GALOIS REPRESENTATIONS AND MODULAR FORMS

Kenneth A. Ribet

University of California, Berkeley

Last fall, the organizers of this Mathfest asked me to discuss “the mathematics behind Andrew Wiles’s solution of the Fermat conjecture.” At the time it was anticipated that Fermat’s Last Theorem would be proved by the methods which Wiles outlined in his lectures last summer. The situation changed dramatically in December when Wiles posted a message to the USENET news group sci.math stating that “the final calculation of a precise upper bound for the Selmer group … is not yet complete as it stands” (see [16]). At this writing, the required upper bound has not been obtained Fermat’s conjecture is not a theorem. Nevertheless, it is my opinion that the methods introduced by Wiles are significant and continue to merit our attention. The most striking consequence of Wiles’s work is that one now has tools for exhibiting infinite families of elliptic curves over the rational field which are attached to modular forms [37]. No such families were known before last summer.

Because of the intense publicity surrounding Fermat’s Last Theorem, most of the background material in these notes has been discussed in news and expository articles which were written on the heels of Wiles’s announcement. Among these are the author’s news item in the Notices [34] and his article with Brian Hayes in American Scientist [13], two pieces in the American Mathematical Monthly [4, 12], the report by K. Rubin and A. Silverberg in the AMS Bulletin [36], and unpublished notes by various authors which can be obtained by gopher from e-math.ams.org. In view of the large literature in this subject, I imagined these notes more as an annotated bibliography than as a survey. In the end, it seems that I have written an abbreviated survey which has a relatively large list of references. I have tried to cite some papers which are still in preparation, or have not yet appeared.

I hope that these notes will be of interest to Mathfest participants. In all likelihood, a corrected version of this text will appear in the AMS Bulletin. So if you find misprints, ambiguities or errors, or if you have suggestions for improvements, please let me know about them either in Minneapolis or by e-mail. My oral presentation on August 17 will provide a general overview, slanted toward developments which have emerged in the last year.

Typeset by AMS-TEX

This work was to some extent supported by NSF Grant #DMS 93-06898. I am grateful to B. Mazur, C. O’Neil, R. Taylor and A. Wilkinson for comments and suggestions on drafts of these notes. A special thanks to S. Ribet for moral support.

1. Modular forms

We begin with some background concerning modular forms. Among reference books in the subject, one might cite [17], [27], and [46]. For a first reading in the subject, a good starting point is [40, Ch. VII].

The modular forms we will consider are holomorphic functions of one variable, defined on the upper half-plane of complex numbers with positive imaginary part. These functions are usually presented as converging Fourier series ∑_{n=1}^{∞} a_{n}q^{n}, where q = e^{2πiz}. For the forms which most interest us, the coefficients a_{n}, which are a priori complex numbers, are in fact algebraic integers and often even ordinary integers.

Suppose that N is a positive integer. Let Γ_{0}(N) be the group of integer matrices with determinant 1 which are upper-triangular mod N. The weight-two cusp forms on Γ_{0}(N) are holomorphic functions f(z) on the complex upper half-plane which obey 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). This equation implies, of course, that f(z) is invariant

under integer translation. In addition to the holomorphy and the functional equation, one imposes a suitable condition at infinity [46, Ch. 2]. The cusp forms of weight two on Γ_{0}(N) form a complex vector space S(N). The dimension of S(N) is finite and may be calculated easily using the Riemann-Roch theorem.

The dimension is zero for N ≤ 10 and is 1 for N = 11. A generator for S(11) may be described in terms of the formal power series with integral coefficients ∑ a_{n}X^{n} defined by the identity

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

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 non-zero element of S(11). Anticipating somewhat, we refer the reader to Shimura’s article [45], which discusses the relationship between h and a certain elliptic curve, and introduces the Galois groups G_{n} which appear below.

For each integer n ≥ 1, the nth Hecke operator on S(N) is an endomorphism T_{n} of S(N), whose action is generally written 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} may be described easily from the perspective of Fourier coefficients. Namely, if 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|N, then f|T_{p} is given by the simpler formula ∑_{n=1}^{∞} a_{np}q^{n}.

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. 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: 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.

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 operators T_{n} are not necessarily diagonalizable. This problem arises from those n which have a common factor with N, and can be repaired by the introduction of newforms [1]. 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 have shown 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. Thus S(N) may be described purely in terms of newforms provided that one includes newforms whose “levels” M run over the divisors of N.

2. The Taniyama-Shimura conjecture

The Taniyama-Shimura conjecture relates elliptic curves over Q and certain modular forms. Here is a compact description of the conjecture; another is provided in the recent article by K. Rubin and A. Silverberg [36, §1]. To describe the T-S conjecture, we need to allude to some fundamental concepts pertaining to elliptic curves. For a serious treatment of these concepts, the reader may consult a large number of textbooks and monographs which focus on elliptic curves. (Several of these books discuss modular forms as well.) Rather than list these books here, we refer the reader to the bibliography of a book review by W. Hearst III and the author [14].

