CQQbarRZGalGalQPrSpGLSLPSLGFprojlimellZlZlmrplrpmTatedetwpJJtilWilliam F. HammondA Lecture at the Everyone Seminar University at Albany, October 22, 1993GELLMU Edition with Retrospective Comments April 21, 2001 Minor revisions: July 15, 2004 Comments for the GELLMU EditionSU-11Introduction.SU-22Elliptic curvesSU-33Elliptic curves over CSU-44Modular formsSU-55Euler productsSU-66Elliptic curves over the rational field QSU-77The Shimura mapSU-88The hypothetical Frey curveSU-99adic representations of GalQQSU-10Appendix Late 1993early 1994 StatusSU-11ReferencesSU-TheBibLiog

SU-1KEY-1Comments for the GELLMU EditionAmong the challenges that I have been facing with my GELLMU project are (1) convincing mathematicians that it is possible to use comfortable like markup in a fully rigorous way to prepare our articles so that they can have formal inclusion in the markup category known as href="http://www.w3.org/XML/"XML and (2) then convincing them that high quality typesetting may be obtained from the ensuing XML document instance Toward this end I have revisited the markup for the official notes on my October 1993 Albany seminar presentation and edited what was source to convert it to like source markup for the article document type that is part of the GELLMU didactic markup production system Information about this system and my reasons for developing it may be found at http://www.albany.edu/hammond/gellmu/After the time of the original talk and the subsequent preparation of my href="http://math.albany.edu:8000/math/pers/hammond/oct93.html"original writeup, there was a time fortunately not long and also not to have been unexpected in the aftermath of so large a new development when Andrew Wiless argument underwent some revision in collaboration with Richard Taylor Questions about its soundness appeared to have ceased by the fall of 1994, and the work announced by Wiles in June 1993, as revised, was published in the May 1995 issue of the Annals of MathematicsThere has also been discussion, at times appearing to approach controversy, about the name of the conjecture arising from the 1955 meeting in Japan What I termed the ShimuraTaniyamaWeil conjecture became known as the modular curve conjecture and then, from the summer of 1999, as the modular curve theorem after the work of Breuil, Conrad, Diamond, and Taylor in the same vein as the work of Wiles and Taylor for the semistable caseI list a few references on these matters for the period since my original talk: A. Wiles, Modular elliptic curves and Fermats Last Theorem, Annals of Mathematics, (second series) vol. 141 (1995), pp. 443551 R. Taylor A. Wiles, Ringtheoretic properties of certain Hecke algebras, Annals of Mathematics, (second series) vol. 141 (1995), pp. 553572 H. Darmon, F. Diamond, R. Taylor, Fermats Last Theorem, Current Developments in Mathematics, 1995, International Press, Cambridge, Massachusetts, 1995 G. Cornell, J. H. Silverman, G. Stevens, Modular Forms and Fermats Last Theorem, SpringerVerlag, 1997 This volume is the record of an instructional conference on number theory and arithmetic geometry held August 918, 1995 at Boston University J. Coates S.T. Yau, Elliptic curves, modular forms, Fermats last theorem, 2nd edition, International Press, Cambridge, MA, 1997 Proceedings of the Conference on Elliptic Curves and Modular Forms held at the Chinese University of Hong Kong, Dec. 1993 B. Conrad, F. Diamond, R. Taylor, Modularity of certain potentially BarsottiTate Galois representations, J. Amer. Math Soc. 12 (1999), no. 2, 521567. In this article the modular curve conjecture is proved for any elliptic curve defined over Q with conductor not divisible by 27 C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves over Q: wild 3adic exercises, J. Amer Math. Soc., to appear.Based on a citation found at http://www.math.harvard.edu/rtaylor/ on 21 April 2001. What follows has the same content as the original writeup except that the title of the Appendix has been changed from Current Status to Late 1993Early 1994 Status

