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The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles Gerd Faltings The proof of the conjecture mentioned in sider Eas those solutions in Q, R, or C, denoted, the title was finally completed in Septem- respectively, E(Q), E(R), and E(C). One usually ber of 1994. A.Wiles announced this result includes in this set an infinitely distant point, in the summer of 1993; however, there was a gap denoted 1. With this addition, the solution set in his work. The paper of Taylor and Wiles does has the structure of an abelian group, with 1 not close this gap but circumvents it. This article as the neutral element. The inverse of (x;y) is is an adaptation of several talks that I have given (x;−y), and the sum of three points vanishes if on this topic and is by no means about my own they lie on a line. The group addition is given by work. I have tried to present the basic ideas to a algebraic functions. As a group E(Q) is finitely wider mathematical audience, and in the process generated (Mordell’s Theorem), E(R) is isomor- I have skipped over certain details, which are in phic to R=Z or to R=Z(cid:2)Z=2Z, and my opinion not so much of interest to the non- E(C)(cid:24)= C=lattice(for example, y2=x3−xyields specialist. The specialists can then alleviate their the lattice Z(cid:8)Zi). For an integer nlet E[n]de- boredom by finding those mistakes and correct- note the n-division points, that is, the kernel of ing them. multiplication by n. Over Cthese are isomorphic to (Z=nZ)2, and the coordinates are algebraic Elliptic Curves numbers. For example, the 2-division points are For our purposes an elliptic curve Eis given as exactly 1and the three zeros of f(where y =0). the set of solutions fx;yg of an equation The absolute Galois group Gal(Q=Q) acts on y2=f(x), where f(x)=x3+:::is a polynomial of degree three. Usually E is defined over the ra- Gerd Faltings is affiliated with the Max-Planck-Insti- tional numbers Q; that is, the coefficients of f tute für Mathematik in Bonn, Germany. are in Q. We also demand that all three zeros of Translated from Testausdruck DMV Mitteilungen 27. fare distinct (Eis “nonsingular”). We may con- März 1995 für 2/95by UweF. Mayer, University of Utah. JULY1995 NOTICESOFTHEAMS 743 is also true at all prime divisors p>2 of abc. Therefore the l-division points behave as if Ehad good reduction at all p>2. However, as we shall see, there are no semistable elliptic curves over Qwith this property, and this is the desired contradiction. In order to reach the goal this way, one has to replace elliptic curves by modular forms. That this can be done follows from the conjecture of Taniyama-Weil (which essentially is due to Shimura). If Esatisfies the conclusion of this conjecture, that is, if E is “modular”, then according to a theorem of K.Ribet one can find a modular form for —0(2) which corresponds to the representation of E[l]. However, there are no such modular forms. The content of the papers by R.Taylor and A.Wiles is exactly the proof of the Taniyama-Weil conjecture for semistable elliptic curves over Q. To explain this we need a few basic facts about modular forms. Modular Forms Andrew Wiles Let H=f(cid:28) 2CjIm((cid:28))>0g be the upper half them, since the determining equations have co- plane, on which SL(2;R) acts by the usual efficients in Q. This yields Galois representations Gal(Q=Q)!GL2(Z=nZ). Using a change of (a(cid:28)+b)=(c(cid:28)+d)-rule. The subgroup —0(N) of SL(2;Z) consists of those matrices coordinates, one can arrange that f has integer ‡ · coefficients. If one then reduces modulo a prime a b number p, one obtains a polynomial over the fi- c d nite field Fp. If the zeros of the reduced polynomial are distinct then this yields an elliptic with c (cid:17)0 mod N. A modular form (of weight curve over Fp. This is true for all prime numbers 2) for —0(N) is a holomorphic function f((cid:28)) on pexcept for the finitely many prime divisors of Hwith the discriminant of f. Also the choice of fis not f((a(cid:28)+b)=(c(cid:28)+d))=(c(cid:28)+d)2f((cid:28)) unique, but we do say that Ehas good reduction ‡ · at pif we can find an fsuch that the zeros mod- for a b ulo p are distinct. (These observations are not c d 2—0(N) completely true at p=2because of the term y2 and f((cid:28)) “holomorphic at the cusps”. This last .) Otherwise Ehas bad reduction at p. If in this statement means in particular for the Fourier se- case only two zeros of fmodulo pcoincide, one ries (since f((cid:28)+1)=f((cid:28))) says Ehas semistable bad reduction. Eis called X semistableif at all pit has either good