Among the challenges that I have been facing with my GELLMU project are (1) convincing mathematicians that it is possible to use comfortable LaTeX-like markup in a fully rigorous way to prepare our articles so that they can have formal inclusion in the markup category known as 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 LaTeX markup for the official notes on my October 1993 Albany seminar presentation and edited what was LaTeX source to convert it to LaTeX-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 original write-up, 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 Wiles's 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 Mathematics.
There 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 “Shimura-Taniyama-Weil” 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 “semi-stable” case.
I list a few references on these matters for the period since my original talk:
What follows has the same content as the original write-up except that the title of the Appendix has been changed from “Current Status” to “Late 1993/Early 1994 Status”.
The purpose of this expository lecture is to explain the basic ideas underlying the final resolution of “Fermat's Last Theorem” after 356 years as a consequence of the reported establishment by Andrew Wiles of a sufficient portion of the “Shimura-Taniyama-Weil” 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  of K. Ribet, and (3) a preprint  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 “Fermat's Last Theorem” is a consequence of sufficient knowledge of the theory of “elliptic curves” has been fully documented in the publications (, ) of K. Ribet.
“Fermat's Last Theorem” is the statement, having origin with Pierre de Fermat in 1637, that there are no positive integers such that for any integer exponent . Obviously, if there are no positive integer solutions for a particular , then there are certainly none for exponents that are multiples of . Since every integer is divisible either by or by some odd prime , it follows that “Fermat's Last Theorem” is true if there are no solutions in positive integers of the equation when and when for each prime . The cases are standard fare for textbooks (e.g., see Hardy & Wright ) in elementary number theory. Therefore, this discussion will focus on the case where is prime.
Very briefly, the idea is that we now know enough about the classification of non-degenerate plane cubic curves 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 with , then the curve 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 with .
The enumerative classification of non-degenerate plane cubic curves defined by polynomials with rational coefficients has been entirely conjectural (variously known as the “Taniyama Conjecture”, the “Weil Conjecture”, the “Taniyama-Shimura 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 representations.
What seems to be believed today2 is that the portion of the enumerative classification pertaining to “semi-stable” elliptic curves has been proved by Andrew Wiles. That the existence of positive integers with would violate the enumerative classification of semi-stable elliptic curves was established by 1987 through the work of G. Frey, J.-P. Serre, and K. Ribet.
The 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.
A polynomial of degree in two variables with coefficients in a field gives rise to what is called an affine plane curve of degree : for each field containing (more generally, for each commutative ring that is a -algebra) one has the set and for each -linear homomorphism one has the induced map . From the polynomial one obtains a homogeneous polynomial of degree d in three variables with coefficients in : and the projective plane curve of degree : where denotes -dimensional projective space, which is the quotient set of obtained by identifying points lying on the same line through the origin of . Since the projective plane is the disjoint union of the affine plane with the “(projective) line at infinity” it follows that is the disjoint union of with the finite set of its points lying on the projective line at infinity.
An elliptic curve defined over is the (projective) plane curve given by a homogeneous polynomial of degree in three variables with coefficients in such that (i) is irreducible over the algebraic closure of , (ii) the gradient vector is a non-vanishing vector at points of where vanishes, and (iii) the set is non-empty.
If is any field, then after an isomorphism (see Silverman ) one may obtain a given elliptic curve with an affine equation of the form Then the homogeneous equation for the intersection of with the line at infinity is Thus, in this case, has a unique point on the line at infinity. If the characteristic of is different from and 3 then one may obtain an equation in “Weierstrass normal form”: which is non-singular if and only if the cubic polynomial in the variable has distinct roots in .
Elliptic curves are the “group objects” in the category of algebraic curves that reside in projective space: for each extension field of the set of “-valued points” of is an abelian group. The group law on is characterized by two conditions:
The origin is a given point of .
The points obtained by intersecting with any line in , counted with multiplicities, add up to zero.
