves and Fermat 's Last Theorem.pdf_大学文库

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 Langlands 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 problem 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 formula 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

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 445 Let f be an eigenform associated to the congruence subgroup ri(N)of SL2(Z)of weight k >2 and character x.Thus if Tn is the Hecke operator ass ociated to an integer n there is an algebraic integer c(n,f)such that Tf c(n,f)f for each n.We let Kt be the number field generated over Q by the fc(n,f)}together with the values of x and let Of be its ring of integers. For any prime A of Of let Ofa be the completion of O at A.The following theorem is due to Eichler and Shimura (for =2)and Deligne(for> 2) The analogous result when k=1 is a celebrated theorem of Serre and Deligne but is more naturally stated in terms of complex representations.The image in that case is finite and a converse is known in many cases. THEOREM 0.1.For each prime pEZ and each prime p of Of there is a continuous representation Pf:Gal(Q/Q)-GL2(Of.) which is unramified outside the primes dividing Np and such that for all primes gt Np, tracepf(Frobq)=c(q,f), det(Frobq)=x(g)qk-1 We will be concerned with trying to prove results in the opposite direction, that is to say,with establishing criteria under which a A-ad representation arises in this way from a modular form.We have not found any advantage in assuming that the representation is part of a compatible system of A-adic representations except that the proof may be easier for some A than for others. Assume po:Gal(Q/Q)一GL2(Fp) is a continuous representation with values in the algebraic closure of a finite field of characteristic p and that det po is odd.We say that po is modular if po and pfa mod A are isomorphic over Fp for some f and A and some embedding of /A in Fp.Serre has conjectured that every irreducible po of odd determinant is modular.Very little is known about this conjecture except when the image of po in PGL2(Fp)is dihedral,A or S4.In the dihedral it is true and due (essentially)to Hecke.and in the A.and S4 cases it is again true and due primarily to Langlands,with one important case due to Tunnell (see Theorem 5.1 for a statement) More precisely these theorems actually associate a form of weight one to the corresponding complex representation but the versions we need are straightforward deductions from the complex Even in the reducible case not much is known about the problem in the form we have described it,and in that case it should be observed that one must also choose the lattice carefully as only the semisimplification of =mod is independent of the choice of lattice inK

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 445 Let f be an eigenform associated to the congruence subgroup Fl(N) of SL2(Z) of weight k 2 2 and character X. Thus if Tn is the Hecke operator associated to an integer n there is an algebraic integer c(n, f) such that Tnf = c(n,f )f for each n. We let Kf be the number field generated over Q by the {c (n, f)) together with the values of x and let Of be its ring of integers. For any prime X of Of let Of,x be the completion of Of at A. The following theorem is due to Eichler and Shimura (for k = 2) and Deligne (for k > 2). The analogous result when k = 1is a celebrated theorem of Serre and Deligne but is more naturally stated in terms of complex representations. The image in that case is finite and a converse is known in many cases. THEOREM 0.1. For each prime p E Z and each prime X I p of Of there is a continuous representation which is unramijied outside the primes dividing Np and such that for all primes 4 t NP, We will be concerned with trying to prove results in the opposite direction, that is to say, with establishing criteria under which a A-adic representation arises in this way from a modular form. We have not found any advantage in assuming that the representation is part of a compatible system of A-adic representations except that the proof may be easier for some X than for others. Assume PO : Gal(Q/Q) +GL2(Fp) is a continuous representation with values in the algebraic closure of a finite field of characteristic p and that det po is odd. We say that po is modular if po and pf,x mod X are isomorphic over F, for some f and X and some embedding of Of/X in F,. Serre has conjectured that every irreducible po of odd determinant is modular. Very little is known about this conjecture except when the image of po in PGL~(F,) is dihedral, A4 or S4.In the dihedral case it is true and due (essentially) to Hecke, and in the A4 and S4cases it is again true and due primarily to Langlands, with one important case due to Tunnel1 (see Theorem 5.1 for a statement). More precisely these theorems actually associate a form of weight one to the corresponding complex representation but the versions we need are straightforward deductions from the complex case. Even in the reducible case not much is known about the problem in the form we have described it, and in that case it should be observed that one must also choose the lattice carefully as only the semisimplification of = pf,~mod X is independent of the choice of lattice in Kf2,x