1SU-2KEY-2Introduction.The purpose of this expository lecture is to explain the basic ideas underlying the final resolution of Fermats Last Theorem after 356 years as a consequence of the reported establishment by Andrew Wiles of a sufficient portion of the ShimuraTaniyamaWeil conjecture As these notes are being written, the work of Wiles is not available, and the sources of information available to the author are (1) reports by electronic mail, (2) the AMS Notices article 16ribetnotices of K. Ribet, and (3) a preprint 18rubinsilvbg by K. Rubin and A Silverberg based on the June, 1993 lectures of Wiles at the Newton Institute in Cambridge, England It should be noted that the fact that Fermats Last Theorem is a consequence of sufficient knowledge of the theory of elliptic curves has been fully documented in the publications (14ribinv, 15ribtoul) of K RibetFermats Last Theorem is the statement, having origin with Pierre de Fermat in 1637, that there are no positive integers x,y,z such that $xnynzn$ for any integer exponent n2 Obviously, if there are no positive integer solutions x,y,z for a particular n, then there are certainly none for exponents that are multiples of n Since every integer n2 is divisible either by 4 or by some odd prime p, it follows that Fermats Last Theorem is true if there are no solutions in positive integers of the equation $xnynzn$ when n4 and when np for each prime p2 The cases n3,4 are standard fare for textbooks (e.g., see Hardy Wright 6hardywright) in elementary number theory Therefore, this discussion will focus on the case np where p3 is primeVery briefly, the idea is that we now know enough about the classification of nondegenerate plane cubic curves $Fx,y0$ in two variables, also known as elliptic curves, with rational coefficients to know how to enumerate them in a logical way so that we may conclude that if there were positive integers a,b,c with $apbpcp$, then the curve y2xxapxbp, which is an elliptic curve known as the Frey curve, would fall inside of the enumeration Because the classification is enumerative, when one is presented with a particular elliptic curve with rational coefficients, one knows where to look for the curve in the classification The curve just written is not to be found within the classification As a consequence there cannot be positive integers a,b,c with apbpcpThe enumerative classification of nondegenerate plane cubic curves defined by polynomials with rational coefficients has been entirely conjectural (variously known as the Taniyama Conjecture, the Weil Conjecture, the TaniyamaShimura Conjecture, ) until June, 1993 This conjecture, even as a conjecture, has served as an important motivating example for the idea of the Langlands Program, or perhaps of an extension of that program, that certain kinds of objects in geometry should give rise to certain group representationsWhat seems to be believed todayAs of the time of this writeup Wiles has stated that a portion of what he announced in June needs further justification and that he expects to be able to complete it See the appendix. is that the portion of the enumerative classification pertaining to semistable elliptic curves has been proved by Andrew Wiles That the existence of positive integers a,b,c with $apbpcp$ would violate the enumerative classification of semistable elliptic curves was established by 1987 through the work of G. Frey, J.P Serre, and K. RibetThe primary purpose of this lecture is to explain the enumerative classification of elliptic curves and to give a brief indication of the mathematics involved in showing that the Frey curve violates that classification

2SU-3KEY-3Elliptic curvesA polynomial fX,Y of degree d in two variables with coefficients in a field k gives rise to what is called an affine plane curve of degree d: for each field K containing k (more generally, for each commutative ring that is a kalgebra) one has the set C0Kx,yK2fx,y0, and for each klinear homomorphism $KK{}^{}$ one has the induced map $C{0}_{}KC{0}_{}K{}^{}$ From the polynomial f one obtains a homogeneous polynomial of degree d in three variables with coefficients in k: FX,Y,ZZdfXZ,YZ, and the projective plane curve of degree d: CKx,y,zP2KFx,y,z0, where PNK denotes Ndimensional projective space, which is the quotient set of $KN10$ obtained by identifying points lying on the same line through the origin of KN1 Since the projective plane P2K is the disjoint union of the affine plane $x,y,1x,yK2$ with the (projective) line at infinity x,y,0x,yP1K, it follows that CK is the disjoint union of C0K with the finite set of its points lying on the projective line at infinityAn elliptic curve defined over k is the (projective) plane curve E given by a homogeneous polynomial F of degree 3 in three variables with coefficients in k such that (i) F is irreducible over the algebraic closure k of k, (ii) the gradient vector F is a nonvanishing vector at points of $k30$ where F vanishes, and (iii) the set Ek is nonempty If k is any field, then after an isomorphism (see Silverman 