When is given by an equation in the form (1), the origin is usually taken to be the unique point on the line at infinity. If two distinct points of are given, they determine a line in ; the intersection of that line with is given by a cubic polynomial in a parameter for the line which has two roots in corresponding to the two given points; hence, there is a third root of that cubic polynomial in ; this root gives rise to a point of , which is the negative of the sum of the two given points. The negative of a given point of is obtained as the third point in the intersection with of the line through the given point and the origin.
For a given field the set of homogeneous cubic polynomials in three variables is a vector space over 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 acts on the space of cubics, and two cubic curves in that are related by this action are isomorphic. Since is 9-dimensional, one is led to think of the family of isomorphism classes of elliptic curves as 1-dimensional since “non-singularity” is an “open” condition.
When is the field of complex numbers, one knows (see, e.g., Ahlfors ) that for each lattice in the set of -periodic meromorphic functions on the complex line is the field , which is a quadratic extension of the rational function field , where is the -function of Weierstrass. Moreover, satisfies the famous Weierstrass differential equation thus, the formula defines a holomorphic map from the punctured complex torus to the affine cubic curve it should hardly be necessary to point out that this map extends to a holomorphic map from the torus 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  or 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 arises from some complex torus. Indeed each non-singular cubic curve in determines a compact connected complex-analytic group. Its universal cover is given by a holomorphic homomorphism which has some lattice as kernel.
Any two lattices in are related by a change of real basis for , i.e., by a matrix in . Consequently, there is only one real-analytic isomorphism class for the complex torus as varies. The tori corresponding to two lattices are complex-analytically isomorphic if and only if the corresponding real-linear isomorphism of satisfies the Cauchy-Riemann partial differential equations, i.e., if and only if the -linear isomorphism is -linear.
A lattice may be represented concretely by an ordered basis . If , then is not real, and after permuting the basis members, if necessary, one may assume that is in the “upper-half plane”4 of . Observing that is the image under the -linear map of the lattice with ordered basis , one may assume that is this latter lattice. Let be the complex torus . Allowing for change of basis subject to these assumptions on the basis, one sees that there is an isomorphism of complex-analytic groups if for some matrix Conversely, the monodromy principle may be used to show that every complex-analytic isomorphism among the complex tori arises in this way.
The coefficients and in the Weierstrass normal form (5) have very explicit constructions as infinite series (see, e.g., Ahlfors  or Serre ) determined by the given lattice; from this it is straightforward to see that is a modular form of weight : if and are related by (6), then Consequently, the map carries the curve given by (5) for isomorphically to the curve given by (5) for . The discriminant of the cubic polynomial in the Weierstrass normal form (5) is a modular form of weight , which up to a multiplicative constant, is: is a non-vanishing holomorphic function in . The modular invariant (,) is defined by: it is a holomorphic function in the upper-half plane with the property that if and only if and are related by (6). Furthermore, assumes every value in at some point of . Consequently, the complex-analytic isomorphism classes of complex tori or, equivalently, the isomorphism classes of elliptic curves defined over , are parameterized via in a one-to-one manner by the complex numbers.
Since 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 and (2) the category of complex tori is the “genus one” case of the coincidence (see Weyl ) of (i) the category of “complete” non-singular algebraic curves over and (ii) the category of compact Riemann surfaces (one-dimensional connected complex-analytic manifolds).
Although the classification of elliptic curves over via the -function is a result that is both beautiful and useful, and although two elliptic curves defined over that are isomorphic as curves defined over give rise to elliptic curves defined over that have the same -invariant, it is not true that any two elliptic curves defined over having the same -invariant are isomorphic over . Thus, the classification of elliptic curves over does not lead directly to the desired enumerative classification of elliptic curves defined over , but it does bring to the fore the notion of modular form, which is central in the study of elliptic curves defined over . What can be said easily is that, according to the Shimura-Taniyama-Weil conjecture, the isogeny classes of elliptic curves defined over are parameterized by certain modular forms.
