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&TableOfContentsFile;

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 &BibRef-ribetnotices;ribetnotices of K. Ribet, and (3) a preprint &BibRef-rubinsilvbg;rubinsilvbg 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 (&BibRef-ribinv;ribinv, &BibRef-ribtoul;ribtoul) 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 n 2 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 n 2 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 n 4 and when n p for each prime p 2 The cases n 3, 4 are standard fare for textbooks (e.g., see Hardy Wright &BibRef-hardywright;hardywright) in elementary number theory Therefore, this discussion will focus on the case n p where p 3 is primeVery briefly, the idea is that we now know enough about the classification of nondegenerate plane cubic curves $F(x,\; y)0$ 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 y2 x (x ap) (x bp), 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 ap bp cpThe 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 f(X,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 C0(K) (x,y) K2 f(x,y) 0 , and for each klinear homomorphism $KK{}^{}$ one has the induced map $C{}_{}(K)C{}_{}(K{}^{})$ From the polynomial f one obtains a homogeneous polynomial of degree d in three variables with coefficients in k: F(X,Y,Z) Zd f(XZ, YZ), and the projective plane curve of degree d: C(K) ((x,y,z)) P2(K) F(x,y,z) 0 , where PN(K) 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 P2(K) is the disjoint union of the affine plane $((x,y,1))(x,y)K2$ with the (projective) line at infinity ((x,y,0)) ((x,y)) P1(K), it follows that C(K) is the disjoint union of C0(K) 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 E(k) is nonempty If k is any field, then after an isomorphism (see Silverman &BibRef-silverman;silverman) one may obtain a given elliptic curve E with an affine equation of the form eq.wnormformeq.wnormformEqnAuto y2 a1 x y a3 y x3 a2 x2 a4 x a6 Then the homogeneous equation for the intersection of E(K) with the line at infinity is eq.wnormforminfeq.wnormforminfEqnAuto x3 0 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.swnormformEqnAuto y2 4 x3 g2 x g3 , 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 E(K) of Kvalued points of E is an abelian group The group law on E(K) is characterized by two conditions: The origin is a given point of E(k)The points obtained by intersecting E(K) with any line in P2(K), 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 E(K) are given, they determine a line in P2(K); the intersection of that line with E(K) 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 E(K), which is the negative of the sum of the two given points The negative of a given point of E(K) is obtained as the third point in the intersection with E(K) 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 GL3(k) acts on the space of cubics, and two cubic curves in P2 that are related by this action are isomorphic Since GL3(k) 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 &BibRef-ahlfors;ahlfors) 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.wdiffeqEqnAuto (z)2 4(z)3 g2()(z) g3() ; thus, the formula $z((z),{}^{}(z))$ defines a holomorphic map from the punctured complex torus $C\mapsto 0$ to the affine cubic curve eq.cwnormformeq.cwnormformEqnAuto y2 4x3 g2()x g3() ; 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 &BibRef-igusa;igusa or Siegel &BibRef-siegel;siegel) 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 P2(C) 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 GL2(R) 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 ${}_{},{}_{}$ 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 $z{}_{}z$ 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.linfracEqnAuto a bc d , for some matrix rra bcd SL2(Z) 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 &BibRef-ahlfors;ahlfors or Serre &BibRef-serrecourse;serrecourse) 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 gw() ()2wgw(), c d . Consequently, the map (x,y) (2 x, 3 y) 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: () g23 27 g32 . is a nonvanishing holomorphic function in H The modular invariant () (&BibRef-serrecourse;serrecourse,&BibRef-shimiaf;shimiaf) is defined by: () (12 g2)3 ; 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 &BibRef-weyl;weyl) 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 SL2(Z) is an infinite group that is generated by the two elements rr0 110 , rr0 111 , which have orders 4 and 6 respectively The action of SL2(Z) on the upperhalf plane H by linear fractional transformations has kernel rr1 001 , and the quotient of SL2(Z) by this kernel is the group