27silverman) one may obtain a given elliptic curve E with an affine equation of the form eq.wnormformeq.wnormformEqnAutoy2a1xya3yx3a2x2a4xa6 Then the homogeneous equation for the intersection of EK with the line at infinity is eq.wnormforminfeq.wnormforminfEqnAutox30 Thus, in this case, E has a unique point on the line at infinity If the characteristic of k is different from 2 and 3Thus, one sees that the primes 2 and 3 play a special role in the theory of elliptic curves. then one may obtain an equation in Weierstrass normal form: eq.swnormformeq.swnormformEqnAutoy24x3g2xg3, which is nonsingular if and only if the cubic polynomial in the variable x has distinct roots in k Elliptic curves are the group objects in the category of algebraic curves that reside in projective space: for each extension field K of k the set EK of Kvalued points of E is an abelian group The group law on EK is characterized by two conditions: The origin is a given point of EkThe points obtained by intersecting EK with any line in P2K, counted with multiplicities, add up to zeroWhen E is given by an equation in the form (iref="eq.wnormform"eq.wnormform), the origin is usually taken to be the unique point on the line at infinity If two distinct points of EK are given, they determine a line in P2K; the intersection of that line with EK is given by a cubic polynomial in a parameter for the line which has two roots in K corresponding to the two given points; hence, there is a third root of that cubic polynomial in K; this root gives rise to a point of EK, which is the negative of the sum of the two given points The negative of a given point of EK is obtained as the third point in the intersection with EK of the line through the given point and the originFor a given field k the set of homogeneous cubic polynomials in three variables is a vector space over k having the set of monomials of degree three in three variables as basis Thus, the dimension of the space of homogeneous cubics is 10 The linear group GL3k acts on the space of cubics, and two cubic curves in P2 that are related by this action are isomorphic Since GL3k is 9dimensional, one is led to think of the family of isomorphism classes of elliptic curves as 1dimensional since nonsingularity is an open condition

3SU-4KEY-4sec.ellipCElliptic curves over CWhen k is the field C of complex numbers, one knows (see, e.g., Ahlfors 1ahlfors) that for each lattice in C the set of periodic meromorphic functions on the complex line C is the field C,, which is a quadratic extension of the rational function field C, where is the function of Weierstrass Moreover, satisfies the famous Weierstrass differential equation eq.wdiffeqeq.wdiffeqEqnAutoz24z3g2zg3; thus, the formula $zz,{}^{}z$ defines a holomorphic map from the punctured complex torus $C\mapsto 0$ to the affine cubic curve eq.cwnormformeq.cwnormformEqnAutoy24x3g2xg3; it should hardly be necessary to point out that this map extends to a holomorphic map from the torus C to the corresponding (projective) elliptic curve by sending the origin of the torus to the unique point of the elliptic curve on the line at infinity The classical theory of theta functions (see, e.g., Igusa 7igusa or Siegel 26siegel) leads to a direct demonstration that this map is a homomorphism from the group law on the complex torus to the group law previously described for an elliptic curve It is not difficult to see that the analytic manifold given by any elliptic curve defined over C arises from some complex torus Indeed each nonsingular cubic curve E in P2C determines a compact connected complexanalytic group Its universal cover is given by a holomorphic homomorphism $CE$ which has some lattice as kernelAny two lattices in C are related by a change of real basis for C, i.e., by a matrix in GL2R Consequently, there is only one realanalytic isomorphism class for the complex torus C as varies The tori corresponding to two lattices are complexanalytically isomorphic if and only if the corresponding reallinear isomorphism of R2 satisfies the CauchyRiemann partial differential equations, i.e., if and only if the Rlinear isomorphism is ClinearA lattice may be represented concretely by an ordered basis $1_{},2_{}$ If 21, then is not real, and after permuting the basis members, if necessary, one may assume that is in the upperhalf planeThe fact that the halfplane is a model of nonEuclidean geometry led a popular columnist in November, 1993 to question the validity of the work being discussed here. H of C Observing that is the image under the