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WILES' PROOF OF FERMAT'S LAST THEOREM K. RUBIN AND A. SILVERBERG

Introduction

On June 23, 1993, Andrew Wiles wrote on a blackboard, before an audience at the Newton Institute in Cambridge, England, that if p is a prime number, u, v, and w are rational numbers, and up + vp + wp = 0, then uvw = 0. In other words, he announced that he could prove Fermat's Last Theorem. His announcement came at the end of his series of three talks entitled \Modular forms, elliptic curves, and Galois representations" at the week-long workshop on \p-adic Galois representations, Iwasawa theory, and the Tamagawa numbers of motives". In the margin of his copy of the works of Diophantus, next to a problem on Pythagorean triples, Pierre de Fermat (1601 - 1665) wrote: Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in in nitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caparet.

(It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.) We restate Fermat's conjecture as follows. Fermat's Last Theorem. If n > 2, then an + bn = cn has no solutions in nonzero integers a, b, and c. A proof by Fermat has never been found, and the problem remained open, spurring number theorists to ever greater heights. For details on the history of Fermat's Last Theorem (last because it is the last of Fermat's questions to be answered) see [5], [6], and [27]. What Andrew Wiles announced in Cambridge was that he could prove \many" elliptic curves are modular, suciently many to imply Fermat's Last Theorem. In this paper we will explain Wiles' result and its connection with Fermat's Last Theorem. In x1 we introduce elliptic curves and modularity, The authors thank the NSF for nancial support. 1

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and give the connection between Fermat's Last Theorem and the TaniyamaShimura Conjecture on the modularity of elliptic curves. In x2 we describe how Wiles reduces the proof of the Taniyama-Shimura conjecture to what we call the Modular Lifting Conjecture (which can be viewed as a weak form of the Taniyama-Shimura Conjecture), by using a theorem of Langlands and Tunnell. In x3 and x4 we show how the Modular Lifting Conjecture is related to a conjecture of Mazur on deformations of Galois representations (Conjecture 4.2), and in x5 we describe Wiles' method of attack on this conjecture. Although he does not prove the full Mazur Conjecture (and thus does not prove the full Taniyama-Shimura Conjecture), Wiles' result (Theorem 5.3) implies enough of the Modular Lifting Conjecture to prove Fermat's Last Theorem. Much of this report is based on notes from Wiles' lectures in Cambridge. The authors apologize for any errors we may have introduced. We also apologize to those whose mathematical contributions we, due to our incomplete understanding, do not properly acknowledge. As this paper is being completed (early November 1993), Wiles' proof is being checked by referees. Because of the great interest in this subject and the lack of a publicly available manuscript, we hope this report will be useful to the mathematics community. In order to make this survey as accessible as possible to non-specialists, the more technical details are postponed as long as possible, some of them to the Appendices. The integers, rational numbers, complex numbers, and p-adic integers will be denoted Z, Q, C, and Zp , respectively. If F is a eld, then F denotes an algebraic closure of F. 1. Connection between Fermat's Last Theorem and elliptic curves

1.1. Fermat's Last Theorem follows from the modularity of elliptic curves. Suppose Fermat's Last Theorem were false. Then there would exist

nonzero integers a, b, c, and n > 2, such that an + bn = cn . It is easy to see that no generality is lost by assuming that n is a prime greater than three (or greater than four million, by [2]; see [14] for n = 3 and 4), and that a and b are relatively prime. Write down the cubic curve: (1)

y2 = x(x + an)(x ? bn ):

In x1.3 we will see that such curves are elliptic curves, and in x1.4 we will explain what it means for an elliptic curve to be modular. Kenneth Ribet [28] proved that if n is a prime greater than three, a, b, and c are nonzero integers, and an + bn = cn, then the elliptic curve (1) is not modular.

WILES' PROOF OF FERMAT'S LAST THEOREM

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Theorem 1.1 (Wiles). If A and B are distinct, non-zero, relatively prime integers, and AB(A ? B) is divisible by 16, then the elliptic curve y2 = x(x + A)(x + B) is modular.

Taking A = an and B = ?bn with a, b, c, and n coming from our hypothetical solution to a Fermat equation as above, we see that the conditions of Theorem 1.1 are satis ed since n  5 and one of a, b, and c is even. Thus Theorem 1.1 and Ribet's result together imply Fermat's Last Theorem!