446 ANDREW WILES If is the ring of integers of a local field(containing Qp)we will say that Gal(Q/Q)-GL2()is a lifting of po if,for a specified embedding of the residue field morphic over Fp.Our point of view will be to assume that po is modular and then to attempt to give conditions under which a representation p lifting po comes from a modular form in the sense for some f,.We will restrict our attention to two cases: (I)po is ordinary (at p)by which we mean that there is a one-dimensional subspace of F2,stable under a decomposition group at p and such that the action on the quotient space is unramified and distinct from the action on the subspace. (II)Po is flat (at p),meaning that as a representation of a decomposition group at p,po is equivalent to one that arises from a finite flat group scheme over Zp,and det po restricted to an inertia group at p is the cyclotomic character. We say similarly that p is ordinary(at p)if,viewed as a representation to Q? there is a one-dimensional subspace of Q2 stable under a decomposition group at p and such that the action on the quotient space is unramified. Let e:Gal(Q/Q)Z denote the cyclotomic character.Conjectural converses toTheorem0.1 have been part of the folklore for many years but have hitherto lacked any evidence.The critical idea that one might dispense with compatible systems was already observed by Drinfeld in the function field case [Dr).The idea that one only needs to make a geometric condition on the restriction to the decomposition group at p was first suggested by Fontaine and Mazur.The following version is a natural extension of serre's coniecture which is convenient for stating ou r results and is,in a slightly modified form,the one proposed by Fontaine and Mazur.(In the form stated this incorporates Serre's conjecture.We could instead have made the hypothesis that po is modular.) CONJECTURE.Suppose that p:Gal(Q/Q)-GL2(O)is an irreducible lifting of po and that p is unramified outside of a finite set of primes.There are two cases (i)Assume that po is ordinary.Then if p is ordinary and det p=ek-1x for some integer k >2 and some x of finite order,p comes from a modular form. (ii)Assume that po is flat and that p is odd.Then if p restricted to a de- composition group at p is equivalent to a representation on a p-divisible group,again p comes from a modular form

446 ANDREW WILES If O is the ring of integers of a local field (containing Q,) we will say that p : Gal(Q/Q) -GL2(0) is a lifting of po if, for a specified embedding of the residue field of O in F, p and po are isomorphic over F,. Our point of view will be to assume that po is modular and then to attempt to give conditions under which a representation - p lifting po comes from a modular form in the sense that p - pf,~over Kf,xfor some f, A. We will restrict our attention to two cases: (I) po is ordinary (at p) by which we mean that there is a one-dimensional subspace of F;, stable under a decomposition group at p and such that the action on the quotient space is unramified and distinct from the action on the subspace. (11) po is flat (at p), meaning that as a representation of a decomposition group at p, po is equivalent to one that arises from a finite flat group scheme over Z, and det po restricted to an inertia group at p is the cyclotomic character. We say similarly that p is ordinary (at p) if, viewed as a representation to QE, there is a one-dimensional subspace of Q; stable under a decomposition group at p and such that the action on the quotient space is unramified. Let E : Gal(Q/Q) -+ Z: denote the cyclotomic character. Conjectural converses to Theorem 0.1 have been part of the folklore for many years but have hitherto lacked any evidence. The critical idea that one might dispense with compatible systems was already observed by Drinfeld in the function field case [Dr]. The idea that one only needs to make a geometric condition on the restriction to the decomposition group at p was first suggested by Fontaine and Mazur. The following version is a natural extension of Serre's conjecture which is convenient for stating our results and is, in a slightly modified form, the one proposed by Fontaine and Mazur. (In the form stated this incorporates Serre's conjecture. We could instead have made the hypothesis that po is modular.) CONJECTURE.Suppose that p: Gal(Q/Q) -+GL2(C3) is an irreducible lifting of po and that p is unramified outside of a finite set of primes. There are two cases: (i) Assume that po is ordinary. Then if p is ordinary and det p = &"IX for some integer k 2 2 and some x of finite order, p comes from a modular form. (ii) Assume that po is flat and that p is odd. Then if p restricted to a decomposition group at p is equivalent to a representation on a p-divisible group, again p comes from a modular form