The group is an infinite group that is generated by the two elements which have orders 4 and 6 respectively. The action of on the upper-half plane by linear fractional transformations has kernel and the quotient of by this kernel is the group . It is not difficult to see that the set is a “fundamental domain” for the action of on . More precisely, this set meets each orbit, and the only redundancies are the boundary identifications arising from the maps and . The quotient is not compact since the fundamental domain is “open at the top”. Beyond that the modular invariant induces a bicontinuous biholomorphic isomorphism of the quotient with the affine line over . Since , and since for one has for , there is a holomorphic function in the punctured unit disk such that . Likewise may be regarded as function of , and one may use the calculus of residues to show that has a simple zero at ; hence, has a simple pole at , or, equivalently, has a simple pole at (the “missing top” of the fundamental domain). Thus, gives rise to a bicontinuous biholomorphic isomorphism
A non-trivial element of has a fixed point in if and only if it has finite order, and one's explicit knowledge of the fundamental domain makes it possible to see that the only elements of finite order are of order or 5. A congruence subgroup of is a subgroup that contains one of the principal congruence subgroups; the principal congruence subgroup of level is the set of all elements of that are congruent to the identity matrix. The group is the congruence subgroup of consisting of all elements for which . It is obvious that each congruence group has finite index in , and, consequently the quotient is a non-compact Riemann surface. Observe that for each level the group contains the parabolic element which gives rise to the holomorphic map that fixes the point .
A modular form6 of weight for is a holomorphic function in that satisfies the functional equation and that is holomorphic at each cusp of . The role of cusps for is to provide a slightly larger set than , where acts such that is a compact Riemann surface containing 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 in = that are fixed by some non-trivial parabolic element of . When = , the set of cusps is . In view of (7) applied to the case one sees that a modular form of any weight for the group satisfies and, therefore, defines a holomorphic function in the variable for . The condition in the definition of modular form that should be holomorphic at means that as a function of is holomorphic at . Consequently, admits an absolutely convergent Fourier expansion which is a Taylor series in .
For any cusp of a congruence group one may define the notion holomorphic at for a modular form by an analogous procedure using an arbitrary parabolic element of that fixes instead of . For a given congruence group two cusps and are equivalent if there is some element in such that . A modular form is holomorphic at any cusp that is equivalent to another where it is holomorphic. The modular form is a cuspform if, in addition to being holomorphic at each cusp, 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 forms a finite-dimensional vector space over 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 there are no modular forms of odd weight, there is an Eisenstein series of every even weight greater than 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 and since is a simple zero of , every cuspform for is divisible by . Thus, in this case, there are no cuspforms of weight less than .
It is not difficult to see that the cuspforms of weight for a congruence group correspond to holomorphic differential -forms (differentials of the first kind) on the compact Riemann surface . Thus, the dimension of the space of cuspforms of weight is the genus of . The fact that there are no cuspforms of weight for the group matches the previously mentioned fact that is . It is certain of the cuspforms of weight two for the groups that, according to the Shimura-Taniyama-Weil conjecture, parameterize the isogeny classes of elliptic curves defined over .
It will be recalled that the infinite series converges for and gives rise by analytic continuation to a meromorphic function in . For admits the absolutely convergent infinite product expansion 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 of integers. It is, as well, the product taken over all the non-archimedean completions of the rational field (which completions are indexed by the set of primes) of the “Mellin transform”7 in of the canonical “Gaussian density” which Gaussian density is equal to its own Fourier transform. For the archimedean completion of the rational field one forms the classical Mellin transform of the classical Gaussian density (which also is equal to its own Fourier transform). Then the function is meromorphic in , and satisfies the functional equation
The connection of Riemann's -function with the subject of modular forms begins with the observation that is essentially the Mellin transform of , where , which is a modular form of weight and level , is defined in the upper-half plane by the formula In fact, one of the classical proofs of the functional equation (10) is given by applying the Poisson summation formula8 to the function , while observing that the substitution for corresponds in the upper-half plane to the substitution for the theta series.
If is a cuspform for a congruence group containing and so, consequently, , then, as previously explained, one has the Fourier expansion (9) The Mellin transform of leads to the Dirichlet series which may be seen to have a positive abscissa of convergence. One is led to the questions:
For which cuspforms does the associated Dirichlet series admit an analytic continuation with functional equation?