or semi- f((cid:28))= an(cid:1)e2(cid:25)in(cid:28) stable reduction. The curve y2=x3−x is not n2Z semistable at p=2(no CM-curve is semistable). An example (which in the end will not exist) that all an vanish for n<0. If additionally of a semistable curve is the Frey curve. To a so- a0=0, then f is called a cusp form. The Hecke lution of Fermat’s equation al+bl =cl (where algebra Tacts on the space of cusp forms. It is - a;b;c are relatively prime, and l(cid:21)3 is prime) generated by Hecke operators Tp (for p N one associates the curve prime) and Up (for pjN). For the Fourier coefficients one has E:y2=x(x−al)(x−cl): an(Tpf)=anp(f)+pan=p(f); This curve has bad reduction exactly at the prime an(Upf)=anp(f): divisors of abc. It has the following noteworthy property: Consider the associated Galois repre- An eigenform is a common eigenform of all sentation Gal(Q=Q)!GL2(Fl). This representa- Hecke operators. One can always normalize it so tion is unramified (the analog of “good reduc- that a1(f)=1; then ap(f) is the corresponding tion”) at all prime numbers p at which E has eigenvalue of Tp orUp. The above equations good reduction. Here one might have to say allow one to determine all an recursively, and “crystalline” for “unramified” if p=l. Because of therefore one can determine the eigenform f. the particular form of the equation for E, this Conversely, one can construct for a given sys- 744 NOTICESOFTHEAMS VOLUME42, NUMBER7 tem fPapg of eigenvalues a Fourier series representation on the 3-division points is al- f((cid:28))= ane2(cid:25)in(cid:28). According to a theorem of ready modular according to (“lifting”) theorems A.Weil this is a mPodular form if and only if the by Langlands and Tunnell. This uses intensively L-series L(s;f)= 1n=1ann−shas a holomorphic the special properties of the prime number l=3. extension to the full s-plane and satisfies a suit- For l=2 the general theory does not work well able functional equation. (This also must be true for various reasons, and for l(cid:21)5 this begin- for twists by Dirichlet characters.) ning is impossible. We now look for a deforma- In case all ap are in Q, the eigenform f has tion argument for the representations modulo an associated elliptic curve E with good reduc- 9, 27, 81, 243, 729, etc., to be successively rec- - tion outside the prime divisors of N. For p N ognized as being modular. For this one uses the the number of the Fp-rationalpointsE(Fp) is universal deformation of the representation equal to ]E(Fp)=p+1−ap. Conversely one can modulo 3: There is a Z3-algebra Rof the form define for each elliptic curve Eover Qa Hasse- R=Z3[[T1;:::;Tr]]=I(Iis an ideal), and a “uni- Weil L-seriesL(s;E), and it is conjectured that versal” Galois representation it has the nice properties from above. Accord- ing to the theorem of A.Weil it should thus be- (cid:26):Gal(Q=Q)!GL2(R) long to an eigenform with rational eigenvalues. with these properties: This is the content of the Taniyama-Weil conjecture. 1. (cid:26)is unramified (or crystalline, respectively) - Even when the coefficients ap are not in Q, for p N (that is, E has good reduction at one can construct a Galois representation asso- p); ciated to the eigenform. 2. (cid:26)has certain local properties at pjN(“cer- The Hecke algebra Tis a finitely generated Z- tain” will not be discussed here); b - module. We now replace it by the completion T 3. det((cid:26)(Frobp))=p for p N; at a suitable maximal ideal m(a “non-Eisenstein 4. (cid:26) mod (3;T1;:::;Tr) is our given repre- ideal”), with (cid:20) =T=mdenoting the residue class sentation on E[3]; field of characteristic l. Then there is a two-di- 5. any other representation Gal(Q=Q)! mensional Galois representation GL2(A)with the properties 1)–4) arises in a (cid:26):Gal(Q=Q)!GL2(Tb); Thuen icqounes wtrauyc tviioan a o hf oRmofomlloorwpsh igsemn eRral! prAin.- which is unramified (or crystalline, respectively) ciples. Basically, one takes a set of generators at p-N, with f(cid:27)1;:::;(cid:27)sgof the Galois group, and considers trace((cid:26)(Frobp))=Tp the ring of power series in 4s variables and di- vides by the smallest ideal I such that modulo det((cid:26)(Frobp))=p: Ione obtains a representation with 1), :::,4), pro- Ahonm eiogmenoforprmhi swmith T bra!tioZnl,a la enigde n(cid:26)vailnudeus cyeiesl dtsh ae vhiadse tdh oe nfeo uars suignnksn otow n(cid:27)si tchoer r2es(cid:2)po2n-mdiantgr itxo w (cid:27)hiicahs l-adic representation that is given by the asso- coefficients. ciated elliptic curve E, describing the Galois ac- After the construction we get the following tion on all ln-division points of E. Conversely, commutative diagram it is possible to show that E is modular if and only if the associated l-adic representation can be constructed in this manner. Deformations The l-adic representation is constructed for l=3,starting with the representation on the 3- division points. This is known to be congruent to a modular representation, and then the universal lifting of this representation is proven modular, which is the core of the proof. The prime 3 is very special here. So one starts with where the two left mappings arise from the the consideration of l=3. modular Galois representation and from the one One can restrict to the case that the 3-division of E. Wiles’s idea is now to show that Ris iso- points yield a surjective map morphic to Tb, because then the elliptic Galois Gal(Q=Q)!GL(2;F3) representation is automatically modular. For this, naturally, one needs information on (in this argument 5-division points are also used Rthat is not supplied by the general construc- once). As PGL(2;F3)(cid:24)= S4 (the symmetric group tion. Let Wn denote the adjoint Galois repre- on the four elements of P1(F3)) is solvable, the sentation of sl(2;Z=3nZ) (2(cid:2)2-matrices with JULY1995 NOTICESOFTHEAMS 745 trace zero). Then, for example, the minimal num- (cid:27)1;:::;(cid:27)r are generators of G. Finally, R=Tb, ber of generators r(R=Z3[[T1;:::;Tr]]=I) is and this is a complete intersection. given by dimF3Hf1(Q;W1), where Hf1 denotes a To reduce to the minimal case, one estimates cohomology group satisfying certain local con- how both sides of the inequality ditions corresponding to 1), 2) from above. This is also called a Selmer group. One sees this by ]p=p2(cid:21)]Z3=(cid:17)(cid:1)Z3 setting A=F3[T]=(T2) in the definitions. One change as one proceeds from level Mto a higher can show (M.Flach) that the orders of Hf1(Q;Wn) level N (MjN). For the left-hand side ainr et huen fiofollromwliyn gb onuunmdeerdic ianl cnr.i tTehrieosne foorrd tehres eoqcucaulr- ]Hf1(Q;Wn) certain local conditions are weak- ened, and one obtains an upper bound. For the ity R=Tb: there is a Z3-homomorphism Tb !O, right-hand side there is the phenomenon of “fu- where O is the integral closure of Z3 in a finite sion”, that is, of congruences between oldforms extension of Q3. For simplicity we will assume and newforms. Here a lower bound has been con- that O=Z3. It is known that Tb is Gorenstein; that structed by Ribet and Ihara. Luckily the two b b jise,c tHioonm TbZ3!(TZ;Z33t)heisn ah afsr eaen Tad-mjooindtu Zle3. !ThTbe, saunrd- bounds agree, and thus everything is shown. the composition of these two maps is multiplication by an element (cid:17)2Z3, which is well defined up to a unit. Furthermore, (cid:17)6=0. On the other hand, let p(cid:18)R be the kernel of the sur- jection R!Tb !Z3. Then one has (“]” = order) ]p=p2(cid:21)]Z3=(cid:17)(cid:1)Z3 and equality if and only if R=Tb and this is also a complete intersection (I can be generated by r elements). The left- hand side ]p=p2is identical to the order of the Selmer group Hf1(Q;Wn), for n>0. The first at- tempt tried to establish equality by using Euler systems (invented by Kolyvagin). However, it was only possible to show that p=p2 is annihi- lated by (cid:17). This is the content of the theorem of M. Flach. The higher levels of the Euler sys- tem, however, could not be constructed. The Proof One first shows the minimal case and then reduces to it. By the minimal case we mean that all primes of bad reduction occur already modulo 3 (and not only modulo higher powers). Ac- cording to the theorem of Ribet and others (used for l=3 and not for lthe exponent of Fermat’s equation), the Galois representation belonging to the curve modulo 3 is modular of level 3. In the minimal case the computation of Euler char- acteristics (Poitou-Tate) shows that Hf1(Q;W1) and Hf2(Q;W1)have the same dimension r. For each none chooses r prime numbers q1;:::;qr (cid:17)1 mod 3n. Then one proceeds to use a subgroup of —0(N). This subgroup contains the intersection with —1(q1(cid:1)(cid:1)(cid:1)qr), and the quotient is isomorphic to G=(Z=3nZ)r. The associated b Hecke algebra T1 is a free module over Z3[G], b with G-coinvariants T, and is the quotient of a representation ring R1=Z3[[T1;:::;Tr]]=I1, which again can be generated by r elements. The ideal I1is small due to the free action of the group G. Now one takes a limit n!1, and in the limit R1 andTb1 become rings of power series and equal. Furthermore, one obtains Rfrom R1and Tb from Tb1 in both cases by putting in the additional r relations “(cid:27)i =1”, where 746 NOTICESOFTHEAMS VOLUME42, NUMBER7