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 447 In case (ii) it is not hard to see that if the form exists it has to be of weight 2; in [i) of course it would have weight k. One can of course enlarge this conjecture in several ways, by weakening the conditions in (i) and (ii), by considering other number fields in place of Q and by considering groups other than GL2. We prove two results concerning this conjecture. The first includes the hypothesis that po is modular. Here and for the rest of the paper we will assume that p is an odd prime. THEOREM 0.2. Suppose that po is irreducible and satisfies either (I) or (11) above. Suppose also that po is modular and that (i) po is absolutely irreducible when restricted to Q (J-a). (ii) If q q -1 modp is ramified in po then either polDq is reducible over the algebraic closure where Dq is a decomposition group at q or polIq is absolutely irreducible where Iq is an inertia group at q. Then any representation p as in the conjecture does indeed come from a modular form. The only condition which really seems essential to our method is the requirement that p~ be modular. The most interesting case at the moment is when p = 3 and po can be defined over F3. Then since PGL2(F3) E S4every such representation is modular by the theorem of Langlands and Tunnel1 mentioned above. In particular, every representation into GL2(Z3) whose reduction satisfies the given conditions is modular. We deduce: THEOREM 0.3. Suppose that E is an elliptic curve defined over Q and that po is the Galois action on the 3-division points. Suppose that E has the following properties: (i) E has good or multiplicative reduction at 3. (ii) po is absolutely irreducible when restricted to Q(-). tii"1) For awg q =- -lm~d3eithe~pQ(Dqis reducible over the algebraic closure or polIq is absolutely irreducible. Then E is modular. We should point out that while the properties of the zeta function follow directly from Theorem 0.2 the stronger version that E is covered by Xo(N)

448 AnDrew WILES requires also the isogeny theorem proved by Faltings (and earlier by Serre when E has nonintegral invariant,acase which includes the semistable curves) We note that if E is modular then so is any twist of E,so we could relax condition (i)somewhat. The important of semistable curves,with square-freecon ductor,satisfies(i)and (iii)but not necessarily (ii).If(ii)fails then in fact po is reducible.Rather surprisingly,Theorem 0.2 can often be applied in this case also by showing that the repre entation on the 5-division points also occurs for another elliptic curve which Theorem 0.3 has already proved modular.Thus Theorem 0.2 is applied this time with p=5.This argument,which is explained in Chapter 5,is the only part of the paper which really uses deformations of the elliptic curve rather than deformations of the Galois representation.The argument works more generally than in the semistable case but in this setting we obtain the following theorem: THEOREM 0.4.Suppose that E is a semistable elliptic curve defined over Q.Then E is modular. More general families of elliptic curves which are modular are given in Chap- ter 5. In 1986,stimulated by an ingenious idea of Frey Fr],Serre conjectured and Ribet proved(in Ril])a property of the Galois representations associated to modular forms which enabled Ribet to show that Theorem 0.4 implies 'Fer- mat's Last Theorem'.Frey's suggestion,in the notation of the follo wing thec rem,was to show that the (hypothetical)elliptic curve y2=x(x+uP)(x-uP) could not be modular.Such elliptic curves had already been studied in He but without the connection with modular forms s.Serre e made precise the ide of Frey by proposing a conjecture on modular forms which meant that the rep- resentation on the p-division points of this particular elliptic curve,if modular, would be associated to a form of conductor 2.This,by a simple inspection could not exist.Serre's conjecture was then proved by Ribet in the summer of 1986.However,one still needed to know that the curve in question would have to be modular,and this is accomplished by Theorem 0.4.We have then (finally!): THEOREM0.5.Suppose that uP+vP+wP=0 with u,v,w∈Qandp≥3， then uvw=0. The prove about the conjecture does not require the assumption that po be modular(since it is already known in this case)

