APPEARED IN BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 4, October 1995, Pages 375-402

GALOIS REPRESENTATIONS AND MODULAR FORMS

KENNETH A. RIBET

Abstract. In this article, I discuss material which is related to the recent proof of Fermat’s Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey’s construction, and the conjectures of Serre and of Taniyama-Shimura.

1. Introduction

This article is a revised version of the notes which were distributed at my Progress in Mathematics lecture at the August 1994 Minneapolis Mathfest. When I was first approached in 1993 by the Progress in Mathematics committee, I was asked to discuss “the mathematics behind Andrew Wiles’s solution of the Fermat conjecture.” As the reader is no doubt aware, Wiles had announced in a series of lectures at the Isaac Newton Institute for Mathematical Sciences that he was able to prove for semistable elliptic curves the conjecture of Shimura and Taniyama to the effect that all elliptic curves over Q are “modular” (i.e., attached to modular forms in a sense which will be explained below). Fermat’s Last Theorem follows from this result, together with a theorem that I proved seven years ago [62].

At the time of the Mathfest, however, a gap which had appeared in Wiles’s work had not yet been repaired (see [33]) Fermat’s “Theorem” was still a conjecture. Nevertheless, it was readily apparent that the methods introduced by Wiles were significant and deserving of attention. Most notably, these methods had been used to construct for the first time an infinite set E of modular elliptic curves over Q with the following property: if E_{1} and E_{2} are elements of E , then E_{1} and E_{2} are non-isomorphic even when viewed as elliptic curves over the complex field C (see [69]). In my lecture at the Mathfest, I stressed this achievement of Wiles and discussed the analogy between the Taniyama-Shimura conjecture and conjectures of Serre [74] about two-dimensional Galois representations.

As I began to revise these notes, I found that the situation had changed dramatically for the better. Wiles announced in October 1994 that the bound he sought in his original proof could be obtained by a method which circumvented the Euler system construction [34]. This method arose from an observation that Wiles made early in his investigation: the required upper bound would follow from a proof that certain Hecke algebras are complete intersection rings. This statement about Hecke algebras is the main theorem of an article written jointly by Richard Taylor and Andrew Wiles [87]; these authors announced their result at the same time that Wiles disseminated a revised version of his original manuscript [90].

It is premature to undertake a detailed analysis of the results of Wiles and Taylor-Wiles. The aim of this survey is more modest: to present an introduction to the

\begin{matrix}c \\ ◯ \end{matrix}1995 American Mathematical Society
0273-0979/95 $1.00 + $.25 per page

Received by the editors November 22, 1994, and, in revised form, April 2, 1995.

1991 Mathematics Subject Classification. Primary 11F, 11D.

This work was to some extent supported by NSF grant #DMS 93-06898.

circle of ideas which form the background for these results. Because of the intense publicity surrounding Fermat’s Last Theorem, a good deal of the material I have chosen has been discussed in news and expository articles which were written in connection with Wiles’s 1993 announcement. Among these are the author’s news item in the Notices [65] and his article with Brian Hayes in American Scientist [30], two pieces in the American Mathematical Monthly [13, 28], the report by K. Rubin and A. Silverberg in this Bulletin [68], a long survey by H. Darmon [14], an article for undergraduates and their teachers by N. Boston [6], and an elementary article by A. Ash and R. Gross [1] which discusses conjectured reciprocity relations between algebraic varieties and automorphic forms. Books containing articles related to Fermat’s Last Theorem are soon to appear [10, 55]. Furthermore, two videotapes related to Fermat have been in circulation for some time: a 1993 lecture by the author is available from the AMS [66], and the Mathematical Sciences Research Institute has been distributing a videotape based on the July 1993 “Fermat Fest”, which was organized by the MSRI and the San Francisco Exploratorium. Readers may also consult material available by gopher from e-math.ams.org.

In view of the burgeoning literature in this subject, I imagined these notes mostly as a somewhat biased guide to reference works and expository articles. In the end, what I have written might best be characterized as an abbreviated survey with a disproportionately large list of references.

2. Modular forms

We begin by summarizing some background material concerning modular forms. Among reference books in the subject, one might cite [36], [53], and [79]. Also, several books on elliptic curves contain substantial material on modular forms; in particular, Knapp’s book [35] has been recommended to the author as a good source for an overview of the Eichler-Shimura theory relating elliptic curves to certain modular forms. For a first introduction to modular forms, a fine starting point is [72, Ch. VII].

The modular forms to be considered are “cusp forms of weight two on Γ_{0}(N) ”, for some integer N ≥ 1 . Here, Γ_{0}(N) is the group of integer matrices with determinant 1 which are upper-triangular mod N . A cusp form on this group is, in particular, a holomorphic function on the upper half-plane H consisting of complex numbers with positive imaginary part.

These functions are usually presented as Fourier series ∑_{n=1}^{∞} a_{n}q^{n} , where q := e^{2πiz} . For the forms which most interest us, the complex numbers a_{n} are algebraic integers; frequently they are even ordinary integers.

The weight-two cusp forms on Γ_{0}(N) are holomorphic functions f(z) on the complex upper half-plane H. One requires principally that f(z) dz be invariant under Γ_{0}(N) , i.e., that f(z) satisfy the functional equation

f (\frac{az + b}{cz + d}) = (cz + d)^{2}f(z)

for all ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) . (In particular, one has f(z + b) = f(z) for all inte-

gers b.) In addition to the holomorphy and the functional equation, one imposes subsidiary conditions at infinity [79, Ch. 2].

The group Γ_{0}(N) acts on H by fractional linear transformations, with ( \begin{matrix}a & b \\ c & d \end{matrix} ) acting as

z ↦ \frac{az + b}{cz + d}.

The invariance of f(z) dz means that f(z) dz arises by pullback from a differential on the quotient Y_{0}(N) := Γ_{0}(N)\H. This quotient is a non-compact Riemann surface with a standard compactification known as X_{0}(N) . The complement of Y_{0}(N) in X_{0}(N) is a finite set, the set of cusps of X_{0}(N) . The conditions on infinity satisfied by a cusp form require the differential on Y_{0}(N) associated to f(z) to extend to X_{0}(N) . This means simply that f(z) is required to vanish at the cusps.

Let us identify the space S(N) of weight-two cusp forms on Γ_{0}(N) with the space of holomorphic differentials on the Riemann surface X_{0}(N) . Then S(N) has finite dimension: dim S(N) is the genus of the curve X_{0}(N) . This integer may be calculated easily via the Riemann-Hurwitz formula (applied to the covering X_{0}(N) → X_{0}(1) which corresponds to the inclusion of Γ_{0}(N) in Γ_{0}(1) ). For example, dim S(N) is zero for N ≤ 10 and is one for N = 11 . (See [79, Ch. 2].)

In order to present an example of a non-zero cusp form, we will exhibit a nonzero element of the 1-dimensional space S(11) . Namely, consider the formal power series with integral coefficients ∑ a_{n}X^{n} which is defined by the identity

X ∏_{m=1}^{∞}(1 − X^{m})^{2}(1 − X^{11m})^{2} = ∑_{n=1}^{∞} a_{n}X^{n}.

It can be shown that the holomorphic function ∑ a_{n}q^{n} (with q = e^{2πiz} ) is a cusp form h of weight two on Γ_{0}(11) , i.e., a generator of the 1-dimensional space S(11) , cf. [79, Ex. 2.28]. One has h(z) = η(z)^{2}η(11z)^{2} , where η(z) , the Dedekind ηfunction, is the standard example of a modular form of weight 1/2 .

For each integer n ≥ 1 , the n th Hecke operator on S(N) is an endomorphism T_{n} of S(N) whose action is generally written on the right: f ↦ f|T_{n} . The various T_{n} commute with each other and are interrelated by identities which express a given T_{n} in terms of the Hecke operators indexed by the prime factors of n. If p is a prime, the operator T_{p} is defined as a composite β^{∘} α, where α: S(N) → S(Np) and β: S(Np) → S(N) are standard homomorphisms, known as “degeneracy maps”. In terms of Fourier coefficients, T_{p} is given by the following rules: If p is not a divisor of N and f = ∑ a_{n}q^{n} , then f|T_{p} has the Fourier expansion

∑_{n=1}^{∞} a_{np}q^{n} + p ∑_{n=1}^{∞} a_{n}q^{pn}.

If p divides N , then f|T_{p} is given by ∑_{n=1}^{∞} a_{np}q^{n} . The T_{n} have a strong tendency to be diagonalizable on S(N) : for n prime to N , T_{n} is self-adjoint with respect to the Petersson pairing on S(N) and is therefore semisimple [79, §3.5].

The elements of S(N) having special arithmetic interest are the normalized eigenforms in S(N) ; these are the non-zero cusp forms f = ∑ a_{n}q^{n} which are eigenvectors for all the T_{n} and which satisfy the normalizing condition a_{1} = 1 . (The latter condition is imposed mostly for convenience, since an arbitrary eigenform is a multiple of a normalized one.) If f is such an eigenform, its Fourier coefficients and its eigenvalues coincide: one has f|T_{n} = a_{n}f for all n ≥ 1 . The normalized eigenforms are “arithmetic” because the a_{n} belong to the realm of algebraic number theory. In fact, if f is a normalized eigenform, then the subfield of C generated by the a_{n} is a finite (algebraic) extension E of Q in C and the elements a_{n} of E are algebraic integers (see [79, Th. 3.52] or [16, Prop. 2.7]).