Clinear map $z{1}_{}z$ of the lattice with ordered basis $1,$, one may assume that is this latter lattice Let E be the complex torus C Allowing for change of basis subject to these assumptions on the basis, one sees that there is an isomorphism of complexanalytic groups $E{}^{}E$ if eq.linfraceq.linfracEqnAutoabcd, for some matrix rrabcdSL2Z Conversely, the monodromy principle may be used to show that every complexanalytic isomorphism among the complex tori E arises in this wayThe coefficients g2 and g3 in the Weierstrass normal form (iref="eq.cwnormform"eq.cwnormform) have very explicit constructions as infinite series (see, e.g., Ahlfors 1ahlfors or Serre 20serrecourse) determined by the given lattice; from this it is straightforward to see that gw is a modular form of weight 2w: if and are related by (iref="eq.linfrac"eq.linfrac), then gw2wgw,cd. Consequently, the map x,y2x,3y carries the curve given by (iref="eq.cwnormform"eq.cwnormform) for isomorphically to the curve given by (iref="eq.cwnormform"eq.cwnormform) for The discriminant of the cubic polynomial in the Weierstrass normal form (iref="eq.cwnormform"eq.cwnormform) is a modular form of weight 12, which up to a multiplicative constant, is: g2327g32. is a nonvanishing holomorphic function in H The modular invariant (20serrecourse,24shimiaf) is defined by: 12g23; it is a holomorphic function in the upperhalf plane H with the property that if and only if and are related by (iref="eq.linfrac"eq.linfrac) Furthermore, assumes every value in C at some point of H Consequently, the complexanalytic isomorphism classes of complex tori or, equivalently, the isomorphism classes of elliptic curves defined over C, are parameterized via in a onetoone manner by the complex numbersSince this is an expository discourse, it is hoped that the reader will not feel patronized by having noted the fact that the coincidence of (1) the category of elliptic curves over C and (2) the category of complex tori is the genus one case of the coincidence (see Weyl 33weyl) of (i) the category of complete nonsingular algebraic curves over C and (ii) the category of compact Riemann surfaces (onedimensional connected complexanalytic manifolds)Although the classification of elliptic curves over C via the function is a result that is both beautiful and useful, and although two elliptic curves defined over Q that are isomorphic as curves defined over Q give rise to elliptic curves defined over C that have the same invariant, it is not true that any two elliptic curves defined over Q having the same invariant are isomorphic over Q Thus, the classification of elliptic curves over C does not lead directly to the desired enumerative classification of elliptic curves defined over Q but it does bring to the fore the notion of modular form, which is central in the study of elliptic curves defined over Q What can be said easily is that, according to the ShimuraTaniyamaWeil conjecture, the isogeny classes of elliptic curves defined over Q are parameterized by certain modular forms

4SU-5KEY-5Modular formsThe group SL2Z is an infinite group that is generated by the two elements rr0110,rr0111, which have orders 4 and 6 respectively The action of SL2Z on the upperhalf plane H by linear fractional transformations has kernel rr1001, and the quotient of SL2Z by this kernel is the group PSL2Z It is not difficult to see that the set H12Re12,1 is a fundamental domain for the action of PSL2Z on H More precisely, this set meets each orbit, and the only redundancies are the boundary identifications arising from the maps $1$ and $1$ The quotient HPSL2Z is not compact since the fundamental domain is open at the top Beyond that the modular invariant j induces a bicontinuous biholomorphic isomorphism of the quotient HPSL2Z with the affine line over C Since $1$, and since for $qe2\pi i$ one has q1 for H, there is a holomorphic function in the punctured unit disk such that q Likewise may be regarded as function of q, and one may use the calculus of residues to show that has a simple zero at q0; hence, has a simple pole at q0, or, equivalently, has a simple pole at (the missing top of the fundamental domain) Thus, gives rise to a bicontinuous biholomorphic isomorphism HPSL2ZP1C. A nontrivial element of PSL2Z has a fixed point in H if and only if it has finite order, and ones explicit knowledge of the fundamental domain makes it possible to see that the only elements of finite order are of order 2 or 3Thus, one sees that the primes 2 and 3 play a special role in the study of the group