To describe the conjecture, we begin with a construction due to Shimura. Suppose that f ∈ S(N) is a normalized eigenform and let E be the field generated by the coefficients of f. Shimura has associated 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 [46, Th. 7.14]. The construction f ↦ A_{f} is sufficiently flabby that it associates to f a “clump” of isogenous abelian varieties, rather than a specific abelian variety which is singled out up to isomorphism. (By definition, two abelian varieties are isogenous if and only if they are obtained from each other by dividing by a finite group. A theorem of Faltings [6, §5] justifies the vague statement that two abelian varieties are isogenous if and only if they are equivalent arithmetically.)

If f has integral Fourier coefficients, then E = Q and A_{f} has dimension 1. This means that A_{f} is an elliptic curve over Q, which we can regard as being defined up to isogeny. The eigenvalues a_{p} can be extracted from the elliptic curve A = A_{f} via the process of “reduction mod p.” More precisely, it is possible to find cubic polynomials with integral coefficients which define A. To each such polynomial, one associates a non-zero discriminant. There is an essentially unique defining equation whose discriminant has smallest absolute value; the discriminant of this equation is said to be the (minimal) discriminant Δ of A. The 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. 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; its rational points, i.e., points with values in Z/pZ, form a finite abelian group which we can call A(Z/pZ). A theorem of H. Hasse states that the integer

b_{p} := p+1 − #(A(Z/pZ))

is bounded in absolute value by 2\sqrt{p}. The relation between f and A = A_{f} is as follows. Firstly, if p does not divide N, then A has good reduction at p. Further, for such p, the pth Fourier coefficient of f coincides with the quantity p+1 − #(A(Z/pZ)). In other words, we have 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 describe, but involves no essentially new ideas.)

Incidentally, 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 the definition, see for example [39, §2].) 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 is always positive, while the minimal discriminant may be positive or negative. Further, while the latter number may be divisible by a large power of a given prime, the conductor of an elliptic curve tends to be divisible only by low powers of its prime divisors. (In fact, a prime p ≥ 5 can divide the conductor at most to the second power.) Finally, the conductor of an elliptic curve depends only on its Q-isogeny class; the discriminant changes under isogeny.

It happens quite frequently that the conductor is square free, i.e., divisible only by the first powers of its prime divisors. The elliptic curve A is then said to be semistable.

Suppose again that f is an eigenform in S(N) with integral Fourier coefficients and that A is the elliptic curve A_{f}. It is natural to ask how the conductor of A is related to N. The answer begins with the fact that f is necessarily built from a newform in S(M), for some 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 [3], following work of Shimura, Igusa, Deligne and Langlands.

The Taniyama-Shimura conjecture has several equivalent formulations. In one form, it states that every elliptic curve over Q is isogenous to a some curve A_{f}. A more precise statement runs as follows: Let A be given over Q and suppose that N is the conductor of A. Define integers b_{p} for p |/ N as above. Then there is a newform f ∈ S(N) whose Fourier coefficients a_{n} satisfy the identity a_{p} = b_{p} for p |/ N.

The theory of newforms shows that f is unique, if it exists. Also, if f exists, we may apply “Tate’s conjecture” now a theorem of Faltings [6, §5] to show that A and A_{f} are isogenous over Q.

If the Taniyama-Shimura conjecture is true for A, we say that A is modular.

Mazur’s article [24] formulates the Taniyama-Shimura conjecture purely as a statement about the Riemann surface associated with an elliptic curve over Q. If A is such an elliptic curve, write A(C) for the space of points of A with complex coordinates, so that A(C) is a complex torus. For each integer N, consider the group Γ(N) consisting of integer matrices with determinant 1 which are congruent to the identity matrix modulo N. This group acts naturally on the complex upper

half-plane H by fractional linear transformations, with ( \begin{matrix}a & b \\ c & d \end{matrix} ) giving rise to

z ↦ \frac{az + b}{cz + d}.

The quotient Γ(N)\H has a natural compactification called X(N). Mazur shows that an elliptic curve A over Q is modular if and only if one can find a non-constant holomorphic map from X(N) to A(C), for some positive integer N.

In [33], I consider the set of elliptic curves A over C for which there is a nonconstant holomorphic map X(N) → A(C) for some positive integer N. Answering a question of Serre, I show that the Taniyama-Shimura conjecture may be generalized to a natural conjectural characterization of such elliptic curves.

3. Galois representations attached to elliptic curves

Let A be an elliptic curve over Q. We view A as a commutative algebraic group over Q. Concretely, one models A as the projective plane curve defined by an affine cubic equation of the form y^{2} = x^{3} + ax + b with a, b ∈ Q. If K is a field containing Q, the set of points of A with values in K is then a subset A(K) of the projective plane over K with a classical group law, the so-called “chord and tangent operation.” There is a unique point O of A outside the affine plane; this is taken to be the identity element in the group. Three distinct points on A then sum to O if and only if they are colinear. The composition law on A(K) can be described in terms of coordinates by a family of polynomial equations with coefficients in Q. (For a recent discussion concerning families of such equations, the reader may consult [2].)