It is a mildly complicating fact that the normalized eigenforms in S(N) do not always form a basis of S(N) . In other words, the commuting operators T_{n} are not necessarily all diagonalizable. This problem, which arises from those n which have a common factor with N , can be repaired by the introduction of newforms [2]. Briefly, a newform is a normalized eigenform f = ∑ a_{n}q^{n} for which the space

{ g ∈ S(N) | g|T_{n} = a_{n}g for all n prime to N }

contains only f and its multiples. Atkin and Lehner showed in 1970 that S(N) has a basis built out of suitable transforms of the newforms in the spaces S(M) , where M runs over the positive divisors of N . Namely, for each M , there are natural embeddings of the space S(M) into S(N) which are indexed by the positive divisors of N/M . To each divisor d of N/M is associated the embedding δ_{d}: S(M) ↪ S(N) which sends ∑ a_{n}q^{n} to the series ∑ a_{n}q^{nd} . The embeddings δ_{d} are degeneracy mappings; one such mapping was introduced briefly above in connection with the definition of the p th Hecke operator T_{p} . With the help of these degeneracy mappings, the full space S(N) may be reconstructed purely in terms of the newforms in the spaces S(M) with M dividing N . Specifically, let S(M)^{new} be the subspace of S(M) spanned by its newforms. Then

S(N) = ⨁_{M|N} ⨁_{d| \frac{N}{M}} δ_{d}(S(M)^{new}).

We conclude this discussion with the remark that the work of Atkin and Lehner was generalized by T. Miyake [52] in the setting of automorphic forms on GL(2) and then by W. Li [42] in the setting of classical modular forms.

3. Elliptic curves

We next introduce some foundational concepts pertaining to elliptic curves. For an in-depth treatment of these concepts, the reader may consult the large number of textbooks and monographs which focus on elliptic curves. (As was indicated above, some of these books discuss modular forms as well.) Rather than list these references here, we refer the reader to the bibliography of a recent book review written by W. R. Hearst III and the author [31]. One book which has appeared since that review was written is the recent Advanced topics in the arithmetic of elliptic curves by J. H. Silverman [84].

Elliptic curves are distinguished by the fact that they are simultaneously curves, i.e., varieties of dimension 1, and abelian varieties, projective varieties which are endowed with a group law. The definition is very simple: an elliptic curve over a field k is a projective non-singular curve A of genus 1 over k which is furnished with a distinguished rational point O. An elliptic curve is thus a pair (A , O) ; an isomorphism between two pairs (A , O) and (A′ , O′) is an isomorphism A →^{∼} A′ which maps O to O′ . By convention, one usually omits explicit mention of the point O and refers simply to an elliptic curve A over k.

One obtains elliptic curves from 5-tuples of elements of k: to (α_{1} , α_{2} , α_{3} , α_{4} , α_{6}) one associates the projective curve whose affine equation is the generalized Weierstraß equation

y^{2} + α_{1}xy + α_{3}y = x^{3} + α_{2}x^{2} + α_{4}x + α_{6}.

This curve has genus 1 if it is non-singular; the non-singularity is detected by the non-vanishing of the discriminant of the equation. (The discriminant is a polynomial in the α_{i} which will not be reproduced here; it can be found, for instance, on page 36 of [85].) A short calculation shows that the plane curve with the indicated equation has exactly one point which does not lie on the affine plane; we take this point to be the distinguished point O. Using the Riemann-Roch theorem, one checks that every elliptic curve over k is isomorphic to one obtained by this process. (See, e.g., [75, Ch. II] for a discussion of the Riemann-Roch theorem and [83, Ch. III, §3] for a proof that elliptic curves are given by generalized Weierstraß equations.)

As we have suggested, an elliptic curve A over k may be viewed as a commutative algebraic group over k. Concretely, suppose that one has chosen a Weierstraß equation for A as above. For each field K containing k, let A(K) be the subset of the projective plane over K which is defined by the Weierstraß equation. This set is endowed with a classical group law, often known as the “chord and tangent operation”. In this law, O is the identity element of A(K) , and three distinct elements of A(K) sum to O if and only if they are collinear. The composition law on A(K) can be described explicitly in terms of coordinates by a family of polynomial equations with coefficients in k; this family depends on the chosen Weierstraß equation but is independent of K . (For a recent discussion concerning families of such equations, see [5].)

If A is an elliptic curve over Q , then one can choose the Weierstraß coefficients α_{i} to be rational integers. The coefficients are essentially unique if we demand that the discriminant of the equation have the smallest possible value; the discriminant is then said to be the (minimal) discriminant Δ of A.

Those prime numbers which divide Δ are the primes at which A has bad reduction; the others are the primes at which A has good reduction. The point is that, for good primes p, the minimal equation, when viewed mod p, yields an elliptic curve Ã_{p} over the finite field Z/pZ . The reduced curve Ã_{p} over Z/pZ is a projective plane curve over Z/pZ ; the group Ã_{p}(Z/pZ) consisting of the points of Ã_{p} with coordinates in Z/pZ is then a finite abelian group. A theorem of H. Hasse states that the order of this group is approximately p. More precisely, the integer

b_{p} := p+1 − #(Ã_{p}(Z/pZ))

is bounded in absolute value by 2\sqrt{p} [83, Ch. V, Th. 1.1].

To work with a concrete example, let us introduce the elliptic curve A over Q defined by the equation y^{2} + y = x^{3} − x^{2} . The group A(Q) has five evident points: the origin O and the four affine points gotten by taking y ∈ {0 , −1} and x ∈ {0 , 1}. One may verify that these points form a subgroup of A(Q) . Since they remain distinct under reduction mod p, Ã_{p}(Z/pZ) has a subgroup of order 5 whenever A has good reduction at p. Accordingly, the number #(Ã_{p}(Z/pZ)) is divisible by 5, so that b_{p} ≡ p+1 mod 5 for all primes p which do not divide the discriminant of A.

Now Table 1 of [4] informs us that the minimal discriminant of A is −11 . Therefore, integers b_{p} are defined for all p ≠ 11 . When p is small, it is not hard to compute b_{p} ; one finds b_{2} = −2 , b_{3} = −1 , b_{5} = 1 , and so on. (The mod 5 congruence on the b_{p} , plus Hasse’s bound |b_{p}| < 2\sqrt{p}, determines the first few of these integers.) Anticipating the conjecture of Shimura and Taniyama which will be discussed below, I would like to point out that the arithmetically defined b_{p} are known to agree with the prime-indexed coefficients a_{p} of the weight-two modular form h on Γ_{0}(11) which was introduced above. The astounding identity b_{p} = a_{p} is a special case of a relation which was discovered by Eicher and Shimura (and which is recalled in [78]). A priori, the Eichler-Shimura relation gives information only about those elliptic curves which are related geometrically to modular curves of the form X_{0}(N) . The Taniyama-Shimura conjecture states that all elliptic curves over Q can be so related.

Next, suppose that A and A′ are elliptic curves over k. An isogeny A → A′ over k is a non-constant map of curves over k which takes the distinguished point O of A to the corresponding point O′ of A. If such a map exists, then it is a map of algebraic groups, and A and A′ are said to be isogenous over k. (As the terminology suggests, the relation of being isogenous is symmetric; it is in fact an equivalence relation.) It is not difficult to show that two isogenous elliptic curves over Q have the same primes of bad reduction. (This follows from the criterion of Néron-Ogg-Shafarevich see [76] or [83, Ch. VII, §7].) Moreover, if A and A′ are isogenous over Q , then for each good reduction prime p, one has

#(Ã_{p}(Z/pZ)) = #(Ã′_{p}(Z/pZ)).

In other words, the numbers b_{p} are the same for A and A′ , so that one might make the informal statement that A and A′ are “equivalent arithmetically”. A striking theorem of G. Faltings [19, §5] states that, conversely, two elliptic curves A and A′ over Q are isogenous if #(Ã_{p}(Z/pZ)) = #(Ã′_{p}(Z/pZ)) for all primes p at which the two curves have good reduction. (In [19], Faltings proves a more general statement about homomorphisms between abelian varieties over number fields, thus confirming conjectures of J. Tate.)

Each elliptic curve A over Q has a conductor, which is a positive integer divisible precisely by the primes at which A has bad reduction. For example, the curve with equation y^{2} + y = x^{3} − x^{2} has conductor 11. Although the definition of the conductor is somewhat unenlightening (see for example [70, §2]), a well-known algorithm of J. Tate [85] makes it possible to compute conductors by hand in specific cases. Alternatively, the computer algebra program gp [3], the Mathematica package [44] and the Maple package apecs [11] all implement Tate’s algorithm and more generally make it easy to perform elliptic curve calculations on a workstation or personal computer. (The latter package is recommended by F. Gouvêa.)

Because the conductor is divisible exactly by the set of “bad primes”, the conductor and the minimal discriminant of A are divisible by the same set of prime numbers. Nevertheless, these integers have a completely different feel. For one thing, the conductor of an elliptic curve depends only on its Q-isogeny class, while the discriminant may change under isogeny. (The curve with equation y^{2} + y = x^{3} − x^{2} − 10x − 20 is isogenous to the conductor-11 curve that we have been discussing; its conductor is 11, but its minimal discriminant is −11^{5} .) For another, the conductor is always positive; the minimal discriminant may be positive or negative. Finally, the latter number may be divisible by a large power of a given prime, whereas the conductor of an elliptic curve tends to be divisible only by low powers of its prime divisors. (In particular, a prime p ≥ 5 can divide the conductor at most to the second power.) A beautiful formula of A. Ogg [57] expresses the conductor of a given curve A in terms of the minimal discriminant and the Néron model of A. (The latter is a curve over Z which may be regarded as the “best possible” model for A. See also [83, App. C, §16] for a statement of Ogg’s formula.)

Suppose that the conductor of A is N . Then A is said to be semistable at a prime number p if p^{2} does not divide N . This means either that p is prime to N , in which case A has good reduction at p, or else that p “exactly” divides N , in which case the reduction of A at p is bad but not too bad (it is said to be multiplicative). If A is semistable at all primes p, i.e., if N is square free, then the elliptic curve A is said to be semistable. Semistable elliptic curves occur in connection with Fermat’s Last Theorem and in other applications.