PSL2(Z) It is not difficult to see that the set H 12 Re() 12, 1 is a fundamental domain for the action of PSL2(Z) 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 HPSL2(Z) 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 HPSL2(Z) with the affine line over C Since $(1)()$, and since for $qe2\pi i$ one has q 1 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 q 0; hence, has a simple pole at q 0, or, equivalently, has a simple pole at (the missing top of the fundamental domain) Thus, gives rise to a bicontinuous biholomorphic isomorphism HPSL2(Z) P1(C) . A nontrivial element of PSL2(Z) 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 SL2(Z). A congruence subgroup of SL2(Z) is a subgroup that contains one of the principal congruence subgroups; the principal congruence subgroup (N) of level N is the set of all elements of SL2(Z) that are congruent N to the identity matrix The group 0(N) is the congruence subgroup of SL2(Z) consisting of all elements rra bcd for which $c0N$ It is obvious that each congruence group has finite index in SL2(Z), and, consequently the quotient H is a noncompact Riemann surface Observe that for each level N the group 0(N) contains the parabolic element T rr1 101 , which gives rise to the holomorphic map $1$ that fixes the point A modular formDetails concerning the discussion in this section may be found in Shimuras book &BibRef-shimiaf;shimiaf. of weight w for is a holomorphic function f in H that satisfies the functional equation eq.modformeq.modformEqnAuto f( ) (c d)w f(), and that is holomorphic at each cusp of The role of cusps for is to provide a slightly larger set H than H, H H cusps , where acts such that H is a compact Riemann surface containing H as the open complement of a finite set of points arising from cusps The cusps of are the points of the closure of the boundary of H in P1(C) $C$ that are fixed by some nontrivial parabolic element of When SL2(Z), the set of cusps is $Q$ In view of (iref="eq.modform"eq.modform) applied to the case $T$ one sees that a modular form f of any weight for the group 0(N) satisfies eq.periodeq.periodEqnAuto f( 1) f() , and, therefore, f defines a holomorphic function in the variable q e2 i for 0 q 1 The condition in the definition of modular form that f should be holomorphic at means that f as a function of q is holomorphic at q 0 Consequently, f admits an absolutely convergent Fourier expansion eq.fouriereq.fourierEqnAuto f() m 0 cm e2 i m , which is a Taylor series in qFor any cusp of a congruence group one may define the notion holomorphic at for a modular form f by an analogous procedure using an arbitrary parabolic element of that fixes instead of T For a given congruence group two cusps and are equivalent if there is some element in such that $$ A modular form f is holomorphic at any cusp that is equivalent to another where it is holomorphic The modular form f is a cuspform if, in addition to being holomorphic at each cusp, f vanishes at each cusp For a given congruence group a modular form vanishes at any cusp that is equivalent to another where it vanishes The set of modular forms of given weight w forms a finitedimensional vector space over C in which the set of cuspforms is a linear subspace of codimension bounded by the number of equivalence classes of cusps In fact, using Eisenstein series one may show that the codimension of the space of cuspforms in the space of modular forms is often equal to the number of equivalence classes of cusps For example, with the group $(1)SL{}_{}(Z)$ there are no modular forms of odd weight, there is an Eisenstein series of every even weight greater than 2 that is not a cuspform, and every cusp is equivalent to Furthermore, since is the only zero of the cusp form (of the preceding section) in the quotient $H(1)$ and since is a simple zero of every cuspform for (1) is divisible by Thus, in this case, there are no cuspforms of weight less than 12It is not difficult to see that the cuspforms of weight 2 for a congruence group correspond to holomorphic differential 1forms (differentials of the first kind) on the compact Riemann surface $XH$ Thus, the dimension of the space of cuspforms of weight 2 is the genus of X The fact that there are no cuspforms of weight 2 for the group (1) matches the previously mentioned fact that X is P1 It is certain of the cuspforms of weight two for the groups 0(N) that, according to the ShimuraTaniyamaWeil conjecture, parameterize the isogeny classes of elliptic curves defined over Q