For which cuspforms does the associated Dirichlet series have an Euler product expansion?
For the “modular group” the Dirichlet series associated to every cuspform of weight admits an analtyic continution with functional equation under the substitution . Since is generated by the two matrices and and since the functional equation of a modular form relative to is reflected in the formation of the Fourier series (9), the condition that an absolutely convergent series (9) is a modular form for is the functional equation for a modular form relative solely to . This is equivalent to the (properly formulated) functional equation for the associated Dirichlet series together with a “growth condition”. For the group , with , the question of a functional equation is more complicated since, although is available, there is no reason for a cuspform to satisfy a law of transformation relative to . 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 , that the cuspforms admitting an Euler product expansion form a vector space.
In a nutshell the cuspforms admitting Euler products are those which arise as eigenforms for an arithmetically defined commutative algebra of semi-simple operators on the space of cuspforms of a given weight introduced by E. Hecke. The theory of Hecke operators is reasonably simple for level but somewhat more complicated in general (see, e.g., Shimura's book ).
Observing that the formula gives a (the hyperbolic) -invariant metric in with associated invariant measure one introduces the Petersson (Hermitian) inner product in the space of cuspforms of weight for with the definition: (Integration over the quotient makes sense since the integrand is -invariant.)
For the modular group the Hecke operator is the linear endomorphism of the space of cuspforms of weight arising from the following considerations. Let be the set of matrices in with determinant . For and for a function in one defines and then, observing that under acts trivially on the modular forms of weight , one may define the Hecke operator by where the quotient refers to the action of by left multiplication on the set . One finds for , coprime that and furthermore one has Consequently, the operators commute with each other, and, therefore, generate a commutative algebra of endomorphisms of the space of cusp forms of weight for . It is not difficult to see that the Hecke operators are self-adjoint for the Petersson inner product on the space of cuspforms. Consequently, the space of cuspforms of weight admits a basis of simultaneous eigenforms for the Hecke algebra. A “Hecke eigencuspform” is said to be normalized if its Fourier coefficient . If is a normalized Hecke eigencuspform, then
The Fourier coefficient of is the eigenvalue of for .
The Fourier coefficients of satisfy
Consequently, the Dirichlet series associated with a simultaneous Hecke eigencuspform of level and weight admits an Euler product For example, when is the unique normalized cuspform of level and weight , one has where is the function of Ramanujan.
For the congruence group a Hecke eigencuspform of weight gives rise to a Dirichlet series that admits an Euler product expansion whose factors at primes coprime to resemble those given by (15). In order for to satisfy a functional equation under the substitution , one needs to require that the eigencuspform admits a functional equation not only with respect to each element of the group but also with respect to the substitution in the upper-half plane given by the matrix A. Weil () showed that the cuspforms of weight for the group satisfying the appropriate functional equation under the mapping of given by correspond precisely to Dirichlet series with certain growth conditions that admit analytic continuations as meromorphic functions in satisfying a finite number of “twisted” functional equations.
The reader will have noticed that it is not extremely easy to characterize the cuspforms of weight that conjecturally (Shimura-Taniyama-Weil) parameterize the isogeny classes of elliptic curves defined over the rational field . 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 satisfies these conditions, i.e., is the Dirichlet series associated to some -compatible Hecke eigencuspform for the group , where is the “conductor” of . This has led to efforts, related to the “Langlands program” to understand the -compatible Hecke eigencuspforms in a more intrinsic way as objects of representation theory over (see, e.g., the survey of Gelbart ).