This is a good point to insert a short digression about abelian varieties, whose arithmetic theory is exposed in various chapters of [12] and the recent graduate-level text [54]; see also the books by such authors as A. Weil, S. Lang [37], D. Mumford, H. P. F. Swinnerton-Dyer, G. Kempf and H. Lange-Ch. Birkenhake. As was mentioned very briefly above, an abelian variety over a field k is a projective algebraic variety over k which is equipped with the structure of an algebraic group over k. The first examples of abelian varieties are obtained by taking Jacobians of curves; the Jacobian of a curve of genus g over k is an abelian variety over k of dimension g. (For the construction of Jacobians, the reader may consult, e.g., Chapter V of [75].) If A is an elliptic curve over k, then the chosen origin O of A enables one to identify A (endowed with its group structure) with the Jacobian of A. In fact, an elliptic curve over k is nothing other than an abelian variety over k of dimension one. Since an abelian variety of dimension one is an elliptic curve (and since an abelian variety of dimension zero is just a single point), one might describe abelian varieties as “higher-dimensional analogues” of elliptic curves.

Abelian varieties play an inevitable role in our story because the conjecture which we are about to discuss (a priori one involving elliptic curves over Q and weight-two newforms with integer coefficients) extends naturally to a conjectural dictionary between arbitrary newforms of weight two and a certain class of abelian varieties over Q (see Conjecture 2 below).

4. The Taniyama-Shimura conjecture

The conjecture of Shimura and Taniyama relates elliptic curves over Q and certain modular forms. Its history is the subject of an engrossing “file” compiled by S. Lang [38].

Before giving a formal description of the conjecture, we refer once again to the elliptic curve y^{2} + y = x^{3} − x^{2} , which has conductor 11. The integers b_{p} which control the numbers of mod p points on this curve might be viewed initially as mysterious quantities for which we seek a “formula”. The relation b_{p} = a_{p} , where a_{p} is the p th coefficient of the normalized eigenform in S(11) , may be regarded as such a formula.

The Taniyama-Shimura conjecture affirms that there is an analogous relation for all elliptic curves over Q . Namely, if A is an elliptic curve over Q of conductor N , then one conjectures that there is a newform f = ∑ a_{n}q^{n} in S(N) such that b_{p} = a_{p} for all primes p |̸ N . All coefficients a_{n} of f are then necessarily rational integers. (For another account of the conjecture, see the article by K. Rubin and A. Silverberg [68, §1].)

A geometric formulation of the Taniyama-Shimura conjecture may be given in terms of the construction presented in Chapter 7 of [79] and, from a different perspective, in [82]. Suppose that f ∈ S(N) is a normalized eigenform, and let E be the field generated by the coefficients of f . Shimura associates to f an abelian variety A_{f} over Q whose dimension is the degree [E: Q] and whose arithmetic incorporates the eigenvalues a_{p} for p prime to N . Although the construction of A_{f} is perfectly precise, it is fruitful in this context to regard the association f ⇝ A_{f} as a flabby one linking f to a “clump” of isogenous abelian varieties rather than a specific abelian variety which is singled out up to isomorphism. (The relevant notion of isogeny is an appropriate generalization of the notion of isogeny for elliptic curves.)

If the Fourier coefficients of f are rational integers, then E = Q and A_{f} has dimension 1. This means that A_{f} is an elliptic curve over Q , which we shall regard as being defined only up to isogeny. According to a theorem of Eichler and Shimura, the eigenvalues a_{p} are reflected in the arithmetic of the elliptic curve A = A_{f} in the following way. If p does not divide N , then A has good reduction at p. Further, for such p, the p th Fourier coefficient of f coincides with the quantity b_{p} := p+1 − #(Ã_{p}(Z/pZ)) . In other words, one has b_{p} = a_{p} for all p |̸ N . (If A_{f} has dimension greater than 1, the relation between f and A_{f} is a bit more complicated to formulate but involves no new ideas.)

In its geometric form, the Taniyama-Shimura conjecture states that every elliptic curve A over Q is modular in the sense that it is isogenous to some curve A_{f} . In other words, the conjecture asserts the surjectivity of the construction f ⇝ A_{f} , viewed as a map from eigenforms with integral coefficients (in some S(N) ) to isogeny classes of elliptic curves over Q . In analogy with the arithmetic formulation, when A has conductor N , A is conjectured to be isogenous to an A_{f} with f ∈ S(N) .

The connection between the two formulations is as follows. Suppose that A is an elliptic curve over Q of conductor N , and assume that A is related arithmetically to an eigenform in S(N) . Specifically, assume that f = ∑ a_{n}q^{n} is a normalized eigenform in S(N) with integral coefficients and that a_{p} = b_{p} for all p prime to N . (The b_{p} have their usual meaning.) Let A′ be the elliptic curve A_{f} , and write b′_{p} for the analogues of the b_{p} for A′ . Then one has b′_{p} = a_{p} for all p |̸ N by the formula of Eichler-Shimura, and hence b_{p} = b′_{p} for all p |̸ N . A theorem of Faltings which was quoted above then ensures that A and A′ = A_{f} are isogenous over Q , so that the geometric form of the Taniyama-Shimura conjecture is true for A. Conversely, if A is isogenous to an A_{f} , then numbers b_{p} computed for A coincide with their analogues for A_{f} . By the Eicher-Shimura formula, these latter numbers are the prime-indexed coefficients of f .

In connection with the construction f ⇝ A_{f} , let us consider the situation where f is a normalized eigenform in S(N) with integral Fourier coefficients but where f is not necessarily a newform. What is the conductor of the elliptic curve A = A_{f} ? The answer to this question begins with the fact that f is necessarily built from a newform in S(M) for some unique divisor M of N . It is then true that the conductor of A is precisely this divisor M . This theorem was proved by H. Carayol in [9], following work of Shimura, Igusa, Deligne and Langlands.

A formulation of the Taniyama-Shimura conjecture with a completely different flavor is provided by Mazur’s article [48]. In this article, Mazur rephrases the conjecture as a statement about the Riemann surface associated with an elliptic curve over Q . If A is such an elliptic curve, we write A(C) for this Riemann surface, which may be realized as the subset of the complex projective plane which is defined by a Weierstraß equation for A. This surface is holomorphically a complex torus. For each integer N , consider the subgroup Γ_{1}(N) of Γ_{0}(N) consisting of

matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) for which a ≡ d ≡ 1 mod N . By considering Γ_{1}(N)

in place of Γ_{0}(N) , one obtains an analogue of X_{0}(N) which is called X_{1}(N) . Mazur shows that an elliptic curve A over Q is modular if and only if there exists a non-constant holomorphic map from X_{1}(N) to A(C) for some positive integer N . As Mazur explains in his article, one can paraphrase this condition as the statement that the arithmetic object A is hyperbolically uniformized, since H is a model for the hyperbolic plane. This circumstance has led to the charge that Wiles’s proof of Fermat’s Last Theorem could not possibly be correct, since its logical structure involves a statement that may be interpreted in terms of hyperbolic geometry [89]. For a refutation, see [7].

Mazur’s observation led Serre to ask for a description of the set of elliptic curves over the complex field which satisfy Mazur’s condition: Which elliptic curves A over C are modular in the sense that one can find a non-constant holomorphic map X_{1}(N) → A(C) for some positive integer N ? (According to Mazur’s theorem, an elliptic curve over Q is modular in the usual sense if and only if it is modular in this new sense when viewed over C .) In [64], I provide a conjectural answer to Serre’s question. Namely, I show that the conjectures made by Serre in [74] imply the following statement, which recalls §10 of [81].

Conjecture 1. Let A be an elliptic curve over C . Then A is modular if and only if A is isogenous to all elliptic curves ^{σ}A obtained by conjugating A by algebraic automorphisms σ of the field C .

Conjecture 1 is related to Serre’s conjectures via a second conjecture, which we will state after some motivating remarks. Consider a normalized eigenform f = ∑ a_{n}q^{n} in the space of weight-two cusp forms on Γ_{1}(N) . Let E = Q(…, a_{n} , …) be the field generated by the coefficients of f , and let d = [E: Q] be the degree of E , i.e., the dimension of E as a Q-vector space. The abelian variety A_{f} is an abelian variety over Q of dimension d which comes equipped with an action of E . To give sense to the last statement, one introduces the ring End(A_{f}) whose elements are maps A_{f} → A_{f} in the category of algebraic varieties over Q which respect the group structure on A_{f} . (Such maps are the endomorphisms of A_{f} .) The ring End(A_{f}) turns out to be a free rank- d module over Z ; the associated Q-algebra End(A_{f}) ⊗ Q is isomorphic to E . Thus A_{f} has many endomorphisms; moreover, it is “modular” in the sense that there is a non-constant map X_{1}(N) → A which is defined over Q .

Conjecture 2. Let A be an abelian variety over Q for which End(A) ⊗ Q is a number field of degree equal to dim A. Then for some N ≥ 1 , there is a nonconstant map X_{1}(N) → A which is defined over Q .

In [64], I prove that Serre’s conjectures imply Conjecture 2 and that Conjecture 2 implies Conjecture 1.

It is natural to regard Conjecture 1 and Conjecture 2 as generalizations of the Taniyama-Shimura conjecture. The first conjecture pertains to elliptic curves which are not necessarily defined over Q , while the second pertains to abelian varieties over Q which are not necessarily elliptic curves. Neither of these conjectures is proved in [90].

The Taniyama-Shimura conjecture can be generalized still further. Indeed, a common generalization of Conjectures 1 and 2 was presented in the 1995 Berkeley Ph.D. thesis of E. Pyle. See also [22].

5. Galois representations attached to elliptic curves

Let A be an elliptic curve over Q . A model for A(K) when K = C is given by the Weierstraß theory of complex analysis: the group A(C) is the complex torus C/L, where L is the lattice of periods associated to the given cubic equation. (Explicitly, L is obtained by integrating the differential \frac{dx}{y} on A over the free abelian group H_{1}(A(C) , Z) of rank two.) Let n be a positive integer, and let A[n] be the group of elements of A(C) whose order divides n. This group of n-division points on A may be modeled as \frac{1}{n}L/L; it is therefore a free module of rank two over Z/nZ , since L is free of rank two over Z .

On reflection, one sees that A[n] in fact lies in A(\overline{Q}) , where \overline{Q} is the subfield of C consisting of all algebraic numbers. Indeed, the group A[n] is a finite subgroup of A(C) which consists of those points satisfying a certain set of polynomial equations with rational coefficients; it follows that the coordinates are algebraic numbers. Moreover, let Gal(\overline{Q}/Q) be the group of automorphisms of \overline{Q} . Then the same reasoning shows that A[n] is stable under the action of Gal(\overline{Q}/Q) on A(\overline{Q}) which results from the action of Gal(\overline{Q}/Q) on \overline{Q} . Thus A[n] comes equipped with a canonical action of the Galois group Gal(\overline{Q}/Q) .

It is important to observe, for each σ ∈ Gal(\overline{Q}/Q) , that the automorphism P ↦ ^{σ}P is a group automorphism of A[n] ; in symbols, ^{σ}(P + Q) = ^{σ}P +^{σ}Q for P , Q ∈ A[n] . This equation is a consequence of the fact that the composition law which expresses P + Q in terms of P and Q involves only polynomials with rational coefficients. Our observation means that the action of Gal(\overline{Q}/Q) on A[n] may be viewed as a (continuous) homomorphism

ρ_{A , n}: Gal(\overline{Q}/Q) → Aut(A[n])

in which Aut(A[n]) stands for the group of automorphisms of A[n] as an abelian group. Since A[n] is isomorphic to the group (Z/nZ)^{2} , one has

Aut(A[n]) ≈ GL(2 , Z/nZ) ,

where the group on the right consists of two-by-two invertible matrices with coefficients in Z/nZ . While there is no canonical isomorphism between these groups, each choice of basis A[n] ≈ (Z/nZ)^{2} determines such an isomorphism; moreover, the various isomorphisms obtained in this way differ by inner automorphisms of GL(2 , Z/nZ) . Therefore, each element of Aut(A[n]) has a well-defined trace and determinant in Z/nZ .

It is often fruitful to fix a choice of basis A[n] ≈ (Z/nZ)^{2} and to view ρ_{A , n} as taking values in the matrix group GL(2 , Z/nZ) . Once this choice is made, ρ_{A , n} becomes the matrix-valued representation of the Galois group Gal(\overline{Q}/Q) ; it is the representation of Gal(\overline{Q}/Q) defined by the group of n-division points of A. The kernel of this representation corresponds, via Galois theory, to a finite Galois extension K_{n} of Q in \overline{Q} . Concretely, this extension is obtained by adjoining to Q the coordinates of the various points in A[n] . The Galois group G_{n} := Gal(K_{n}/Q) may thus be identified with the image of ρ_{A , n} , which is a subgroup of the target group GL(2 , Z/nZ) . The elliptic curve A and the positive integer n have given rise to a finite Galois extension K_{n}/Q whose Galois group is a subgroup of the group of two-by-two invertible matrices with coefficients in Z/nZ .

It is natural to ask for a description of G_{n} as a subgroup of GL(2 , Z/nZ) , cf. [78]. There is a (relatively rare) special case to consider: that where A has complex multiplication (over C ). When A is viewed as C/L, the complex multiplication case is that for which there is a complex number α ∉ Z such that αL ⊆ L. The group G_{n} then has an abelian subgroup of index ≤ 2 , so it is much smaller than the ambient group GL(2 , Z/nZ) . In the more common case where A has no complex multiplication, Serre showed in [71] that the index of G_{n} in GL(2 , Z/nZ) is bounded as a function of n. In particular, G_{p} = GL(2 , Z/pZ) for all but finitely many primes p.

As background, we point out that the Taniyama-Shimura conjecture was proved for complex multiplication elliptic curves over Q by Shimura in 1971 [80]. The result of [80] is suggestive and may be regarded as evidence for the general case of the Taniyama-Shimura conjecture. As we recall below, however, the elliptic curves which appear in connection with Fermat’s Last Theorem are semistable. And it is a fact that semistable elliptic curves over Q never have complex multiplication. (One possible proof can be summarized as follows: An elliptic curve with complex multiplication has an integral j-invariant, i.e., potentially good reduction. Hence if it is semistable, it has good reduction everywhere. However, a theorem of Tate states that there is no elliptic curve over Q with everywhere good reduction, cf. [21].) Accordingly, the theorem of [80] cannot be used to prove Fermat’s Last Theorem.

A key piece of information about the extension K_{n}/Q (which depends on A as well as on n) is that its discriminant is divisible only by those prime numbers which divide either n or the conductor of A. In other words, if p |̸ n is a prime number at which A has good reduction, then K_{n}/Q is unramified at p; one says frequently that the representation ρ_{A , n} is unramified at p. Whenever this occurs, a familiar construction in algebraic number theory produces a Frobenius element σ_{p} in G_{n} which is well defined up to conjugation.

We shall now summarize this construction with K_{n} replaced by an arbitrary finite Galois extension K of Q . Let O_{K} be the ring of algebraic integers in K . The Galois group Gal(K/Q) leaves O_{K} invariant, so that one obtains an induced action of Gal(K/Q) on the set of ideals of O_{K} . The set of prime ideals 𝔭 of O_{K} which contain the prime number p is permuted under this action. For each 𝔭, the subgroup D_{𝔭} of Gal(K/Q) consisting of those elements in Gal(K/Q) which fix 𝔭 is called the decomposition group of 𝔭 in Gal(K/Q) . Meanwhile, the finite field F_{𝔭} := O_{K}/𝔭 is a finite extension of the prime field F_{p} . The extension F_{𝔭}/F_{p} is necessarily Galois; its Galois group is the cyclic group generated by the Frobenius automorphism

φ_{p}: x ↦ x^{p}

of F_{𝔭} . There is a natural map D_{𝔭} → Gal(F_{𝔭}/F_{p}) , gotten by associating a given δ ∈ D_{𝔭} to the automorphism of O_{K}/𝔭 induced by δ. This map is surjective; its injectivity is equivalent to the assertion that p is unramified in the extension K/Q . Therefore, whenever this assertion is true, there is a unique σ_{𝔭} ∈ D_{𝔭} whose image in Gal(F_{𝔭}/F_{p}) is φ_{p} . The automorphism σ_{𝔭} is then a well-defined element of Gal(K/Q) known as the Frobenius automorphism for 𝔭. It is easy to show that the various 𝔭 are all conjugate under Gal(K/Q) and that the Frobenius automorphism for the conjugate of 𝔭 by g is gσ_{𝔭}g^{−1} . In particular, the various σ_{𝔭} are all conjugate; this justifies the practice of writing σ_{p} for any one of them and stating that σ_{p} is well defined up to conjugation.

For later use, we prolong this digression and introduce the concept of Frobenius elements in Gal(\overline{Q}/Q) . Let p again be a prime number, and let 𝔭 now be a prime of \overline{Q} lying over p. (One can think of 𝔭 as a coherent set of choices of primes lying over p in the rings of integers of all finite extensions of Q in \overline{Q} .) To 𝔭 we associate: (1) its residue field F_{𝔭} , which is an algebraic closure of the finite field F_{p} , and (2) a decomposition subgroup D_{𝔭} of Gal(\overline{Q}/Q) . There is again a surjective map D_{𝔭} → Gal(F_{𝔭}/F_{p}) . The Frobenius automorphism φ_{p}: x ↦ x^{p} generates the target group in the topological sense: the subgroup of Gal(F_{𝔭}/F_{p}) consisting of powers of φ_{p} is dense in Gal(F_{𝔭}/F_{p}) . We shall use the symbol Frob_{p} to denote any preimage of φ_{p} in D_{𝔭} and refer to Frob_{p} as a Frobenius element for p in Gal(\overline{Q}/Q) .

One thinks of Frob_{p} as a specific element of Gal(\overline{Q}/Q) , albeit one which is doubly ill defined. The ambiguities in Frob_{p} result from the circumstance that 𝔭 needs to be chosen and from the fact that D_{𝔭} → Gal(F_{𝔭}/F_{p}) has a large kernel, the inertia subgroup I_{𝔭} of D_{𝔭} . The usefulness of Frob_{p} stems from the fact that the various 𝔭 are conjugate by elements of Gal(\overline{Q}/Q) , so that all the subgroups D_{𝔭} of Gal(\overline{Q}/Q) are conjugate and similarly all I_{𝔭} are conjugate. Thus if ρ is a homomorphism mapping Gal(\overline{Q}/Q) to some other group, the kernel of ρ contains one I_{𝔭} if and only if it contains all I_{𝔭} . In this case, one says that ρ is unramified at p; the image of Frob_{p} is then an element of the target group of ρ which is well defined up to conjugation. Note that one may write ρ_{A , n}(Frob_{p}) = σ_{p} for all primes p at which ρ_{A , n} is unramified. As has been stated, these include all primes which divide neither n nor the conductor of A.

Let us return now to the matrix group G_{n} . We pointed out above that each element of G_{n} has a trace and determinant in Z/nZ which are independent of any choice of basis. The Frobenius element σ_{p} is an element of G_{n} which is well defined only up to conjugation. Nevertheless, the trace and determinant of σ_{p} are well defined, since conjugate matrices have the same traces and determinant. The number det σ_{p} is the residue class of p mod n. On the other hand, one has the striking congruence

tr(σ_{p}) ≡ b_{p} mod n ,

where b_{p} is the number p+1−#(A(Z/pZ)) introduced above. This means that the representation ρ_{A , n} encapsulates information about the numbers b_{p} (for prime numbers p which are primes of good reduction and which are prime to n); more precisely, it determines the numbers b_{p} mod n.