5SU-6KEY-6Euler productsIt will be recalled that the infinite series n 1 1ns converges for Re(s) 1 and gives rise by analytic continuation to a meromorphic function (s) in C For Re(s) 1 (s) admits the absolutely convergent infinite product expansion p 11 ps, taken over the set of primes This Euler product may be regarded as an analtyic formulation of the principle of unique factorization in the ring Z of integers It is, as well, the product taken over all the nonarchimedean completions of the rational field Q (which completions Qp are indexed by the set of primes) of the Mellin transformThe Mellin transform is, more or less, Fourier transform on the multiplicative group Classically, the Mellin transform of f is given formally by (s) 0 f(x) xs (dxx) in Qp p(s) 11 ps , of the canonical Gaussian density p(x) ll1 if x closure ofZ in Qp 0otherwise , which Gaussian density is equal to its own Fourier transform For the archimedean completion $Q{}_{}R$ of the rational field Q one forms the classical Mellin transform (s) (s2) (s2) of the classical Gaussian density (x) e x2 , (which also is equal to its own Fourier transform) Then the function (s) (s) (s) p p(s) is meromorphic in C, and satisfies the functional equation eq.zetafnleq.zetafnlEqnAuto (1s) (s) The connection of Riemanns function with the subject of modular forms begins with the observation that (2s) is essentially the Mellin transform of I(x) (ix) 1 where , which is a modular form of weight 12 and level 8, is defined in the upperhalf plane H by the formula () m Z exp( i m2) . In fact, one of the classical proofs of the functional equation (iref="eq.zetafnl"eq.zetafnl) is given by applying the Poisson summation formulaOn the other hand, (iref="eq.zetafnl"eq.zetafnl) may be regarded directly as a divergent model of the Poisson summation formula. to the function $xexp(\pi ix2)$, while observing that the substitution $s(12)s$ for (2s) corresponds in the upperhalf plane to the substitution $1$ for the theta seriesIf f is a cuspform for a congruence group containing T rr1 101 , and so, consequently, $f(1)f()$, then, as previously explained, one has the Fourier expansion (iref="eq.fourier"eq.fourier) f() m 1 cm e2 i m . The Mellin transform (s) of fI leads to the Dirichlet series eq.dirserieseq.dirseriesEqnAuto (s) m 1 cm ms, which may be seen to have a positive abscissa of convergence One is led to the questions: For which cuspforms f does the associated Dirichlet series (s) admit an analytic continuation with functional equationFor which cuspforms f does the associated Dirichlet series (s) have an Euler product expansionFor the modular group (1) the Dirichlet series associated to every cuspform of weight w admits an analtyic continution with functional equation under the substitution $sws$ Since (1) is generated by the two matrices T and W rr0 110 and since the functional equation of a modular form f relative to T is reflected in the formation of the Fourier series (iref="eq.fourier"eq.fourier), the condition that an absolutely convergent series (iref="eq.fourier"eq.fourier) is a modular form for (1) is the functional equation for a modular form relative solely to W This is equivalent to the (properly formulated) functional equation for the associated Dirichlet series together with a growth condition For the group 0(N), with N 1, the question of a functional equation is more complicated since, although T is available, there is no reason for a cuspform to satisfy a law of transformation relative to W But note that for any the set of cuspforms of given weight for which the associated Dirichlet series have analytic continuations satisfying a given finite set of functional equations is a vector space On the other hand, there is no reason to believe, even for level 1, that the cuspforms admitting an Euler product expansion form a vector spaceIn a nutshell the cuspforms admitting Euler products are those which arise as eigenforms for an arithmetically defined commutative algebra of semisimple operators on the space of cuspforms of a given weight introduced by E. Hecke The theory of Hecke operators is reasonably simple for level 1 but somewhat more complicated in general (see, e.g., Shimuras book &BibRef-shimiaf;shimiaf)Observing that the formula ds2 dx2 dy2y2,forxiy H, gives a (the hyperbolic) SL2(R)invariant metric in H with associated invariant measure d dx dyy2, one introduces the Petersson (Hermitian) inner product in the space of cuspforms of weight w for with the definition: eq.innerprodeq.innerprodEqnAuto f,g H f() g() Im()w d() (Integration over the quotient H makes sense since the integrand f() g() yw is invariant.)For the modular group (1) the nth Hecke operator $T(n)T{}_{}(n)$ is the linear endomorphism of the space of cuspforms of weight w arising from the following considerations Let Sn be the set of 2 2 matrices in Z with determinant n For M rra bcd Sn and for a function f in H one defines eq.actonfnseq.actonfnsEqnAuto (M w f)() det(M)w1 (c d)w f() , and then, observing that (1) under w acts trivially on the modular forms of weight w, one may define the Hecke operator Tw(n) by eq.heckeopeq.heckeopEqnAuto Tw(n)(f) M Sn(1) (M w f)() , where the quotient Sn(1) refers to the action of (1) by left multiplication on the set Sn One finds for m, n coprime that T(m n) T(m) T(n) , and furthermore one has T(pe1) T(pe)T(p) pw1T(pe1) . Consequently, the operators T(n) commute with each other, and, therefore, generate a commutative algebra of endomorphisms of the space of cusp forms of weight w for (1) It is not difficult to see that the Hecke operators are selfadjoint for the Petersson inner product on the space of cuspforms Consequently, the space of cuspforms of weight w admits a basis of simultaneous eigenforms for the Hecke algebra A Hecke eigencuspform is said to be normalized if its Fourier coefficient c1 1 If f is a normalized Hecke eigencuspform, then The Fourier coefficient cm of f is the eigenvalue of f for T(m)The Fourier coefficients c(m) cm of f satisfy c(m n) c(m) c(n)for m, n coprime, and c(pe1) c(pe)c(p)pw1c(pe1)for p prime Consequently, the Dirichlet series associated with a simultaneous Hecke eigencuspform of level 1 and weight w admits an Euler product eq.modeulerpeq.modeulerpEqnAuto (s) p 11 cp ps pw12s For example, when f is the unique normalized cuspform of level 1 and weight 12 one has (s) p 11 (p) ps p112s , where cp (p) is the function of RamanujanFor the congruence group 0(N) a Hecke eigencuspform of weight w gives rise to a Dirichlet series (s) that admits an Euler product expansion whose factors at primes p coprime to N resemble those given by (iref="eq.modeulerp"eq.modeulerp) In order for (s) to satisfy a functional equation under the substitution $sws$,one needs to require that the eigencuspform f admits a functional equation not only with respect to each element of the group 0(N) but also with respect to the substitution in the upperhalf plane H given by the matrix WN rr0 1N0 . A. Weil (&BibRef-weilmathann;weilmathann) showed that the cuspforms of weight 2 for the group 0(N) satisfying the appropriate functional equation under the mapping of H given by WN correspond precisely to Dirichlet series with certain growth conditions that admit analytic continuations as meromorphic functions in C satisfying a finite number of twisted functional equationsThe reader will have noticed that it is not extremely easy to characterize the cuspforms of weight 2 that conjecturally (ShimuraTaniyamaWeil) parameterize the isogeny classes of elliptic curves defined over the rational field Q The Euler product is an extremely important part of the characterization since the Dirichlet series given by such an elliptic curve, as will be made explicit in the next section, is, by its very nature, an Euler product Weil conjectures explicitly that the Dirichlet series with Euler product given by each elliptic curve defined over Q satisfies these conditions, i.e., is the Dirichlet series associated to some WNcompatible Hecke eigencuspform for the group 0(N), where N is the conductor of E This has led to efforts, related to the Langlands program to understand the WNcompatible Hecke eigencuspforms in a more intrinsic way as objects of representation theory over Q (see, e.g., the survey of Gelbart &BibRef-gelbartadv;gelbartadv)

6SU-7KEY-7Elliptic curves over the rational field QLet E be an elliptic curve defined over Q One may clear denominators from its cubic equation, if necessary, in order to arrive at an equation with integer coefficients having no common factor While the Weierstrass normal form (iref="eq.swnormform"eq.swnormform) is available to represent the isomorphism class of any elliptic curve over a field of characteristic different from 2 and 3, one needs the generalized Weierstrass form eq.gwnormformeq.gwnormformEqnAuto y2 a1 x y a3 y x3 a2 x2 a4 x a6 over an arbitrary field, and, moreover, for each elliptic curve E defined over Q there is a best possible equation (e.g., see Silverman &BibRef-silverman;silverman) of the form (iref="eq.gwnormform"eq.gwnormform) with integer coefficients called the Neron model of E With an abuse of notation E will denote the Neron model, which may be regarded as a curve over Z (One would want to call it an elliptic curve over Z if it were smooth over Z, i.e., if it had good reduction at each prime p; the fact that every Neron model has bad reduction at least once corresponds under the dictionary to the fact that there are no cuspforms of weight two and level 1.) It then may be observed that for each prime p the Neron model gives rise to a cubic equation over the finite field Fp For all but a finite number of p the equation over Fp is nonsingular over Fp, i.e., determines an elliptic curve Ep defined over Fp One says in this case that E has good reduction mod p Following Tate (&BibRef-tatesurvey;tatesurvey) one introduces b2 a12 4 a2 , b4 a1 a3 2 a4 , b6 a32 4 a6 , and b8 b2 a6 a1 a3 a4 a2 a32 a42 . Then one has b22 b8 8 b43 27 b62 9 b2 b4 b6 . The nonvanishing of mod p is necessary and sufficient for E to have good reduction mod p It follows that a prime p divides if and only if E does not have good reduction mod p If p is a prime for which E has bad reduction, then there is a single singular point of the reduced curve Ep and either (a) Ep has distinct tangent lines at the singular point (semistable reduction) or (b) Ep has a single tangent line occurring with multiplicity 2 E is called semistable if it has either good or semistable reduction at each prime The conductor of E is the integer N defined by N p pp , where p ll0 ifEhas good reduction at p .1ifEhas semistable reduction atp.2 p 2otherwise. The nonnegative integer p cannot be positive unless p is 2 or 3 Tautologically, E is semistable if and only if its conductor N is squarefreeOne defines the Lseries of E by LseriesLseriesEqnAuto L(E,s) p N 11 cp ps p N 11 cp ps p1 2s , where cp is defined when E has good reduction mod p by the formula cp p 1 E(Fp) , and cp is defined when E has bad reduction mod p by cp ll1 ifp 1and the tangents are defined over Fp 1if p 1 with irrational tangents0if p 1 One observes readily that the Lfunction of E codifies information about the number of points of E in the finite field Fp Quite generally for an algebraic variety defined over Q the analogous codification of information obtained by counting points in the various reductions mod p of the variety yields the HasseWeil zeta function, which reflects cohomological information about E The Lseries of E is the essential part, corresponding to cohomology in dimension 1, of the HasseWeil zeta function of E The HasseWeil zeta function is a special case of the general notion (Serre &BibRef-serrepurdue;serrepurdue) of zeta function for a scheme of finite type over ZOne observes that L(E,s) resembles, at least insofar as one considers its Euler factors for primes p corresponding to good reductions of E, the Dirichlet series associated to a cuspform of weight 2 that admits an Euler product expansion The observation of this resemblance is the beginning of an appreciation of the ShimuraTaniyamaWeil conjecture One is led to ask to what extent the two classes of Dirichlet series with Euler products coincide The conjecture states that the Lfunction of an elliptic curve defined over Q with conductor N arises from a cuspform for the group 0(N) that is compatible with the substitution in the upperhalf plane H given by WN Isogenous elliptic curves have the same Lfunction, and, conversely (cf. Tate &BibRef-tateendom;tateendom and Faltings &BibRef-storrsconf;storrsconf) two elliptic curves with the same Lfunction must be isogenous Thus, the idea of the conjecture is that the isogeny classes of elliptic curves defined over Q with conductor N are in bijective correspondence with the set of Hecke eigencuspforms for the group 0(N) of level N compatible with the extension of that group by the substitution arising from WN having rational Fourier coefficients and not arising from levels dividing N

7SU-8KEY-8The Shimura mapShimura (&BibRef-shimnagoya;shimnagoya, &BibRef-shimiaf;shimiaf, &BibRef-shimjmsj;shimjmsj) showed for a given WNcompatible Hecke eigencuspform f of weight 2 for the group 0(N) with rational Fourier coefficients how to construct how to construct an elliptic curve Ef defined over Q such that the Dirichlet series (s) associated with f is the same as the Lfunction L(Ef,s) Thus, the ShimuraTaniyamaWeil conjecture becomes the statement that Shimuras map from the set of such cuspforms to the set of elliptic curves defined over Q is surjective up to isogeny A rough description of the Shimura map followsLet be a congruence subgroup of SL2(Z) and let X() denote the compact Riemann surface H The inclusion of in (1) induces a branched covering X() X(1) P1 . One may use the elementary RiemannHurwitz formula from combinatorial topology to determine the Euler number, and consequently the genus, of X() The genus is the dimension of the space of cuspforms of weight 2 Even when the genus is zero one obtains embeddings of X() in projective spaces Pr through holomorphic maps (f0(), f1(), , fr()) , where $f{}_{},\; f{}_{},,\; f{}_{}$ is a basis of the space of modular forms of weight w with w sufficiently large For example, any w 12 will suffice for (1) For 0(N) (but not for arbitrary ) one may find a basis of the space of modular forms of weight w having rational Fourier coefficients Using the corresponding projective embedding one finds a model for $X{}_{}(N)X({}_{}(N))$ over Q i.e., an algebraic curve defined over Q in projective space that is isomorphic as a compact Riemann surface to X0(N)Associated with any complete nonsingular algebraic curve (i.e., after Weyl &BibRef-weyl;weyl, any compact Riemann surface) X of genus g is a complex torus, the Jacobian J(X) of X, that is the quotient of gdimensional complex vector space Cg by the lattice generated by the period matrix, which is the g 2g matrix in C obtained by integrating each of the g members i of a basis of the space of holomorphic differentials over each of the 2g loops in X representing the members of a homology basis in dimension 1 Furthermore, if one picks a base point z0 in X, then for any z in X, the path integral from z0 to z of each of the g holomorphic differentials is welldefined modulo the periods of the differential One obtains a holomorphic map $XJ(X)$ from the formula z (z0z 1, , z0z g) . This map is, in fact, universal for pointed holomorphic maps from X to complex tori Furthermore, the Jacobian J(X) is an algebraic variety that admits definition over any field of definition for X and z0, and the universal map also admits definition over any such field The complex tori that admit embeddings in projective space are the abelian group objects in the category of projective varieties They are called abelian varieties Every abelian variety is isogenous to the product of simple abelian varieties: abelian varieties having no abelian subvarieties Shimura showed that one of the simple isogeny factors of J(X0(N)) is an elliptic curve Ef defined over Q characterized by the fact that its onedimensional space of holomorphic differentials induces on X0(N) via the composition of the universal map with projection on Ef the onedimensional space of differentials on X0(N) determined by the cuspform f He showed further that L(Ef, s) is the Dirichlet series (s) with Euler product given by f An elliptic curve E defined over Q is said to be modular if it is isogenous to Ef for some WNcompatible Hecke eigencuspform of weight 2 for 0(N) Equivalently E is modular if and only if L(E,s) is the Dirichlet series given by such a cuspform The ShimuraTaniyamaWeil Conjecture states that every elliptic curve defined over Q is modular Shimura &BibRef-shimnagoya;shimnagoya showed that this conjecture is true in the special case where the Zmodule rank of the ring of endomorphisms of E is greater than one In this case the point (notation of section iref="sec.ellipC"sec.ellipC) of the upperhalf plane corresponding to E(C) is a quadratic imaginary number, and L(E, s) is a numbertheoretic Lfunction associated with the corresponding imaginary quadratic number field

8SU-9KEY-9The hypothetical Frey curveLet p 5 be a prime Based on the assumption, which presumably is false, that there are nonzero integers a, b, c such that $apbpcp0$ G. Frey observed that the elliptic curve given by the equation eq.freyeq.freyEqnAuto y2 x(x ap)(x bp) , which is certainly defined over Q, would not be likely to be modular Thus, if the ShimuraTaniyamaWeil Conjecture were true, then Fermats Last Theorem would also be true By 1987 it had been shown through the efforts of Frey, Ribet and Serre that the Frey curve (iref="eq.frey"eq.frey) is not modular The proof involves the systematic study of what is known as the adic representation of an elliptic curve defined over Q which is described in the next section This same technique is what has been reported to be the basis of the proof of Wiles that every semistable elliptic curve defined over Q is modular The Frey curve (iref="eq.frey"eq.frey) has discriminant (abc)p It is only slightly difficult to see that it is semistable, and, therefore, that its conductor N is the squarefree integer abc If the Frey curve is modular, one is led to a cuspform of weight 2 for 0(abc) The theory of adic representations leads one along a path of reductions of the level N from the initial level abc that enables one to conclude that there is a cuspform of weight 2 for 0(2); but the genus of X0(2) is 0, and, consequently, there is no such cuspform

9SU-10KEY-10adic representations of Gal(QQ)Let E be an elliptic curve defined over Q Inasmuch as the group law $EEE$ is defined over Q it follows that for each integer m the group (scheme) E[m] of mtorsion points, i.e., for any field K containing Q the group E[m](K) consisting of all x in E(K) such that mx 0, is defined by equations with rational coefficients Consequently, any automorphism of K must carry the group E[m](K) into itself Since E(C) is the quotient of C by a lattice, it is clear that E[m](C) is isomorphic to $ZmZZmZ$; in fact, this latter group is isomorphic to E[m](K) for each algebraically closed field of characteristic 0 There is a unique ring homomorphism $ZmnZZmZ$ for each integer n 1, and the family of these ring homomorphisms gives rise to an inverse system in the category of commutative rings If one specializes to the case m r where is prime, the projective limit is the ring Z of adic integers The groups E[m] form a direct system with respect to the inclusions $E[m]E[mn]$, but, corresponding to the inverse system of the groups ZmZ form an inverse system (the Tate system) with respect to the family of homorphisms $E[mn]E[m]$ defined by $xnx$ If one specializes to the case m r where is prime, one obtains the projective limit eq.tategpeq.tategpEqnAuto T(E) proj limr E[r](Q) Z Z , which is isomorphic to the cohomology module H1(E, Z) . The action of Gal(QQ) on the torsion groups E[m] induces an action of Gal(QQ) on the projective limit T(E) This action gives rise to a representation : Gal(QQ) GL2(Z) , which is called the adic representation of E In considering one is reminded of the action of the automorphism group of a manifold M on the cohomology H(M) and, more particularly, the action of Gal(CR) on the cohomology of M when M is an algebraic manifold in Pn(C) defined by equations with real coefficients, but one must keep in mind that the transformations of E(Q) arising from the elements of Gal(QQ) are not even remotely continuous in the classical topology on E(C) More generally, there is an algebraic way of defining the cohomology ring H(M, Z) (see Tate &BibRef-tatepurdue;tatepurdue) when M is an algebraic variety with the property that automorphisms fixing the field of definition act on H(M, Z) An introduction to the study of may be found in Serres Montreal Notes &BibRef-serremontreal;serremontrealThe canonical ring homomorphism from the ring Z of adic integers to the field ZZ induces a group homomorphism $AA$, called reduction mod from the group GL2(Z) to the finite group GL2(ZZ) An adic representation of Gal(QQ) is called modular if it is isomorphic to the representation arising from the elliptic curve Ef that is the image under the Shimura map of a modular form f A representation Gal(QQ) GL2(ZZ) is called modular if it is isomorphic to for some modular adic representation In the extensive detailed study of representations of Gal(QQ) particular attention has been paid to the question of when a representation in GL2(Z) is modular and also to the question of when a representation of GL2(ZZ) is modular Under certain conditions (see Serre &BibRef-serreduke;serreduke and Ribet &BibRef-ribinv;ribinv, &BibRef-ribtoul;ribtoul) one can show that is modular if is modular, i.e., is modular if it is congruent mod to a modular adic representation Such arguments are central both to the work of Ribet in showing that the ShimuraTaniyamaWeil conjecture implies Fermats Last Theorem and to the reported work of Wiles in proving that semistable elliptic curves are modular In the work of Ribet the basic idea is that the modularity of the Frey curve, which has squarefree conductor N abc, implies the existence of a cusp form of weight 2 and level N By using an argument at the scene of the mod representations, Ribet shows that one may split each odd prime divisor out of the level N and arrive at the conclusion that there is a cusp form of weight 2 and level 2, which is not possible pagebreak

Appendix SU-11KEY-11Late 1993early 1994 StatusAndrew Wiles posted the following announcement in the UseNet electronic news group called sci.math:FromwilesrugolaPrincetonEDUAndrewWilesNewsgroupsscimathSubjectFermatstatusMessageID1993Dec401365012700PrincetonEDUDate4Dec93013650GMTSendernewsPrincetonEDUUSENETNewsSystemOrganizationPrincetonUniversityLines21OriginatornewsnimasterNntpPostingHostrugolaprincetoneduInviewofthespeculationonthestatusofmyworkontheTaniyamaShimuraconjectureandFermatsLastTheoremIwillgiveabriefaccountofthesituationDuringthereviewprocessanumberofproblemsemergedmostofwhichhavebeenresolvedbutoneinparticularIhavenotyetsettledThekeyreductionofmostcasesoftheTaniyamaShimuraconjecturetothecalculationoftheSelmergroupiscorrectHoweverthefinalcalculationofapreciseupperboundfortheSelmergroupinthesemistablecaseofthesymmetricsquarerepresentationassociatedtoamodularformisnotyetcompleteasitstandsIbelievethatIwillbeabletofinishthisinthenearfutureusingtheideasexplainedinmyCambridgelecturesThefactthatalotofworkremainstobedoneonthemanuscriptmakesitstillunsuitableforreleaseasapreprintInmycourseinPrincetonbeginninginFebruaryIwillgiveafullaccountofthisworkAndrewWiles pagebreak

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