Let be an elliptic curve defined over . 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 (3) is available to represent the isomorphism class of any elliptic curve over a field of characteristic different from and , one needs the generalized Weierstrass form over an arbitrary field, and, moreover, for each elliptic curve defined over there is a “best possible” equation (e.g., see Silverman ) of the form (16) with integer coefficients called the Neron model of . With an abuse of notation will denote the Neron model, which may be regarded as a “curve over ”. (One would want to call it an “elliptic curve over ” if it were “smooth over ”, i.e., if it had good reduction at each prime ; 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 the Neron model gives rise to a cubic equation over the finite field . For all but a finite number of the equation over is non-singular over , i.e., determines an elliptic curve defined over . One says in this case that has “good reduction” mod . Following Tate () one introduces Then one has The non-vanishing of mod is necessary and sufficient for to have good reduction mod . It follows that a prime divides if and only if does not have good reduction mod . If is a prime for which has “bad reduction”, then there is a single singular point of the reduced curve , and either (a) has distinct tangent lines at the singular point (semi-stable reduction) or (b) has a single tangent line occurring with multiplicity . is called semi-stable if it has either good or semi-stable reduction at each prime. The conductor of is the integer defined by where The non-negative integer cannot be positive unless is or . Tautologically, is semi-stable if and only if its conductor is square-free.
One defines the “L-series” of by where is defined when has good reduction mod by the formula and is defined when has bad reduction mod by One observes readily that the L-function of codifies information about the number of points of in the finite field . Quite generally for an algebraic variety defined over the analogous codification of information obtained by counting points in the various reductions mod of the variety yields the “Hasse-Weil zeta function”, which reflects “cohomological” information about . The L-series of is the essential part, corresponding to cohomology in dimension , of the Hasse-Weil zeta function of . The Hasse-Weil zeta function is a special case of the general notion (Serre ) of “zeta function” for a scheme of finite type over .
One observes that resembles, at least insofar as one considers its Euler factors for primes corresponding to good reductions of , the Dirichlet series associated to a cuspform of weight that admits an Euler product expansion. The observation of this resemblance is the beginning of an appreciation of the Shimura-Taniyama-Weil conjecture. One is led to ask to what extent the two classes of Dirichlet series with Euler products coincide. The conjecture states that the L-function of an elliptic curve defined over with conductor arises from a cuspform for the group that is compatible with the substitution in the upper-half plane given by . Isogenous elliptic curves have the same -function, and, conversely (cf. Tate  and Faltings ) two elliptic curves with the same -function must be isogenous. Thus, the idea of the conjecture is that the isogeny classes of elliptic curves defined over with conductor are in bijective correspondence with the set of Hecke eigencuspforms for the group of level , compatible with the extension of that group by the substitution arising from , having rational Fourier coefficients and not arising from levels dividing .
Shimura (, , ) showed for a given -compatible Hecke eigencuspform of weight for the group with rational Fourier coefficients how to construct how to construct an elliptic curve defined over such that the Dirichlet series associated with is the same as the L-function . Thus, the Shimura-Taniyama-Weil conjecture becomes the statement that Shimura's map from the set of such cuspforms to the set of elliptic curves defined over is surjective up to isogeny. A rough description of the Shimura map follows.
Let be a congruence subgroup of , and let denote the compact Riemann surface . The inclusion of in induces a “branched covering” One may use the elementary Riemann-Hurwitz formula from combinatorial topology to determine the Euler number, and consequently the genus, of . The genus is the dimension of the space of cuspforms of weight . Even when the genus is zero one obtains embeddings of in projective spaces through holomorphic maps where is a basis of the space of modular forms of weight with sufficiently large. For example, any will suffice for . For (but not for arbitrary ) one may find a basis of the space of modular forms of weight having rational Fourier coefficients. Using the corresponding projective embedding one finds a model for over , i.e., an algebraic curve defined over in projective space that is isomorphic as a compact Riemann surface to .