448 ANDREW WILES requires also the isogeny theorem proved by Faltings (and earlier by Serre when E has nonintegral j-invariant, a case which includes the semistable curves). We note that if E is modular then so is any twist of E, so we could relax condition (i) somewhat. The important class of semistable curves, i.e., those with square-free conductor, satisfies (i) and (iii) but not necessarily (ii). If (ii) fails then in fact po is reducible. Rather surprisingly, Theorem 0.2 can often be applied in this case also by showing that the representation on the 5-division points also occurs for another elliptic curve which Theorem 0.3 has already proved modular. Thus Theorem 0.2 is applied this time with p = 5. This argument, which is explained in Chapter 5, is the only part of the paper which really uses deformations of the elliptic curve rather than deformations of the Galois representation. The argument works more generally than in the semistable case but in this setting we obtain the following theorem: THEOREM0.4. Suppose that E is a semistable elliptic curve defined over Q. Then E is modular. More general families of elliptic curves which are modular are given in Chap ter 5. In 1986, stimulated by an ingenious idea of Frey [Fr], Serre conjectured and Ribet proved (in [Rill) a property of the Galois representations associated to modular forms which enabled Ribet to show that Theorem 0.4 implies 'Fermat's Last Theorem'. Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve y2 = x(x +up)(x -up) could not be modular. Such elliptic curves had already been studied in [He] but without the connection with modular forms. Serre made precise the idea of Frey by proposing a conjecture on modular forms which meant that the rep resentation on the pdivision points of this particular elliptic curve, if modular, would be associated to a form of conductor 2. This, by a simple inspection, could not exist. Serre's conjecture was then proved by Ribet in the summer of 1986. However, one still needed to know that the curve in question would have to be modular, and this is accomplished by Theorem 0.4. We have then (finally!): THEOREM0.5. Suppose that up+vp+wp = 0 with u,v,w E Q andp > 3, then uvw = 0. The second result we prove about the conjecture does not require the assumption that po be modular (since it is already known in this case)

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 449 THEOREM 0.6.Suppose that po is irreducible and satisfies the hypotheses of the conjecture,including(I)above.Suppose further that (i)po=Ind ko for a character ko of an imaginary quadratic extension L of Q which is unramified at p. (ii)det=w Then a representation p as in the conjecture does indeed come from a modular form. This theorem can also be used to prove that certain families of elliptic curves are modular.In this summary we have only described the principal theorems associated to Galois representations and elliptic curves.Our results conc erning generalized class groups are described in Theorem 3.3 The following is an account of the origins of this work and of the more result.For several years I had been working on the Iwasawa conjecture for totally real fields and some applications of it.In the process,I had been using and developing results on (-ad representations associ iated to Hilbert modular forms.It was therefore natural for me to consider the problem of modularity from the point of view of (-adic representations.I began with the assumption that the reduction of a given ordinary -adic representation was reducible and tried to prove under this hypothesis that the representation itself would have to be modular.I hoped rather naively that in this situation I could apply the techniques of Iwasawa theory.Even more optimistically I hoped that the case =2 would be tractable as this would suffice for the study of the curves used by Frey.From now on and in the main text,we write p for e because of the connections with Iwasawa theory. After several months studying the -adic representation,I made the firs real breakthrough in realizing that I could use the 3-adic representation instead: the Langlands-Tunnell theorem meant that pa,the mod 3 representation of any given ellin Ptic curve over Q,would necessarily be modular. This enabled m to try inductively to prove that the GL2(Z/3"Z)representation would be modular for each n.At this time I considered only the ordinary case.This led quickly to the study of H(Gal(F/Q),W)for i=1 and 2,where F is the splitting field of the m-adic torsion on the Jacobian of a suitable modular curve. m being the maximal ideal of a Hecke ring associated to ps and Wr the module associated to a modular form f described in Chapter 1.More specifically.I needed to compare this cohomology with the cohomology of Gal(Q/Q)acting on the same module. I tried to apply some ideas from Iwasawa theory to this problem.In my fields [Wi,Ihad introduced

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 449 THEOREM 0.6. Suppose that po is irreducible and satisfies the hypotheses of the conjecture, including (I) above. Suppose further that (i) po = 1ndF KO for a character KO of an imaginary quadratic extension L of Q which is unramified at p. (ii) detpOllp= W. Then a representation p as in the conjecture does indeed come from a modular form. This theorem can also be used to prove that certain families of elliptic curves are modular. In this summary we have only described the principal theorems associated to Galois representations and elliptic curves. Our results concerning generalized class groups are described in Theorem 3.3. The following is an account of the origins of this work and of the more specialized developments of the 1980's that affected it. I began working on these problems in the late summer of 1986 immediately on learning of Ribet's result. For several years I had been working on the Iwasawa conjecture for totally real fields and some applications of it. In the process, I had been using and developing results on t-adic representations associated to Hilbert modular forms. It was therefore natural for me to consider the problem of modularity from the point of view of t-adic representations. I began with the assumption that the reduction of a given ordinary C-adic representation was reducible and tried to prove under this hypothesis that the representation itself would have to be modular. I hoped rather naively that in this situation I could apply the techniques of Iwasawa theory. Even more optimistically I hoped that the case t = 2 would be tractable as this would suffice for the study of the curves used by Frey. From now on and in the main text, we write p for t because of the connections with Iwasawa theory. After several months studying the 2-adic representation, I made the first real breakthrough in realizing that I could use the 3-adic representation instead: the Langlands-Tunnel1 theorem meant that p3, the mod 3 representation of any given elliptic curve over Q, would necessarily be modular. This enabled me to try inductively to prove that the GL2 (Z/3n Z) representation would be modular for each n. At this time I considered only the ordinary case. This led quickly to the study of ~~(Gal(p,/~), Wf) for i = 1and 2, where F, is the splitting field of the m-adic torsion on the Jacobian of a suitable modular curve, m being the maximal ideal of a Hecke ring associated to p3 and Wf the module associated to a modular form f described in Chapter 1. More specifically, I needed to compare this cohomology with the cohomology of Gal(Qc/Q) acting on the same module. I tried to apply some ideas from Iwasawa theory to this problem. In my solution to the Iwasawa conjecture for totally real fields [Wi4], I had introduced

450 ANDREW WILES a new technique in order to deal with the trivial zeroes.It involved replacing the standard Iwasawa theory method of considering the fields in the cyclotomic Z-extension by a similar analysis based on a choice of infinitely many distinct Some aspects of this method suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of Wr,might be to make a comparison between the cohomology groups as varies but with the field Q fixed.The new principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones.After reading the paper [Grel],I realized that the duality theorems in Galois cohomology of Poitou and Tate would be useful for this.The crucial extract tfrom this latter theory is in Section 2 of Chapter In order to put these ideas into practice I developed in a naive form the techniques of the first two sections of Chapter 2.This drew in particular on a detailed study of all the c ngruen ces betw en f and other m dular form of differing levels,a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first cohomology group well under two assumptions,first that a certain subgroup of the second cohomology vanished and sond that the form choeat the minimal level These assumptions were much too restrictive to be really effective but at least they pointed in the right direction.Some of these arguments are to be found in the second section of Chapter 1 and some form the first weak approximatior to the argument in Chapter 3.At that time.however.I used auxiliary primes g=-1 modp when varying E as the geometric techniques I worked with did mot apply in general for mes g=1modp (This was as for much the sam reason that the reduction of level argument in Ril]is much more difficult when g=1modp.)In all this work I used the more general assumption that Pp was modular rather than the assumption that p=3. In the late 1980's,I translated these ideas into ring-theoretic language.A few years previously Hida had constructed some explicit one-parameter fam- ilies of Galois representations.In an attempt to understand this.Mazur had been developing the language of Galois representati More- over,Mazur realized that the universal deformation rings he found should be given by Hecke rings,at least in certain special cases.This critical conjecture refined the exp ation that all ordinary liftings of modular repre ntations should be modular.In making the translation to this ring-theoretic language I realized that the vanishing assumption on the subgroup of H2 which I had needed should be replaced by the stronger condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one could estimate the number of generators and relations and so made the original assumption more plausible. To be of use,the defo rmation th ory required some development.Apart from some special examples examined by Boston and Mazur there had been

450 ANDREW WILES a new technique in order to deal with the trivial zeroes. It involved replacing the standard Iwasawa theory method of considering the fields in the cyclotomic Zp-extension by a similar analysis based on a choice of infinitely many distinct primes qi r 1modpnt with ni -+ co as i -t co. Some aspects of this method suggested that an alternative to the standard technique of Iwasawa theory, which seemed problematic in the study of Wf, might be to make a comparison between the cohomology groups as C varies but with the field Q fixed. The new principle said roughly that the unramified cohomology classes are trapped by the tamely ramified ones. After reading the paper [Grel], I realized that the duality theorems in Galois cohomology of Poitou and Tate would be useful for this. The crucial extract from this latter theory is in Section 2 of Chapter 1. In order to put these ideas into practice I developed in a naive form the techniques of the first two sections of Chapter 2. This drew in particular on a detailed study of all the congruences between f and other modular forms of differing levels, a theory that had been initiated by Hida and Ribet. The outcome was that I could estimate the first cohomology group well under two assumptions, first that a certain subgroup of the second cohomology group vanished and second that the form f was chosen at the minimal level for m. These assumptions were much too restrictive to be really effective but at least they pointed in the right direction. Some of these arguments are to be found in the second section of Chapter 1and some form the first weak approximation to the argument in Chapter 3. At that time, however, I used auxiliary primes q - -1 modp when varying C as the geometric techniques I worked with did not apply in general for primes q - lmodp. (This was for much the same reason that the reduction of level argument in [Rill is much more difficult when q E 1modp.) In all this work I used the more general assumption that pp was modular rather than the assumption that p = 3. In the late 1980's, I translated these ideas into ring-theoretic language. A few years previously Hida had constructed some explicit one-parameter families of Galois representations. In an attempt to understand this, Mazur had been developing the language of deformations of Galois representations. Moreover, Mazur realized that the universal deformation rings he found should be given by Hecke rings, at least in certain special cases. This critical conjecture refined the expectation that all ordinary liftings of modular representations should be modular. In making the translation to this ring-theoretic language I realized that the vanishing assumption on the subgroup of H2 which I had needed should be replaced by the stronger condition that the Hecke rings were complete intersections. This fitted well with their being deformation rings where one could estimate the number of generators and relations and so made the original assumption more plausible. To be of use, the deformation theory required some development. Apart from some special examples examined by Boston and Mazur there had been

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 451 little work on it.I checked that one could make the appropriate adiustments to the theory in order to describe deformation theories at the minimal level.in the fall of 1989,I set Ramakrishna,then a student of mine at Princeton,the task of proving the existence of a deformation theory associated to representations arising from finite flat group schemes over Zn.This was needed in order to e the restriction to the ordinary case.These developments are described in the first section of Chapter 1 although the work of Ramakrishna was not completed until the fall of 1991.For a long time the ring-theoretic version of the problem,although more natural,did not look any simpler.The usual methods of Iwasawa theory when translated into the ring-theoretic language seemed to require unknown principles of base change.One needed to know the exact relations between the Hecke rings for different fields in the cyclotomic Zp-extension of Q,and not just the relations up to torsion The turning point in this and indeed in the whole proof came in the spring of 1991.In searching for a clue from commutative algebra I had been particularly struck some y s earlier by a paper of Kunz [Ku2).I had already needed to verify that the Hecke rings were Gorenstein in order to compute the congruences developed in Chapter 2.This property had first been proved by Mazur in the case of prime level and his argument had already been extended by other authors as the need arose. Kunz's paper suggested the use of an invariant (the n-invariant of the appendix)which I saw could be used to test for isomorphisms between Gorenstein rings.A different invariant (the p/p2. invarian of the appendix)I had already o observed could be used to test fo isomorphisms between complete intersections.It was only on reading Section 6 of [Ti2]that I learned that it followed from Tate's account of Grothendieck dualty theory for complete intersections that these two invariants were eq for such rings.Not long afterwards I realized that,unlikely though it seemed at first,the equality of these invariants was actually a criterion for a Gorenstein ring to be a complete intersection.These argu nts are given in the appendix The impact of this result on the main problem was enormous.Firstly,th relationship between the hecke rings and the deformation rings could be tested just using these two invariants.In particular I could provide the inductive ar gument of Section 3 of Chapter 2 to show that if all lifting with restricted ramification are modular then all liftings are modular.This i had been trying to do for a long time but without success until the breakthrough in commuta tive algebra.Secondly, by means of a calculation of Hida sur amarized in Hi2 the main problem could be transformed into a problem about class numbers of a type well-known in Iwasawa theory.In particular,I could check this in the ordinary CM case using the recent theorems of Rubin and Kolyvagin.This is the content of Chapter 4.Thirdly,it meant that for the first time it could be verified that infinitely many j-invariants were modular.Finally,it meant that i could focus on the minimal level where the estimates given by my earlier

MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 451 little work on it. I checked that one could make the appropriate adjustments to the theory in order to describe deformation theories at the minimal level. In the fall of 1989, I set Ramakrishna, then a student of mine at Princeton, the task of proving the existence of a deformation theory associated to representations arising from finite flat group schemes over Zp. This was needed in order to remove the restriction to the ordinary case. These developments are described in the first section of Chapter 1 although the work of Ramakrishna was not completed until the fall of 1991. For a long time the ring-theoretic version of the problem, although more natural, did not look any simpler. The usual methods of Iwasawa theory when translated into the ring-theoretic language seemed to require unknown principles of base change. One needed to know the exact relations between the Hecke rings for different fields in the cyclotomic Zp-extension of Q, and not just the relations up to torsion. The turning point in this and indeed in the whole proof came in the spring of 1991. In searching for a clue from commutative algebra I had been particularly struck some years earlier by a paper of Kunz [Ku~]. I had already needed to verify that the Hecke rings were Gorenstein in order to compute the congruences developed in Chapter 2. This property had first been proved by Mazur in the case of prime level and his argument had already been extended by other authors as the need arose. Kunz's paper suggested the use of an invariant (the rl-invariant of the appendix) which I saw could be used to test for isomorphisms between Gorenstein rings. A different invariant (the p/p2- invariant of the appendix) I had already observed could be used to test for isomorphisms between complete intersections. It was only on reading Section 6 of [Ti21 that I learned that it followed from Tate's account of Grothendieck duality theory for complete intersections that these two invariants were equal for such rings. Not long afterwards I realized that, unlikely though it seemed at first, the equality of these invariants was actually a criterion for a Gorenstein ring to be a complete intersection. These arguments are given in the appendix. The impact of this result on the main problem was enormous. Firstly, the relationship between the Hecke rings and the deformation rings could be tested just using these two invariants. In particular I could provide the inductive argument of Section 3 of Chapter 2 to show that if all liftings with restricted ramification are modular then all liftings are modular. This I had been trying to do for a long time but without success until the breakthrough in commutative algebra. Secondly, by means of a calculation of Hida summarized in [Hi21 the main problem could be transformed into a problem about class numbers of a type well-known in Iwasawa theory. In particular, I could check this in the ordinary CM case using the recent theorems of Rubin and Kolyvagin. This is the content of Chapter 4. Thirdly, it meant that for the first time it could be verified that infinitely many j-invariants were modular. Finally, it meant that I could focus on the minimal level where the estimates given by my earlier

452 ANDREW WILES Galois cohomology calculations looked more promising.Here I was also using the work of Ribet and others on Serre's conjecture (the same work of Ribet that had linked Fermat's Last Theorem to modular forms in the first place)to know that there was a minimal level The class number problem was of a type well-known in Iwasawa theory and in the ordinary case had already been conjectured by Coates and Schmidt. However,the traditional methods of Iwasawa theory did not seem quite suf ficient in this case and,as explained earlier,when translated into the ring- theoretic language seemed to require unknown principles of base change.So instead I developed further the idea of using auxiliary primes to replace the change of field that is used in iwasawa theory.The galois cohomology esti mates described in Chapter 3 were now much stronger,although at that time I was still using primes g -1modp for the argument.The e main difficulty was that although I knew how the n-invariant changed as one Dassed to an auxiliary level from the results of Chapter 2,I did not know how to estimate the change in the p/p2-invariant precisely.However,the method did give the right bound for the generalised class group,or Selmer group as it is often called in this context,under the additional assumption that the minimal Hecke ring was a complete intersection. I had earlier realized that ideally what I needed in this method of auxiliary primes was a replacement for the power series ring construction one obtains in the more natural approach based on Iwasawa theory.In this more usual setting. the projective limit of the Hecke rings for the varying fields in a cyclotomi tower would be expected to be a power series ring,at least if one assumed the vanishing of the u-invariant.However,in the setting with auxiliary primes where one would change the lev el but not the field,the natural limiting process did not appear to be helpful,with the exception of the closely related and very important construction of Hida Hill.This method of Hida often gave one step wards a power series ring in the ordinary case. ofn ment in thry (Schol W without success for the key. Then,in August,1991,I learned of a new construction of Flach [Fl]and quickly became convinced that an extension of his method was more plausi ble.Flach's approach seemed to be the first step towards the construction of an Euler system,an approach which would give the precise upper bound for the size of the Selmer group if it could be completed.By the fall of 1992,I believed I had achieved this and began then to consider the remaining case where the mod3 representation was assumed reducible.For several months I tried simply to repeat the methods using deformation rings and Hecke rings Then unexpectedly in May 1993,on reading of a construction of twisted forms of modular curves in a paper of Mazur [Ma3),I made a crucial and surprising breakthrough:I found the argument using families of elliptic curves with 8

452 ANDREW WILES Galois cohomology calculations looked more promising. Here I was also using the work of Ribet and others on Serre's conjecture (the same work of Ribet that had linked Fermat's Last Theorem to modular forms in the first place) to know that there was a minimal level. The class number problem was of a type well-known in Iwasawa theory and in the ordinary case had already been conjectured by Coates and Schmidt. However, the traditional methods of Iwasawa theory did not seem quite sufficient in this case and, as explained earlier, when translated into the ringtheoretic language seemed to require unknown principles of base change. So instead I developed further the idea of using auxiliary primes to replace the change of field that is used in Iwasawa theory. The Galois cohomology estimates described in Chapter 3 were now much stronger, although at that time I was still using primes q = -1 modp for the argument. The main difficulty was that although I knew how the 7-invariant changed as one passed to an auxiliary level from the results of Chapter 2, I did not know how to estimate the change in the p/p2-invariant precisely. However, the method did give the right bound for the generalised class group, or Selmer group as it is often called in this context, under the additional assumption that the minimal Hecke ring was a complete intersection. I had earlier realized that ideally what I needed in this method of auxiliary primes was a replacement for the power series ring construction one obtains in the more natural approach based on Iwasawa theory. In this more usual setting, the projective limit of the Hecke rings for the varying fields in a cyclotomic tower would be expected to be a power series ring, at least if one assumed the vanishing of the p-invariant. However, in the setting with auxiliary primes where one would change the level but not the field, the natural limiting process did not appear to be helpful, with the exception of the closely related and very important construction of Hida [Hill. This method of Hida often gave one step towards a power series ring in the ordinary case. There were also tenuous hints of a patching argument in Iwasawa theory ([Scho], [Wi4, §lo]), but I searched without success for the key. Then, in August, 1991, I learned of a new construction of Flach [Fl] and quickly became convinced that an extension of his method was more plausible. Flach's approach seemed to be the first step towards the construction of an Euler system, an approach which would give the precise upper bound for the size of the Selmer group if it could be completed. By the fall of 1992, I believed I had achieved this and began then to consider the remaining case where the mod 3 representation was assumed reducible. For several months I tried simply to repeat the methods using deformation rings and Hecke rings. Then unexpectedly in May 1993, on reading of a construction of twisted forms of modular curves in a paper of Mazur [Ma3], I made a crucial and surprising breakthrough: I found the argument using families of elliptic curves with a