Annals of Mathematics,142(1995),443-551 Modular elliptic curves and Fermat's Last Theorem By ANDREW WILES* For Nada,Clare,Kate and Olivia Cubum autem in duos cubos,aut quadratoguadratum in duos m in infinitum cujus ret demonstrationem mirabilem sane deteri.Hanc marginis exiguitas non caperet. Pierre de fermat Introduction An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form Xo(N).Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a ctional equation of the standard type.If an elliptic curve over Q with a given j-invariant is modular it ise ves with the same j-invariant are modular (in which case we say that the j-invariant is modular).A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950's and 1960's asserts that every elliptic curve over Q is modular.However,it only became widely known through its publication in a paper of Weil in 1967 Wel (as an xercise for the interested r der!),in which. moreover,Weil gave co eptual evidence for the conjecture.Alth ugh it hac been numerically verified in many cases,prior to the results described in this paper it had only been known that finitely many j-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should PeratsLast Theorem.The precise mechanism relting the err as the e-conj ctur e and this was then proved by Ribet in the summer of 1986.Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem. The workon this paper was supported byan NSF grant
Annals of Mathematics, 142 (1995), 443-551 Modular elliptic curves and Fermat 's Last Theorem For Nada, Clare, Kate and Olivia Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. Pierre de Femnat Introduction An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form Xo(N).Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a functional equation of the standard type. If an elliptic curve over Q with a given j-invariant is modular then it is easy to see that all elliptic curves with the same j-invariant are modular (in which case we say that the j-invariant is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950's and 1960's asserts that every elliptic curve over Q is modular. However, it only became widely known through its publication in a paper of Weil in 1967 [We] (as an exercise for the interested reader!), in which, moreover, Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the results described in this paper it had only been known that finitely many j-invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should imply Fermat's Last Theorem. The precise mechanism relating the two was formulated by Serre as the &-conjecture and this was then proved by Ribet in the summer of 1986. Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem. *The work on this paper was supported by an NSF grant
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 ove 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 (II above.Suppose also that po is modular and that )）oisabseredube when restricted oQ（V-l)号p】 (ii)If q=-1modp is ramified in po then either polD is reducible over the algebraic closure where De is a decomposition group at g or po is absolutely irreducible where Ig is an inertia group atq Then any representation p as in the conjecture does indeed come from a mod ular form. The only condition which really seems essential to our method is the re- quirement that po be modular. The most interesting case at the moment is when p=3 and po can be de fined over F3.Then since PGL2(Fa)Severy such representation is modular by the theorem of Langlands and Tunnell mentioned above.In particular,ev- ery 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(3 For any=-1mod3either is reducible over the algebraic closure or po 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)
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)