Associated with any “complete non-singular” algebraic curve (i.e., after Weyl , any compact Riemann surface) of genus is a complex torus, the “Jacobian” of , that is the quotient of -dimensional complex vector space by the lattice generated by the “period matrix”, which is the matrix in obtained by integrating each of the members of a basis of the space of holomorphic differentials over each of the loops in representing the members of a homology basis in dimension . Furthermore, if one picks a base point in , then for any in , the path integral from to of each of the holomorphic differentials is well-defined modulo the periods of the differential. One obtains a holomorphic map from the formula This map is, in fact, universal for pointed holomorphic maps from to complex tori. Furthermore, the Jacobian is an algebraic variety that admits definition over any field of definition for and , 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 is an elliptic curve defined over characterized by the fact that its one-dimensional space of holomorphic differentials induces on , via the composition of the universal map with projection on , the one-dimensional space of differentials on determined by the cuspform . He showed further that is the Dirichlet series with Euler product given by . An elliptic curve defined over is said to be modular if it is isogenous to for some -compatible Hecke eigencuspform of weight for . Equivalently is modular if and only if is the Dirichlet series given by such a cuspform. The Shimura-Taniyama-Weil Conjecture states that every elliptic curve defined over is modular. Shimura  showed that this conjecture is true in the special case where the -module rank of the ring of endomorphisms of is greater than one. In this case the point (notation of section 3) of the upper-half plane corresponding to is a quadratic imaginary number, and is a number-theoretic -function associated with the corresponding imaginary quadratic number field.
Let be a prime. Based on the assumption, which presumably is false, that there are non-zero integers such that , G. Frey observed that the elliptic curve given by the equation which is certainly defined over , would not be likely to be modular. Thus, if the Shimura-Taniyama-Weil Conjecture were true, then “Fermat's Last Theorem” would also be true. By 1987 it had been shown through the efforts of Frey, Ribet and Serre that the Frey curve (18) is not modular. The proof involves the systematic study of what is known as the “-adic representation” of an elliptic curve defined over , 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 semi-stable elliptic curve defined over is modular. The Frey curve (18) has discriminant . It is only slightly difficult to see that it is semi-stable, and, therefore, that its conductor is the square-free integer . If the Frey curve is modular, one is led to a cuspform of weight for . The theory of -adic representations leads one along a path of reductions of the level from the initial level that enables one to conclude that there is a cuspform of weight for ; but the genus of is , and, consequently, there is no such cuspform.
Let be an elliptic curve defined over . Inasmuch as the group law is defined over it follows that for each integer the group (scheme) of -torsion points, i.e., for any field containing the group consisting of all in such that , is defined by equations with rational coefficients. Consequently, any automorphism of must carry the group into itself. Since is the quotient of by a lattice, it is clear that is isomorphic to ; in fact, this latter group is isomorphic to for each algebraically closed field of characteristic . There is a unique ring homomorphism for each integer , 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 , where is prime, the projective limit is the ring of -adic integers. The groups form a direct system with respect to the inclusions , but, corresponding to the inverse system of the groups , form an inverse system (the Tate system) with respect to the family of homorphisms defined by . If one specializes to the case , where is prime, one obtains the projective limit which is isomorphic to the cohomology module The action of on the torsion groups induces an action of on the projective limit . This action gives rise to a representation which is called the -adic representation of . In considering one is reminded of the action of the automorphism group of a manifold on the cohomology and, more particularly, the action of Gal() on the cohomology of when is an algebraic manifold in defined by equations with real coefficients, but one must keep in mind that the transformations of arising from the elements of are not even remotely continuous in the classical topology on . More generally, there is an algebraic way of defining the cohomology ring (see Tate ) when is an algebraic variety with the property that automorphisms fixing the field of definition act on . An introduction to the study of may be found in Serre's “Montreal Notes” .
The canonical ring homomorphism from the ring of -adic integers to the field induces a group homomorphism , called reduction mod , from the group to the finite group . An -adic representation of is called modular if it is isomorphic to the representation arising from the elliptic curve that is the image under the Shimura map of a modular form . A representation is called modular if it is isomorphic to for some modular -adic representation . In the extensive detailed study of representations of particular attention has been paid to the question of when a representation in is modular and also to the question of when a representation of is modular. Under certain conditions (see Serre  and Ribet , ) 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 Shimura-Taniyama-Weil conjecture implies “Fermat's Last Theorem” and to the reported work of Wiles in proving that semi-stable elliptic curves are modular. In the work of Ribet the basic idea is that the modularity of the Frey curve, which has square-free conductor , implies the existence of a cusp form of weight and level . 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 and arrive at the conclusion that there is a cusp form of weight and level , which is not possible.
Andrew Wiles posted the following announcement in the “UseNet” electronic news group called “sci.math”: