# 大学数学 费马大定理的证明 英文版.pdf

Annals of Mathematics, 141 (1995), 443-552 Pierre de Fermat Andrew John Wiles Modular elliptic curves and Fermat’s Last Theorem By Andrew John Wiles* ForNada,Claire,KateandOlivia Cubumauteminduoscubos,autquadratoquadratuminduosquadra- toquadratos, et generaliter nullam in inﬁnitum ultra quadratum potestatum in duos ejusdem nominis fas est dividere: cujes rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas noncaperet. -PierredeFermat ∼ 1637 Abstract. When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed byit that he decided that he would be the ﬁrst person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a,b,c,n with n2 such that a n + b n = c n . This object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollarybyvirtue of work byFrey, Serre and Ribet. Introduction An elliptic curve over Q is said to be modular if it has a ﬁnite covering by a modular curve of the form X 0 (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 ε-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 byan NSF grant. 444 ANDREW JOHN WILES Our approach to the study of elliptic curves is via their associated Galois representations. Suppose that ρ p is the representation of Gal( ¯ Q/Q)onthe p-division points of an elliptic curve over Q, and suppose for the moment that ρ 3 is irreducible. The choice of 3 is critical because a crucial theorem of Lang- lands and Tunnell shows that if ρ 3 is irreducible then it is also modular. We then proceed by showing that under the hypothesis that ρ 3 is semistable at 3, together with some milder restrictions on the ramiﬁcation of ρ 3 at the other primes, every suitable lifting of ρ 3 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 ε-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 ρ 3 be irreducible at 3 is bypassed by means of an intriguing argument with families of elliptic curves which share a common ρ 5 . 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 Γ 1 (N)of SL 2 (Z) of weight k ≥ 2 and character χ. Thus if T n is the Hecke operator associated to an integer n there is an algebraic integer c(n,f) such that T n f = c(n,f)f for each n. We let K f be the number ﬁeld generated over Q by the {c(n,f)} together with the values of χ and let O f be its ring of integers. For any prime λ of O f let O f,λ be the completion of O f at λ. The following theorem is due to Eichler and Shimura (for k = 2) and Deligne (for k2). 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 p ∈ Z and each prime λ|p of O f there isacontinuousrepresentation ρ f,λ : Gal( ¯ Q/Q) −→ GL 2 (O f,λ ) whichisunramiﬁedoutsidetheprimesdividingNpandsuchthatforallprimes qnotbarNp, trace ρ f,λ (Frob q)=c(q,f), det ρ f,λ (Frob q)=χ(q)q k−1 . We will be concerned with trying to prove results in the opposite direction, that is to say, with establishing criteria under which a λ-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 λ-adic representations except that the proof may be easier for some λ than for others. Assume ρ 0 : Gal( ¯ Q/Q) −→ GL 2 ( ¯ F p ) is a continuous representation with values in the algebraic closure of a ﬁnite ﬁeld of characteristic p and that det ρ 0 is odd. We say that ρ 0 is modular if ρ 0 and ρ f,λ mod λ are isomorphic over ¯ F p for some f and λ and some embedding of O f /λ in ¯ F p . Serre has conjectured that every irreducible ρ 0 of odd determinant is modular. Very little is known about this conjecture except when the image of ρ 0 in PGL 2 ( ¯ F p ) is dihedral, A 4 or S 4 . In the dihedral case it is true and due (essentially) to Hecke, and in the A 4 and S 4 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 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 ρ f,λ = ρ f,λ mod λ is independent of the choice of lattice in K 2 f,λ . 446 ANDREW JOHN WILES If O is the ring of integers of a local ﬁeld (containing Q p ) we will say that ρ : Gal( ¯ Q/Q) −→ GL 2 (O) is a lifting of ρ 0 if, for a speciﬁed embedding of the residue ﬁeld of O in ¯ F p , ¯ρ and ρ 0 are isomorphic over ¯ F p . Our point of view will be to assume that ρ 0 is modular and then to attempt to give conditions under which a representation ρ lifting ρ 0 comes from a modular form in the sense that ρ similarequal ρ f,λ over K f,λ for some f,λ. We will restrict our attention to two cases: (I) ρ 0 is ordinary (at p) by which we mean that there is a one-dimensional subspace of ¯ F 2 p , 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) ρ 0 is ﬂat (at p), meaning that as a representation of a decomposition group at p,ρ 0 is equivalent to one that arises from a ﬁnite ﬂat group scheme over Z p , and detρ 0 restricted to an inertia group at p is the cyclotomic character. We say similarly that ρ is ordinary (at p), if viewed as a representation to ¯ Q 2 p , there is a one-dimensional subspace of ¯ Q 2 p stable under a decomposition group at p and such that the action on the quotient space is unramiﬁed. Let ε : Gal( ¯ Q/Q) −→ Z × p 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 Drinﬁeld 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 ρ 0 is modular.) Conjecture. Suppose that ρ : Gal( ¯ Q/Q) −→ GL 2 (O) is an irreducible lifting of ρ 0 and that ρ is unramiﬁed outside of a ﬁnite set of primes. There aretwocases: (i) Assumethat ρ 0 isordinary. Thenif ρ isordinaryand detρ = ε k−1 χ for some integer k ≥ 2 and some χ of ﬁnite order, ρ comes from a modular form. (ii) Assume that ρ 0 is ﬂat and that p is odd. Then if ρ restricted to a de- composition group at p is equivalent to a representation on a p-divisible group, again ρ comesfromamodularform. 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 of Q and by considering groups other than GL 2 . We prove two results concerning this conjecture. The ﬁrst includes the hypothesis that ρ 0 is modular. Here and for the rest of this paper we will assume that p is an odd prime. Theorem 0.2. Suppose that ρ 0 is irreducible and satisﬁes either (I) or (II) above. Supposealsothat ρ 0 ismodularandthat (i) ρ 0 isabsolutelyirreduciblewhenrestrictedto Q parenleftbigg radicalBig (−1) p−1 2 p parenrightbigg . (ii) If q ≡−1modp is ramiﬁed in ρ 0 then either ρ 0 | D q is reducible over the algebraic closure where D q is a decomposition group at q or ρ 0 | I q is absolutelyirreduciblewhere I q isaninertiagroupat q. Thenanyrepresentation ρ asintheconjecturedoesindeedcomefromamod- ularform. The only condition which really seems essential to our method is the re- quirement that ρ 0 be modular. The most interesting case at the moment is when p = 3 and ρ 0 can be de- ﬁned overF 3 . Then since PGL 2 (F 3 ) similarequal S 4 every such representation is modular by the theorem of Langlands and Tunnell mentioned above. In particular, ev- ery representation into GL 2 (Z 3 ) 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 ρ 0 is the Galois action on the 3-division points. Suppose that E has the followingproperties: (i) E hasgoodormultiplicativereductionat 3. (ii) ρ 0 isabsolutelyirreduciblewhenrestrictedto Q parenleftbig√ −3 parenrightbig . (iii)Forany q ≡−1mod3either ρ 0 | D q isreducibleoverthealgebraicclosure or ρ 0 |I q isabsolutelyirreducible. Then E shouldbemodular. 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 X 0 (N) 448 ANDREW JOHN 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 con- ductor, satisﬁes (i) and (iii) but not necessarily (ii). If (ii) fails then in fact ρ 0 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 the semistable case but in this setting we obtain the following theorem: Theorem 0.4. Supposethat E isasemistableellipticcurvedeﬁnedover Q. Then E ismodular. 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 [Ri1]) a property of the Galois representation 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 following theo- rem, was to show that the (hypothetical) elliptic curve y 2 = x(x+u p )(x−v p ) 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 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