1.2. History. The story of the connection between Fermat's Last Theorem and elliptic curves begins in 1955, when Yutaka Taniyama (1927 - 1958) posed problems which may be viewed as a weaker version of the following conjecture (see [39]).

Taniyama-Shimura Conjecture. Every elliptic curve over Q is modular. The conjecture in the present form was made by Goro Shimura around 196264, and has become better understood due to work of Shimura [34], [35], [36], [37], [38] and of Andre Weil [43] (see also [7]). Beginning in the late 1960's ([16], [17], [18], [19]), Yves Hellegouarch connected Fermat equations an + bn = cn with elliptic curves of the form (1), and used results about Fermat's Last Theorem to prove results about elliptic curves. The landscape changed abruptly in 1985 when Gerhard Frey stated in a lecture at Oberwolfach that elliptic curves arising from counterexamples to Fermat's Last Theorem could not be modular [11]. Shortly thereafter Ribet [28] proved this, following ideas of Jean-Pierre Serre [33] (see [25] for a survey). In other words, \Taniyama-Shimura Conjecture ) Fermat's Last Theorem". Thus, the stage was set. A proof of the Taniyama-Shimura Conjecture (or enough of it to know that elliptic curves coming from Fermat equations are modular) would be a proof of Fermat's Last Theorem.

1.3. Elliptic curves. De nition. An elliptic curve over Q is a nonsingular curve de ned by an equation of the form

(2)

y2 + a1xy + a3 y = x3 + a2 x2 + a4 x + a6

where the coecients ai are integers. The solution (1; 1) will be viewed as a point on the elliptic curve.

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Remarks. (i) A singular point on a curve f(x; y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points. (ii) Two elliptic curves over Q are isomorphic if one can be obtained from the

other by changing coordinates x = A2 x0 + B, y = A3y0 + Cx0 + D, with A, B, C, D 2 Q, and dividing through by A6 . (iii) Every elliptic curve over Q is isomorphic to one of the form y 2 = x 3 + a 2 x2 + a 4 x + a 6 with integers ai . A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots. Example. The equation y2 = x(x+32 )(x ? 42 ) de nes an elliptic curve over Q. 1.4. Modularity. Let H denote the complex upper half plane fz 2 C : Im(z) > 0g where Im(z) is the imaginary part of z. If N is a positive integer, de ne a group of matrices
? ?0 (N) = ac db 2 SL2 (Z) : c is divisible by N : ?
The group ?0(N) acts on H by linear fractional transformations ac db (z) = az+b cz+d : The quotient space H=?0(N) is a (non-compact) Riemann surface. It can be completed to a compact Riemann surface, denoted X0 (N), by adjoining the cusps, which are the nitely many equivalence classes of Q [ fi1g under the action of ?0 (N) (see Chapter 1 of [36]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface. De nition. An elliptic curve E is modular if, for some integer N, there is a holomorphic map from X0 (N) onto E. Example. There is a (holomorphic) isomorphism from X0 (15) onto the elliptic curve y2 = x(x + 32)(x ? 42). Remark. There are many equivalent de nitions of modularity (see xII.4.D of [25] and appendix of [23]). In some cases the equivalence is a deep result. For Wiles' proof of Fermat's Last Theorem it suces to use only the de nition given in x1.7 below.

1.5. Semistability. De nition. An elliptic curve over Q is semistable at the prime q if it is isomorphic to an elliptic curve over Q which modulo q either is nonsingular or has a singular point with two distinct tangent directions. An elliptic curve over Q is called semistable if it is semistable at every prime. Example. The elliptic curve y2 = x(x + 32)(x ? 42) is semistable because it is isomorphic to y2 + xy + y = x3 + x2 ? 10x ? 10, but the elliptic curve y2 = x(x + 42)(x ? 32) is not semistable (it is not semistable at 2).

WILES' PROOF OF FERMAT'S LAST THEOREM

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In x5.5 (Theorem 5.3) we state Wiles' main result, and explain how it implies the following theorem.

Theorem 1.2 (Wiles). Every semistable elliptic curve over Q is modular. If A and B are distinct, nonzero, relatively prime integers write EA;B for the elliptic curve de ned by y2 = x(x + A)(x + B). Since EA;B and E?A;?B are isomorphic over the complex numbers (i.e., as Riemann surfaces), EA;B is modular if and only if E?A;?B is modular. If further AB(A ? B) is divisible by 16, then either EA;B or E?A;?B is semistable (this is easy to check directly; see for example xI.1 of [25]), and therefore both are modular by Theorem 1.2. Thus Theorem 1.2 implies Theorem 1.1, and hence Fermat's Last Theorem.

1.6. Modular forms. In this paper we will work with a de nition of modu-

larity which uses modular forms.

De nition. If N is a positive integer, a modular form f of weight k for ?0(N) is a holomorphic function f : H ! C which satis es
? f( (z)) = (cz + d)k f(z) for every = ac db 2 ?0 (N) and z 2 H; and is holomorphic at the cusps (see Chapter 2 of [36]).

(3)

?




A modular form f satis es f(z) =Pf(z + 1) (apply (3) to 10 11 2 ?0(N)), 2inz , with complex numbers a so it has a Fourier expansion f(z) = 1 n n=0 ane and with n  0 because f is holomorphic at the cusp i1. We say f is a cusp form if it vanishes at all the cusps; in particular for a cusp form the coecient a0 (the value at i1) is zero. Call a cusp form normalized if a1 = 1. For xed N there are commuting linear operators (called Hecke operators) Tm , for integers m  1, on the vector space of cusp forms of weight two for P 2inz then ?0(N) (see Chapter 3 of [36]). If f(z) = 1 n=1 an e (4)

Tm f(z) =

1? X X )=1 n=1 (dd;N j(n;m)




danm=d e2inz 2

where (a; b) denotes the greatest common divisor of a and b and a j b means that a divides b. The Hecke algebra T(N) is the ring generated by these operators.

De nition. In this paper an eigenform will mean a normalized cusp form of weight two for some ?0 (N) which is an eigenfunction for all the Hecke operators.

P 2inz is an eigenform then T f = a f for all m. By (4), if f(z) = 1 m m n=1 an e

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1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the nite eld with p elements, and let E(Fp ) denote the Fp-solutions of the equation for E (including the point at in nity). We now

give a second de nition of modularity for an elliptic curve. De nition. An elliptic curve E over Q is modular if there exists an eigenform P1 2inz such that for all but nitely many primes q, a e n n=1 aq = q + 1 ? #(E(Fq )):

(5)

2. An overview The owchart (Figure 1) shows how Fermat's Last Theorem would follow if one knew the Modular Lifting Conjecture (Conjecture 2.1 below) for the primes 3 and 5. In x1.1 we discussed the upper arrow, i.e., the implication \Taniyama-Shimura Conjecture ) Fermat's Last Theorem". In this section we will discuss the other implications in the owchart. The implication given by the lowest arrow is straightforward (Proposition 2.3), while the middle one uses an ingenious idea of Wiles (Proposition 2.4). Fermat's Last Theorem

6 Taniyama-Shimura Conjecture

Taniyama-Shimura for E;3 irreducible

Tunnell-Langlands Theorem

6 ??@@

6 ??@@

Modular Lifting Conjecture for p = 5

Modular Lifting Conjecture for p = 3

Figure 1. Modular Lifting Conjecture ) Fermat's Last Theorem

The Modular Lifting Conjecture is still an open problem, even for the primes 3 and 5. However, Wiles proves enough of the Modular Lifting Conjecture so

WILES' PROOF OF FERMAT'S LAST THEOREM

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that, with some additional work, he can still obtain enough of the TaniyamaShimura Conjecture to prove Fermat's Last Theorem (see x5.5). 2.1. Modular Lifting Conjecture. Let Q denote the algebraic closure of Q  =Q). If p is a prime, write in C, and let GQ be the Galois group Gal(Q !p : GQ ! Fp for the character giving the action of GQ on the p-th roots of unity. For the facts about elliptic curves stated below see [40]. If E is an elliptic curve over Q, and F is a sub eld of the complex numbers, there is a natural commutative group law on the set of F-solutions of E, with the point at in nity as the identity element. Denote this group E(F). If p is a prime, write E[p] for the subgroup of points in E(Q ) of order dividing p. Then E[p]  = F2p . The action of GQ on E[p] gives a continuous representation E;p : GQ ! GL2 (Fp ) (de ned up to isomorphism) such that (6) det(E;p ) = !p and for all but nitely many primes q, (7) trace(E;p (Frobq ))  q + 1 ? #(E(Fq )) (mod p): (See Appendix A for the de nition of the Frobenius elements Frobq 2 GQ attached to each prime number q.) P 2inz is an eigenform, let O denote the ring of integers If f(z) = 1 f n=1 an e of the number eld Q(a2; a3; : : :). The following conjecture is in the spirit of a conjecture of Mazur (see Conjecture 3.2). Conjecture 2.1 (Modular Lifting Conjecture). Suppose p is a prime and E is an elliptic curve over Q satisfying (a) E;p is irreducible, P 2inz and a prime ideal  of O (b) there are an eigenform f(z) = 1 f n=1 ane such that p 2  and for all but nitely many primes q, aq  q + 1 ? #(E(Fq )) (mod ): Then E is modular. Wiles does not prove the full Modular Lifting Conjecture, but proves it subject to some additional hypotheses on E;p . The Modular Lifting Conjecture is a priori weaker than the TaniyamaShimura Conjecture because of the extra hypotheses (a) and (b). The more serious condition is (b); there is no known way to produce such a form in general. But when p = 3 the existence of such a form follows from the theorem

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below of Tunnell [42] and Langlands [21]. Wiles then gets around condition (a) by a clever argument (described below) which, when E;3 is not irreducible, allows him to use p = 5 instead. 2.2. Langlands-Tunnell Theorem. In order to state the Langlands-Tunnell Theorem, we need weight one modular forms for a subgroup of ?0 (N). Let
? ?1(N) = ac db 2 SL2 (Z) : c  0 (mod N); a  d  1 (mod N) : Replacing ?0 (N) by ?1 (N) in x1.6, one can de ne the notion of cusp forms on ?1(N). See Chapter 3 of [36] for the de nitions of the Hecke operators on the space of weight one cusp forms for ?1 (N). Theorem 2.2 (Langlands-Tunnell). Suppose  : GQ ! GL2(C) is a continuous irreducible representation whose image in PGL2(C) is a subgroup of S4 (the symmetric group on four elements),  is complex conjugation, and P 2inz for some det(()) = ?1. Then there is a weight one cusp form 1 b n=1 n e ?1(N), which is an eigenfunction for all the corresponding Hecke operators, such that for all but nitely many primes q, (8) bq = trace((Frobq )): The theorem as stated by Langlands [21] and by Tunnell [42] produces an automorphic representation, rather than a cusp form. Using the fact that det(()) = ?1, standard techniques (see for example [12]) show that this automorphic representation corresponds to a weight one cusp form as in Theorem 2.2.

2.3. Modular Lifting Conjecture ) Taniyama-Shimura Conjecture. Proposition 2.3. Suppose the Modular Lifting Conjecture is true for p = 3,

E is an elliptic curve, and E;3 is irreducible. Then E is modular. Proof. It suces to show that hypothesis (b) of the Modular Lifting Conjecture is satis ed with the given curve E, for p = 3. There is a faithful representation p : GL2(F3 ) ,! GL2 (Z[ ?2])  GL2(C) such that for every g 2 GL2(F3 ), p (9) trace( (g))  trace(g) (mod (1 + ?2)) and (10) det( (g))  det(g) (mod 3):
?
?p (Explicitly, can be given by () = ??11 10 and ( ) = 1?2 10 where
? ?1 1
?  = ?1 0 and = 11 ?11 generate GL2(F3 ).) Let  =  E;3 . If  is complex conjugation then it follows from (6) and (10) that det(()) = ?1.

WILES' PROOF OF FERMAT'S LAST THEOREM

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The image of in PGL2 (C) is a subgroup of PGL2(F3 )  = S4 . Using that E;3 is irreducible one can show that  is irreducible. p P 2inz be Let p be a prime of Q containing 1 + ?2. Let g(z) = 1 n=1 bn e a weight one cusp form for some ?1(N) obtained by applying the LanglandsTunnell Theorem (Theorem 2.2) to . The function

E(z) = 1 + 6

8