6. Galois representations attached to modular forms

Suppose that f ∈ S(N) is a normalized eigenform. The coefficients a_{n} of f are always algebraic integers but not necessarily ordinary integers. If it happens that the a_{n} all lie in Z , then the abelian variety A = A_{f} is an elliptic curve. By considering the family ρ_{A , n} , one obtains a series of representations of the Galois group Gal(\overline{Q}/Q) . These representations are related to f by the congruence tr(σ_{p}) ≡ a_{p} mod n, valid for the n th representation and all primes p |̸ nN . We are especially interested in the case where n is a prime number ℓ; the ring Z/nZ is then the finite field F_{ℓ} .

The representations ρ are associated to A, which in turn arises from f . Hence it is tempting to write ρ_{f , ℓ} for the representations ρ_{A , ℓ} . The obstacle to doing this arises from the circumstance that A is determined up to isogeny but not always up to isomorphism. If one replaces A by an isogenous elliptic curve, the representations ρ_{A , ℓ} may change! To circumvent this difficulty, we introduce the “semisimplifications” of the ρ_{A , ℓ} .

These representations are defined as follows. If ρ is a two-dimensional representation of a group over a field, ρ is either irreducible, or else “upper-triangular”, i.e., an extension of a one-dimensional representation α by another, β . In the case where ρ is irreducible, we declare its semisimplification to be ρ itself. In the reducible case, the semisimplification of ρ is the direct sum of the two onedimensional representations α and β . Clearly, the trace and determinant are the same for ρ and for its semisimplification.

When A is fixed, the results of [71] show that ρ_{A , ℓ} can be reducible only for a finite number of ℓ. In fact, a theorem of Mazur [46] shows that ρ_{A , ℓ} is irreducible for all ℓ not in the set { 2 , 3 , 5 , 7 , 13 , 11 , 17 , 19 , 37 , 43 , 67 , 163 }. Hence the replacement of ρ_{A , ℓ} by its semisimplification can be thought of as “fine tuning” which affects only a small number of the representations. One shows easily for all ℓ that the semisimplification of ρ_{A , ℓ} depends only on f and on ℓ (but not on the choice of A). Introducing

ρ_{f , ℓ} := semisimplification of ρ_{A , ℓ} ,

one obtains a sequence of semisimple representations of Gal(\overline{Q}/Q) which are well defined up to isomorphism. The characteristic property of ρ_{f , ℓ} may be summarized in terms of Frobenius elements Frob_{p} in the Galois group Gal(\overline{Q}/Q) , elements which were introduced above. Namely, if p is a prime number not dividing ℓN , then ρ_{f , ℓ}(Frob_{p}) has trace a_{p} mod ℓ and determinant p mod ℓ.

It is natural to generalize this process by considering the situation where f = ∑ a_{n}q^{n} is a normalized eigenform whose coefficients a_{n} are algebraic integers but not necessarily rational integers. As we indicated above, the field E = E_{f} generated by the a_{n} is a number field, i.e., a finite extension of Q . Moreover, the coefficients a_{n} of f lie in the integer ring O_{f} of E . It is perhaps worth noting that the ring generated by the a_{n} inside E , while a subring of finite index in O_{f} , is not necessarily equal to O_{f} .

Using the abelian variety A_{f} , one constructs representations indexed not by the prime numbers but rather by the non-zero prime ideals of O_{f} . (For details, see [79, Ch. 7].) If λ is such a prime, its residue field F_{λ} is a finite field, say, of characteristic ℓ. The prime field F_{ℓ} = Z/ℓZ is then canonically embedded in F_{λ} . For each λ, one finds a semisimple representation ρ_{f , λ}: Gal(\overline{Q}/Q) → GL(2 , F_{λ}) which is characterized up to isomorphism by the following property: if p is a prime number not dividing ℓN , then ρ_{f , λ}(Frob_{p}) has trace a_{p} mod λ and determinant p mod λ.

The assertion concerning the determinant of the matrices ρ_{f , λ}(Frob_{p}) may be rephrased as the statement that the determinant of the representation ρ_{f , λ} is the mod ℓ cyclotomic character χ_{ℓ} . This character is defined by considering the group μ_{ℓ} of ℓ th roots of unity in \overline{Q} ; the action of the Galois group Gal(\overline{Q}/Q) on the cyclic group μ_{ℓ} gives rise to a continuous homomorphism

χ_{ℓ}: Gal(\overline{Q}/Q) → Aut(μ_{ℓ}).

Since μ_{ℓ} is a cyclic group of order ℓ, its group of automorphisms is canonically the group (Z/ℓZ)^{∗} = F_{ℓ}^{∗} . We emerge with a map Gal(\overline{Q}/Q) → F_{ℓ}^{∗} , which is the character in question. The equality

det ρ_{f , λ} = χ_{ℓ}

is to be interpreted by viewing both homomorphisms as taking values in F_{λ}^{∗} .

Suppose now that c ∈ Gal(\overline{Q}/Q) is the automorphism “complex conjugation”. Then the determinant of ρ_{f , λ}(c) is χ_{ℓ}(c) . Now c operates on roots of unity by the map ζ ↦ ζ^{−1} , since roots of unity have absolute value 1. Accordingly,

det(ρ_{f , λ}(c)) = −1 ;

one says that ρ_{f , λ} is odd.

This parity statement generalizes to modular forms “with Nebentypus” whose weights are not necessarily two. Here is a quick synopsis of the situation; some relevant references are provided in [60]. For integers k ≥ 1 , N ≥ 1 and characters ϵ: (Z/NZ)^{∗} → C^{∗} , one considers the space S_{k}(N , ϵ) of weight- k cusp forms with character ϵ on Γ_{0}(N) ; we have S(N) = S_{2}(N , 1) . This space is automatically zero unless ϵ(−1) = (−1)^{k} , so we will assume that this parity condition is satisfied. The space S_{k}(N , ϵ) admits an operation of Hecke operators T_{n} , and we again have the concept of a normalized eigenform in S_{k}(N , ϵ) . If f = ∑ a_{n}q^{n} is such a form, the numbers a_{n} ( n ≥ 1 ) and the values of ϵ all lie in a single integer ring O_{f} . For each non-zero prime ideal λ of O_{f} , one constructs a semisimple representation

ρ_{f , λ}: Gal(\overline{Q}/Q) → GL(2 , F_{λ}).

Let ℓ again denote the characteristic of F_{λ} . Then for all p |̸ ℓN , the trace of ρ_{f , λ}(Frob_{p}) is again a_{p} mod λ. The determinant of this matrix is p^{k−1}ϵ(p) mod λ.

Once the proper definition is made, the determinant of the map ρ_{f , λ} becomes the product χ_{ℓ}^{k−1}ϵ. In writing det ρ_{f , λ} = χ_{ℓ}^{k−1}ϵ, we use χ_{ℓ} to denote the mod ℓ cyclotomic character and employ a standard construction to regard ϵ as a map Gal(\overline{Q}/Q) → F_{λ}^{∗} . The construction in question begins with the map Gal(\overline{Q}/Q) → (Z/NZ)^{∗} giving the action of Gal(\overline{Q}/Q) on the N th roots of unity. Composing this map with the character ϵ, we obtain a homomorphism Gal(\overline{Q}/Q) → O_{f}^{∗} . On reducing this homomorphism mod λ, we obtain the desired variant of ϵ.

Evaluating the formula det ρ_{f , λ} = χ_{ℓ}^{k−1}ϵ on c = “complex conjugation” , one finds

det(ρ_{f , λ}(c)) = (−1)^{k−1}ϵ(c) = −1.

In these equalities, we exploit the fact that ϵ(c) is another name for ϵ(−1) and remember the parity condition ϵ(−1) = (−1)^{k} . The upshot of this is that the representations ρ_{f , λ} are always odd, even in the generalized setup.

In fact, it is possible in this situation to find a normalized eigenform f′ of weight two with some character ϵ′ , along with a maximal ideal λ′ of the integer ring for f′ , so that the representations ρ_{f , λ} and ρ_{f′ , λ′} are isomorphic, cf. [67, Th. 2.2 and Cor. 3.2]. (To compare these representations, it is necessary to embed the residue fields of λ and of λ′ in a suitably chosen common finite field of characteristic ℓ.) Hence the “generalized setup” can be reduced to the case of weight two, provided that one considers eigenforms for which the associated characters may be non-trivial. The process of reduction to weight two is a very powerful one in the theory, since the representations arising from forms of this weight are constructed directly from points of finite order on abelian varieties. To the best of my knowledge, the idea of reducing systematically to weight two originated with an unpublished 1968 manuscript of Shimura [77].

Before leaving this topic, we should mention that the cases k = 1 and k > 1 are quite distinct in flavor. In the former case, the representations ρ_{f , λ} may be viewed as the set of reductions of a single continuous representation ρ_{f} with finite image. For details, see [16].

7. Serre’s conjectures

We shall give a brief summary of conjectures made by J.-P. Serre in [74]. A recent article by H. Darmon [15] discusses the conjectures more extensively and emphasizes applications and numerical examples.

Let ℓ be a prime number, and let F be an algebraic closure of the prime field F_{ℓ} . Suppose that ρ: Gal(\overline{Q}/Q) → GL(2 , F) is an odd continuous representation. It seems natural to ask whether or not ρ is the mod λ representation attached to a suitable normalized eigenform. Since the representations ρ_{f , λ} are semisimple by definition, it is necessary to limit our discussion to the case where ρ is semisimple. In fact, the case where ρ is semisimple and reducible is sometimes awkward, so we will assume from now on that ρ is irreducible. However, the excluded case where ρ is reducible is quite interesting; see [8] for some observations in this case.

Let us say then that ρ is modular if one can find: (i) a normalized eigenform f in some space S_{k}(N , ϵ) , (ii) a prime λ dividing ℓ in the ring of integers O_{f} associated to f , and (iii) an embedding F_{λ} ↪ F such that ρ is isomorphic to the representation obtained by composing ρ_{f , λ} with the inclusion

GL(2 , F_{λ}) ↪ GL(2 , F)

associated with (iii). A weak form of the conjectures made by Serre in [74]is the following statement: Every irreducible continuous odd representation ρ: Gal(\overline{Q}/Q) → GL(2 , F) is modular.

This statement was first formulated by Serre in the 1970s for modular forms of level 1 (i.e., on SL(2 , Z) ). For such forms, the representations ρ_{f , λ} are ramified only at ℓ; Serre asked whether an odd irreducible ρ which is unramified outside ℓ is necessarily associated to a normalized eigenform on SL(2 , Z) . Tate confirmed this for ℓ = 2 [86] by showing that there are no such representations. (As Tate remarks at the end of his article, Serre treated the case ℓ = 3 in a similar manner by exploiting the discriminant bounds of Odlyzko and Poitou.) The case where ρ may be ramified at primes other than ℓ was taken up by Serre in the mid-1980s, when computer calculations by J.-F. Mestre convinced Serre that the conjectured statement was plausible.

The qualitative conjecture to this effect is supplemented in [74] by an intricate recipe which pinpoints the space S_{k}(N , ϵ) where one should find an eigenform f giving rise to a specific representation ρ. (As Serre later observed, the recipe for ϵ needs to be modified in certain cases when ℓ = 2 or 3.) The conjunction of the qualitative statement that ρ is modular and the precise recipe fingering the space S_{k}(N , ϵ) is sometimes called the Strong Serre Conjecture. One possible justification for this name is the fact that the “strong” conjecture immediately implies the Taniyama-Shimura conjecture, Fermat’s Last Theorem, and a host of other assertions! It has gradually emerged that the qualitative statement and its strong cousin are in fact equivalent, at least when ℓ > 2 ; see [67] and [17] for a proof of the equivalence. Hence it is now possible to use the singular term “Serre’s conjecture” to refer to what was initially a package of interrelated conjectures.

Perhaps I should close this section by expressing the sentiment that the conjecture of Serre, while visibly important, currently seems intractable. Given an irreducible representation ρ with odd determinant, one is at a loss for a strategy which will lead to a proof that ρ is modular. In my Progress in Mathematics lecture, I engaged in a certain amount of philosophical speculation, stressing the parallel between the Taniyama-Shimura conjecture and Serre’s conjecture. Each conjecture states that all objects of a certain type are modular; in both cases one has a unidirectional “arrow” a means of constructing objects from modular forms. Thus one’s impulse is to try to attack these conjectures by an appropriate form of “counting”. While Serre’s conjecture is broader than the Taniyama-Shimura conjecture (the former implies the latter), one might suspect that Galois representations might be easier to count than elliptic curves. To the extent that this is so, one could imagine attacking the geometric conjecture about elliptic curves via the Galois-theoretic conjecture of Serre.

This philosophy is perhaps not far removed in spirit from the strategy used by Andrew Wiles in approaching the Taniyama-Shimura conjecture. Certainly, however, the analogy should not be taken too seriously; in fact, Wiles introduced his approach in 1993 with the statement that it was “orthogonal” to Serre’s conjecture.

8. Frey’s construction

Like most recent work on Fermat’s Last Theorem, the connection between Serre’s conjecture and FLT begins with constructions linking solutions to Fermat’s equation with elliptic curves. Although Y. Hellegouarch and others had noted such constructions, a decisive step was taken by G. Frey in an unpublished 1985 manuscript entitled “Modular elliptic curves and Fermat’s conjecture”.

Frey’s idea goes as follows. Suppose that there is a non-trivial solution to Fermat’s equation X^{ℓ}+Y^{ℓ} = Z^{ℓ} . We can assume that the exponent is a prime number different from 2 and 3 and that the solution is given by a triple of relatively prime integers a, b, and c. The equation y^{2} = x(x − a^{ℓ})(x + b^{ℓ}) then defines an elliptic curve A with unexpected properties.

These properties are catalogued in §4.1 of Serre’s article [74]: After performing some elementary manipulations, we arrive at a triple (a , b , c) in which b is even and c is congruent to 1 mod 4. Frey’s construction yields for this triple an elliptic curve A whose conductor is the product of the prime numbers which divide abc (each occurring to the first power). In particular, A is semistable. On the other hand, the minimal discriminant Δ of A is the quotient of (abc)^{2ℓ} by the factor 2^{8} . From Frey’s point of view, the main “unexpected” property of A is that Δ is the product of a power of 2 and a perfect ℓ th power, where ℓ is a prime ≥ 5 . Frey translated this property into a statement about the Néron model for A: if p is an odd prime at which A has bad reduction, the number of components in the mod p reduction of the Néron model is divisible by ℓ. Frey’s idea was to compare this number to the corresponding number for the Jacobian of the modular curve X_{0}(N) , where N is the conductor of A. Frey predicted that a discrepancy between the two numbers would preclude A from being modular. In other words, Frey concluded heuristically that the existence of A was incompatible with the Taniyama-Shimura conjecture, which asserts that all elliptic curves over Q are modular.

Frey’s construction spawned several lines of inquiry, in which mathematicians sought either to prove Fermat’s Last Theorem outright or else to link it to established or emerging conjectures such as the abc conjecture and Szpiro’s conjecture. These latter conjectures are treated by such articles as [23, 24, 32, 39, 50] and [56]. From the point of view of Szpiro’s conjecture and the abc conjecture, the surprising feature of Frey’s curve is the size of its discriminant rather than any special properties of the discriminant’s factorization. For Frey’s curve A, the absolute value of the discriminant can be bounded from below by a high power of the conductor of A.

To illustrate the force of Frey’s construction, we now sketch the deduction of Fermat’s Last Theorem from Serre’s conjectures [74]. (A word of caution: these conjectures are still conjectures!) Suppose that a, b and c are relatively prime integers which satisfy Fermat’s equation with exponent ℓ. After performing the manipulations mentioned above, we may use the equation y^{2} = x(x − a^{l})(x + b^{l}) to define a Frey curve A whose discriminant is the product of a power of 2 and a perfect lth power. If ℓ is greater than 3, a theorem of Mazur [46] asserts that the representation ρ = ρ_{A , ℓ} defined by A[ℓ] is an irreducible representation of Gal(\overline{Q}/Q) . Because of the hypothesis on the discriminant of A, ρ is unramified outside 2 and ℓ; moreover it is “finite” at ℓ in a sense which is explained in [74]. The recipes of Serre’s article require ρ to arise from a normalized eigenform in the space S(2) . However, as was mentioned above, this space has dimension 0.

9. Conjecture “epsilon”

A first step toward justifying Frey’s heuristic conclusion was taken in August 1985 by Serre in a letter to J.-F. Mestre [73]. In this letter, the writer formulated two related conjectures about modular forms, which he called C_{1} and C_{2} . (These conjectures predate the conjectures of [74]; they are now special cases of the latter conjectures.) Serre pointed out that Fermat’s Last Theorem is a consequence of the Taniyama-Shimura conjecture together with the two new conjectures. As it was thought initially that C_{1} and C_{2} would be easy to establish, the two statements immediately acquired the collective nickname “Conjecture ϵ ”. One thus had

Taniyama-Shimura + ϵ ⟹ Fermat’s Last Theorem.

Serre’s “ ϵ ” conjecture is in fact a subtle statement about the mod ℓ Galois representations arising from eigenforms in the various spaces S(N) . Specifically, suppose that f ∈ S(N) is a normalized eigenform, and let λ be a prime ideal in the ring of integers of the field generated by the coefficients of f . Then ρ_{f , λ} is a semisimple representation of Gal(\overline{Q}/Q) with values in a finite field, whose characteristic we will call ℓ. This representation is unramified at all primes p ≠ ℓ which do not divide N ; in other words, ρ_{f , λ} has the right to be ramified at ℓ and at those primes which divide N . Serre’s conjecture concerns the case where ρ_{f , λ} is unramified at such a prime: it predicts that this behavior can be attributed to the existence of a normalized eigenform f′ of level lower than N which gives rise to ρ_{f , λ} .

Specifically, suppose that ρ_{f , λ} is irreducible, that ℓ is different from 2, and that ρ_{f , λ} is unramified at a prime number p ≠ ℓ which divides N but whose square does not divide N . Serre’s conjecture predicts that there is an eigenform f′ ∈ S(N/p) , together with a prime ideal λ′ in the integer ring of the field of coefficients of f , such that ρ_{f , λ} and ρ_{f , λ′} are isomorphic. A variant of this conjecture concerns the case p = ℓ. (Note that, as in a situation discussed earlier, one must embed the residue fields of λ and of λ′ in a suitably chosen common finite field of characteristic ℓ before comparing ρ_{f , λ} and ρ_{f , λ′} .)

I proved Serre’s level-lowering conjecture in a 1990 article [62], thereby establishing the implication

Conjecture of Taniyama-Shimura ⟹ Fermat’s Last Theorem

which was the goal of Frey’s construction. (See also [58] and [63] for expository accounts of this work.) Since the Frey curves associated with Fermat solutions are semistable elliptic curves, I proved that the semistable case of the Taniyama-Shimura conjecture implies Fermat’s Last Theorem.

To illustrate the logic used in establishing this implication, we consider a semistable elliptic curve A over Q , together with a prime ℓ ≥ 3 for which the representation ρ_{A , ℓ} is irreducible. Suppose that ρ_{A , ℓ} is unramified at all primes p ≠ 2 , ℓ and moreover that ρ_{A , ℓ} is “finite” at ℓ. Then the results of [62] assert that A cannot be modular. To apply these results to Fermat’s Last Theorem, we suppose that A is the Frey curve associated to a hypothetical solution to the degree- ℓ Fermat equation with ℓ > 3 . Then, as was noted above, A is semistable, and ρ_{A , ℓ} is an irreducible representation with the indicated ramification and “finiteness” properties. Accordingly, A cannot be modular. Consequently, if all elliptic curves over Q are modular, then there can be no solution to Fermat’s equation.

Further light can be shed on [62] if we focus on the simplest situation in which its results apply. Suppose that A is an elliptic curve over Q whose conductor is a prime number p. Let ℓ be a prime number different from 2 and p for which ρ_{A , ℓ} is irreducible. The representation ρ_{A , ℓ} is unramified at all primes other than p and ℓ, and the contribution of [62] is to show that ρ_{A , ℓ} is indeed ramified at p if A is modular. To prove this, one supposes that A is modular, so that ρ_{A , ℓ} is connected up with the space of weight-two cusp forms on Γ_{0}(p) . It is possible to show that ρ_{A , ℓ} is similarly connected with other discrete subgroups of SL(2 , R) , coming from indefinite quaternion division algebras over Q . More precisely, there are prime numbers q ≠ p such that ρ_{A , ℓ} arises from the quaternion algebra over Q of discriminant pq. The desired result about ρ_{A , ℓ} follows from a detailed comparison of the mod p and mod q reductions of the three modular curves X_{0}(p) , X_{0}(q) and X_{0}(pq) with the mod p and mod q reductions of the Shimura curve associated with the quaternion algebra of discriminant pq. The latter curve is an analogue of X_{0}(pq) in which the group Γ_{0}(pq) is replaced by the group of norm-1 elements in a maximal order of the quaternion algebra of discriminant pq.

10. Wiles’s strategy

Suppose that A is an elliptic curve over Q . To verify the Taniyama-Shimura conjecture for A is to link A to modular forms. In an approach inspired by Serre’s conjecture, one might begin by considering the representations ρ_{A , ℓ} obtained from the action of Gal(\overline{Q}/Q) on A[ℓ] , when ℓ is a prime number. If one could show that an infinite number of these representations are modular (in the broadest possible sense), one would go on to prove that A is modular. Alas, as was indicated above, it is not clear how to translate this approach into a proof.

We mentioned previously that Wiles’s approach to the Taniyama-Shimura conjecture is “orthogonal” to one based on consideration of the varying ρ_{A , ℓ} . Here is the nub of the idea: One first fixes a prime ℓ and considers the family of groups A[ℓ^{ν}] for ν = 1 , 2 , …. The resulting sequence of representations

ρ_{A , ℓ^{ν}}: Gal(\overline{Q}/Q) → GL(2 , Z/ℓ^{ν}Z)

may be packaged as a single representation

ρ_{A , ℓ^{∞}}: Gal(\overline{Q}/Q) → GL(2 , Z_{ℓ}) ,

where Z_{ℓ} is the ring of ℓ-adic integers, i.e., the projective limit of the rings Z/ℓ^{ν}Z . To prove that A is a modular elliptic curve, it suffices to show that ρ_{A , ℓ^{∞}} is modular in an appropriate sense. Indeed, the trace of ρ_{A , ℓ^{∞}}(Frob_{p}) coincides with the rational integer b_{p} for all p |̸ ℓN , where N is the conductor of A. As soon as one finds an eigenform f in S(N) whose eigenvalues are related to the traces of ρ_{A , ℓ^{∞}}(Frob_{p}) , one has essentially proved that A is modular.

Needless to say, if ρ_{A , ℓ^{∞}} is modular, then so, in particular, is ρ_{A , ℓ} . Relating ρ_{A , ℓ^{∞}} to modular forms is thus no easier than the formidable task of proving that ρ_{A , ℓ} is modular! On the other hand, to prove that A is modular by the ℓ-adic method, we need only work with a single prime ℓ. The approach of [90] capitalizes on the fact that the finite groups GL(2 , F_{2}) and GL(2 , F_{3}) are solvable. This circumstance enables one to apply deep results of Langlands [40] and Tunnell [88] to prove that ρ_{A , ℓ} is modular for ℓ ≤ 3 , cf. [68, §2.3]. (The relevant results of Langlands are those concerning the theory of base change à la Saito-Shintani. For expositions of these results, see [25] and [26].)

Wiles’s basic idea is to prove that if ℓ is a prime for which ρ_{A , ℓ} is modular, then ρ_{A , ℓ^{∞}} is automatically modular (and hence A is a modular elliptic curve). In thinking about the jump from ρ_{A , ℓ} to ρ_{A , ℓ^{∞}} , one ignores A as much as possible the aim is to prove results about ℓ-adic representations which can be applied to ρ_{A , ℓ^{∞}} .

11. The language of deformations

We now introduce the machinery which underlies the jump from the modularity of ρ_{A , ℓ} to the modularity of ρ_{A , ℓ^{∞}} . We suppose for simplicity that A is a semistable elliptic curve over Q , and we let ℓ ≥ 3 be a prime number for which ρ_{A , ℓ} is both modular and irreducible. Choosing a basis of A[ℓ] , we regard ρ_{A , ℓ} as taking values in the matrix group GL(2 , F_{ℓ}) . As was suggested above, we seek to establish the modularity of ρ_{A , ℓ^{∞}} by a method which treats simultaneously all lifts of ρ_{A , ℓ} which are plausibly modular.

In this context, lifts are continuous homomorphisms

ρ̃: Gal(\overline{Q}/Q) → GL(2 , B) ,

where B is a complete local Noetherian Z_{ℓ}-algebra with residue field F_{ℓ} . They are constrained to lift ρ_{A , ℓ} in the obvious sense. Namely, we require that ρ_{A , ℓ} coincide with the composite of ρ̃ and the homomorphism GL(2 , B) → GL(2 , F_{ℓ}) induced by the residue map B → F_{ℓ} . (This depiction ignores a technical wrinkle: it might to necessary later on to replace Z_{ℓ} by the integer ring of a finite extension of the ℓ-adic field Q_{ℓ} .) The lifts which are “plausibly modular” are those which obey a set of local properties. The word “local” is meant to allude to the restrictions of ρ̃ to the subgroups Gal(\overline{Q}_{p}/Q_{p}) of Gal(\overline{Q}/Q) obtained as decomposition groups for prime numbers p (or, more precisely, for primes of \overline{Q} ). Wiles imposes conditions on these restrictions which lift conditions already satisfied by ρ_{A , ℓ} . These conditions are summarized in §3.4 of [68]; we shall evoke them below as well. There is flexibility and tension implicit in the choice of these conditions. They should be broad enough to be satisfied by ρ_{A , ℓ^{∞}} and tight enough to be satisfied only by lifts that can be related to modular forms. Roughly speaking, in order to prove the modularity of all lifts satisfying a fixed set of conditions, one needs to specify in advance a space of modular forms S so that the normalized eigenforms in S satisfy the conditions and such that, conversely, all lifts satisfying the conditions are plausibly related to forms in S . It is intuitively clear that this program will be simplest to carry out when the conditions are the most stringent and progressively harder to carry out as the conditions are relaxed.

A theme which emerges rapidly is that there are at least two sets of conditions of special interest. Firstly, one is especially at ease when dealing with the most stringent possible set of conditions which are satisfied by ρ_{A , ℓ} ; this leads to what Wiles calls the “minimal” problem. Secondly, one needs at some point to consider some set of conditions which allows treatment of the lift ρ_{A , ℓ^{∞}} this lift is, after all, our main target. It would be natural to consider the most stringent such set. The two sets of conditions may coincide, but there is no guarantee that they do; in general, the second set of conditions is more generous than the first.

In [90], Wiles provides a beautiful “induction” argument which enables him to pass from the minimal set of conditions to a non-minimal set. Heuristically, this argument requires keeping tabs on the set of those normalized eigenforms whose Galois representations are compatible with an incrementally relaxing set of conditions. As the conditions loosen, the set of forms must grow to keep pace with the increasing number of lifts. The increase in the number of lifts can be estimated from above by a local cohomological calculation. A sufficient supply of modular forms is then furnished by the theory of congruences between normalized eigenforms of differing level. This latter theory may be viewed as a vast generalization of what went on in the author’s article [61].

At the risk of slightly distorting the theory, I will index the shifting set of local conditions by a finite set of prime numbers Σ which contains the set Σ_{0} of primes at which ρ_{A , ℓ} is ramified. This set may be viewed concretely as the set of primes which divide the discriminant of the number field obtained by adjoining to Q the coordinates of all points in A[ℓ] . It is not hard to see that Σ_{0} contains ℓ and is contained in the union of {ℓ} and the set of primes at which A has bad reduction. Indeed, this union is the set of primes Σ^{∗} at which ρ_{A , ℓ^{∞}} is ramified, according to the well-known criterion of Néron-Ogg-Shafarevich [76]. In this perspective, the minimal set of conditions corresponds to the choice Σ = Σ_{0} , while a set of conditions broad enough to include ρ_{A , ℓ^{∞}} is obtained by choosing Σ = Σ^{∗} .

As promised, we shall now give the flavor of the conditions that one imposes on the lifts ρ̃ of type Σ . Firstly, we demand that ρ̃ be unramified outside Σ . Secondly, we ask that ρ̃ have the same qualitative behavior at each prime p ∈ Σ as the representation ρ_{A , ℓ} . (The imposition of this condition can be traced back to Gouvêa’s article [27].) Since ℓ lies in Σ_{0} , a condition is imposed on ρ̃ locally at the prime ℓ this condition requires that ρ̃ be “ordinary” if A has ordinary or multiplicative reduction at ℓ and that ρ̃ be “flat” if A has supersingular reduction at ℓ, cf. [68]. For convenience, a final (global) condition may be imposed on ρ̃, to the effect that the determinant of ρ̃ be the composite of the ℓ-adic cyclotomic character

χ̃_{ℓ}: Gal(\overline{Q}/Q) → Z_{ℓ}^{∗}

and the structural map Z_{ℓ}^{∗} → A^{∗} . This supplementary condition has the effect of allowing one to work with the spaces S(N) rather than with spaces of modular forms on groups of the form Γ_{1}(N) .

To prove that all lifts ρ̃ of type Σ are modular, one first passes to equivalence classes with respect to the relation in which two representations with values in GL(2 , B) are equivalent whenever they are conjugate by an element of GL(2 , B) which maps to the identity matrix in GL(2 , F_{ℓ}) . The equivalence classes of lifts are called deformations the terminology is borrowed from algebraic geometry, where deformation theory has been developed extensively. The idea of introducing deformation theory into the subject of Galois representations is due to Mazur [47]. As F. Gouvêa reports in his recent survey [29], the deformation viewpoint has evolved considerably since [47] first appeared. In particular, a 1993 article by R. Ramakrishna [59] introduces foundational tools which Wiles requires in [90]; Ramakrishna shows that deformation theory can be applied to study the family of lifts which satisfy the local conditions to which we have been alluding.

More precisely, the work of Mazur and Ramakrishna proves that there is a universal deformation of type Σ . This is a lift

ρ_{Σ} : Gal(\overline{Q}/Q) → GL(2 , R_{Σ})

which is characterized by the property that for each lift

ρ̃: Gal(\overline{Q}/Q) → GL(2 , B)

of type Σ there is a unique homomorphism ω: R_{Σ} → B of local Z_{ℓ}-algebras so that the deformation defined by ρ̃ agrees with the one obtained from ρ_{Σ} and ω. A common initial impression is that R_{Σ} is a relatively mysterious object whose existence stems from an abstract representability theorem one comes to understand it only gradually.

Wiles seeks to compare R_{Σ} with a concrete ring 𝕋_{Σ} , which he defines directly as a completion of a classical ring of Hecke operators. A theorem of Carayol constructs a Galois representation

ρ′_{Σ} : Gal(\overline{Q}/Q) → GL(2 , 𝕋_{Σ})

with the property that the trace of ρ′_{Σ}(Frob_{p}) is the Hecke operator T_{p} for all but finitely many primes p. One thinks of ρ′_{Σ} as the universal modular deformation of type Σ . The problem is then to prove that ρ_{Σ} and ρ′_{Σ} coincide, so that all deformations of type Σ are modular. To come to grips with this problem, Wiles begins with the canonical homomorphism

φ_{Σ} : R_{Σ} → 𝕋_{Σ}

which results from the universality of ρ_{Σ} . It is relatively easy to show that φ_{Σ} is surjective; the coincidence of ρ_{Σ} and ρ′_{Σ} means that φ_{Σ} is an isomorphism, cf. [68, §4.2].

My reaction to Wiles’s 1993 announcement was astonishment that one could prove the modularity of Galois representations by working directly with φ_{Σ} . Despite conjectures by Mazur [51, p. 85] and Gouvêa [27, p. 108] to the effect that φ_{Σ} is an isomorphism in the ordinary case, and Theorem 2 of [20], I was not prepared for the revelation that φ_{Σ} could be studied fruitfully.

12. Gorenstein and complete intersection conditions

The conjecture that φ_{Σ} is an isomorphism is proved in [87, 90]. The proof relies on standard notions of commutative algebra which are discussed in [43] and in [41].

The argument of Wiles and Taylor-Wiles proceeds from the definition of 𝕋_{Σ} as a completion of the ring generated by the Hecke operators T_{n} acting on a specific space of classical cusp forms. In particular, 𝕋_{Σ} is free of finite rank over Z_{ℓ} . It has been known for some time that Hecke rings such as 𝕋_{Σ} look special from the vantage point of commutative ring theory. For example, 𝕋_{Σ} tends to be Gorenstein, which means that the 𝕋_{Σ}-module Hom(𝕋_{Σ} , Z_{ℓ}) is free of rank 1 over 𝕋_{Σ} . (The Gorenstein property was first noted in a special case by Mazur [45, Ch. II, §15] and then established in ever-widening generality by others, including the author.) Chapter 2 of [90] includes a section which summarizes and improves on the known Gorenstein assertions; it proves, in particular, that 𝕋_{Σ} is Gorenstein.

Using commutative algebra techniques, [90] presents a number of conditions, each of which is sufficient to show that φ_{Σ} is an isomorphism. As stated in the introduction above, Wiles became aware while studying the problem that φ_{Σ} is an isomorphism whenever the ring 𝕋_{Σ} is a complete intersection ring. (This implication is proved from another point of view in [49].) Later, judging that it would be difficult to prove directly that 𝕋_{Σ} is a complete intersection ring, Wiles focused on a numerical inequality [68, Th. 5.2] and showed that φ_{Σ} is an isomorphism whenever it is satisfied.

At the time of his 1993 Cambridge lectures, Wiles believed that he had proved the numerical inequality through the construction of a “geometric Euler system”, thereby generalizing work of M. Flach [20]. Later analysis showed that the construction envisaged by Wiles was flawed. Interestingly, it is not yet clear whether it can be modified so as to yield an Euler system with the desired properties. At the minimum, one feels that this avenue of inquiry is likely to remain extremely active. In particular, Mazur’s course notes [49] extract new information from Flach’s original construction.

At the time of this revision, the arguments of [87] and [90] are being studied and internalized by the mathematical community. Follow-up work is already beginning to appear. For example, we mentioned above that the main theorem of [90] has been strengthened by a recent manuscript of F. Diamond [18]. Also, the arguments given in [87] and [90] have been shortened somewhat by simplifications due to G. Faltings; these are explained in an appendix to [87].

13. Toward the Taniyama-Shimura conjecture

We conclude with a short survey of results which have been obtained by Wiles [91], Taylor-Wiles [87], and Diamond [18]. Two theorems which have the flavor

ρ_{A , ℓ} modular ⟹ ρ_{A , ℓ^{∞}} modular

are presented in [91]. In each of the theorems, the prime ℓ is taken to be odd, and the representation ρ_{A , ℓ} is required to be irreducible.

One of the theorems (namely, [90, Th. 4.8]) has no application to Fermat’s Last Theorem, since it does not apply to semistable elliptic curves. This theorem does apply in situations where ρ_{A , ℓ} is isomorphic to the representation obtained from the ℓ-division points of a complex multiplication elliptic curve A′ over Q . In these cases, A need not have complex multiplication itself; it is merely “linked mod ℓ ” to a CM curve. For a discussion of this theorem, including an explicit determination of the family of curves which can be linked to a fixed curve A′ , see [69].

The theorem in [90] which applies to Fermat’s Last Theorem is the one whose proof depends on the new work of Taylor-Wiles [87]:

Theorem 1. Suppose that A is a semistable elliptic curve over Q . Let ℓ be an odd prime. Assume that the representation ρ_{A , ℓ} is both irreducible and modular. Then A is a modular elliptic curve.

Fermat’s Last Theorem may then be proved by combining the author’s theorem [62] with the following result, which may be viewed as a highly non-obvious corollary of Theorem 1.

Theorem 2. Let A be a semistable elliptic curve over Q . Then A is a modular elliptic curve.

Wiles deduces Theorem 2 from Theorem 1 by an ingenious argument, which we will now describe. (The argument, which is presented in [90, Ch. 5], has been sketched in [68].) Let A be a semistable elliptic curve, and consider the representation ρ_{A , 3} . If this representation happens to be irreducible, then it is also modular, by the results of Langlands and Tunnell which were cited above. Thus Theorem 1 proves that A is modular.

What happens if ρ_{A , 3} is reducible? In this case, we examine ρ_{A , 5} . If this latter representation is reducible as well, then Wiles shows directly that A is modular. Hence we can, and do, suppose that ρ_{A , 5} is irreducible. Wiles shows then that one can find a second semistable elliptic curve A′ whose mod 5 representation is isomorphic to that of A and whose mod 3 representation is irreducible (cf. [68, App. B]). Two applications of Theorem 1 then suffice to show that A is modular. Indeed, applying the theorem to A′ with ℓ = 3 , we find that A′ is modular. In particular, the irreducible representation ρ_{A′ , 5} is modular. Since this representation coincides with ρ_{A , 5} , we may apply the theorem to A with ℓ = 5 to conclude that A is modular, as desired.

A preprint of F. Diamond [18] generalizes Theorem 1 to the case where A is an elliptic curve over Q whose conductor is not divisible by ℓ^{2} (i.e, one which is semistable at ℓ):

Theorem 3. Let ℓ be an odd prime number. Suppose that A is an elliptic curve over Q which is semistable at ℓ. Assume that the representation ρ_{A , ℓ} is both irreducible and modular. Further, if ℓ = 3, assume that ρ_{A , 3}( Gal ( \overline{Q}/Q(\sqrt{−3} ) )) is non-abelian. Then A is a modular elliptic curve.

The condition concerning Gal( \overline{Q}/Q(\sqrt{−3} )) occurs already in Wiles’s work. How-

ever, in Theorem 1, Wiles proves that ρ_{A , 3}( Gal( \overline{Q}/Q(\sqrt{−3} ) )) is irreducible and

non-abelian when ℓ = 3 ; in other words, the assumption relative to Gal( \overline{Q}/Q(\sqrt{−3} )) has been omitted from Theorem 1 because it may be proved unconditionally.

Using a variant of the Wiles argument we have just sketched, Diamond deduces the following generalization of Theorem 2.

Theorem 4. Suppose that A is an elliptic curve over Q which is semistable both at 3 and at 5 . Then A is modular.

Acknowledgments

I wish to thank N. Boston, H. Darmon, F. Diamond, F. Q. Gouvêa, B. Mazur, V. K. Murty, C. O’Neil, S. Ribet, R. Taylor and A. Wilkinson for feedback on drafts of these notes. I am especially grateful to B. Conrad and L. Goldberg for copious comments.

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Mathematics Department, University of California, Berkeley, California 94720-3840

E-mail address : ribet@math.berkeley.edu