A model for A(K) when K = C is given by Weierstraß theory: 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.

Further analysis shows that A[n] in fact lies in A(\overline{Q}), where \overline{Q} is the subfield of C consisting of all algebraic numbers. Indeed, that A[n] is finite and stable under all automorphisms of C implies that none of its elements has a transcendental coordinate. Let Gal(\overline{Q}/Q) be the group of automorphisms of \overline{Q}. Then A[n] is stable under the natural action of Gal(\overline{Q}/Q) on A(\overline{Q}), since a point P ∈ A(\overline{Q}) has order dividing n if and only if its coordinates satisfy a certain set of polynomial equations with rational coefficients. This means that A[n] is equipped with a canonical action of the Galois group Gal(\overline{Q}/Q).

Notice that for each σ ∈ Gal(\overline{Q}/Q) the automorphism P ↦ ^{σ}P is a group automorphism of A[n]: We have ^{σ}(P + Q) = ^{σ}P +^{σ}Q for P, Q ∈ A[n]. This means that when we view the action of Gal(\overline{Q}/Q) on A[n] as a (continuous) homomorphism

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

we are entitled to let Aut(A[n]) stand for the group of automorphisms of A[n] as an abelian group. We do this. Then since A[n] is isomorphic to the group (Z/nZ)^{2}, Aut(A[n]) is isomorphic to the group GL(2, Z/nZ) of two-by-two invertible matrices with coefficients in Z/nZ. While there is no canonical isomorphism Aut(A[n]) ≈ GL(2, Z/nZ), each choice of basis A[n] ≈ (Z/nZ)^{2} determines such an isomorphism, and 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). The kernel of ρ_{A,n} corresponds, via Galois theory, to a finite Galois extension K of Q in \overline{Q}. Concretely, K is the extension of Q obtained by adjoining to the rational field the coordinates of the various points in A[n]. The Galois group Gal(K/Q) is the image of ρ_{A,n}; it is therefore embedded in 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/Q whose Galois group is a subgroup G_{n} of the group of two-by-two invertible matrices with coefficients in Z/nZ.

It is natural to ask for a description of this G_{n} as a subgroup of GL(2, Z/nZ). There is a (relatively rare) special case to consider: that where A has complex multiplication (over C). Viewing A as C/L, we may define the complex multiplication case to be that for which there is a non-integral complex number α such that αL ⊆ L. The group G_{n} then has an abelian subgroup of index ≤ 2, so it is quite far from the ambient group GL(2, Z/nZ). In the much more common case where A has no complex multiplication, Serre has shown that the index of G_{n} in GL(2, Z/nZ) is bounded as a function of n. An essentially equivalent piece of information is that one has the equality G_{p} = GL(2, Z/pZ) for all but finitely many primes p (see [41]).

To orient the reader, we might point out that the Taniyama-Shimura conjecture was proved for complex multiplication elliptic curves over Q by Shimura in 1971 [47]. On the other hand, it is relatively easy to show that semistable elliptic curves over Q never have complex multiplication. As we shall see below, the elliptic curves which appear in connection with Fermat’s Last Theorem are semistable elliptic curves. Accordingly, the result of [47] sheds no light on the question of whether or not they are modular.

A key piece of information about the extension K/Q (which depends on n and A) is that its discriminant is divisible only by those prime numbers which divide the product of n and the conductor of A. In other words, if p |/ n is a prime number at which A has good reduction, then K/Q is unramified at p. In this case, one can introduce a Frobenius element σ_{p} in G_{n} which is well defined up to conjugation. Since G_{n} is a group of matrices, we may view σ_{p} as a matrix whose trace and determinant are well-defined elements of Z/nZ. It is easy to prove that det σ_{p} is the number 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 p prime to n); more precisely, it determines the numbers b_{p} mod n.

4. Galois representations attached to modular forms

Suppose that f ∈ S(N) is a normalized eigenform. If the coefficients a_{n} of f happen to be integers, the abelian variety A = A_{f} is an elliptic curve. By considering the family ρ_{A,n}, we obtain 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 nth 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 we are tempted 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 we replace A by an isogenous elliptic curve, the representations ρ may change! To circumvent this difficulty, we introduce the process of “semisimplification.” 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 one-dimensional representations α and β. Clearly, the trace and determinant are the same for ρ and for its semisimplification. One shows easily 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},

we obtain 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): 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.

Our aim will be to reverse this process: Given a two-dimensional representation of Gal(\overline{Q}/Q) over Z/ℓZ, we would like to show in some circumstances that this representation arises from some eigenform f. Examples show that it is unreasonable to expect f to have integral coefficients; we must confront the situation where the coefficients of f are algebraic integers but not ordinary integers.

Suppose then that f = ∑ a_{n}q^{n} is a normalized eigenform, and let E be the field generated by the a_{n}. Let O be the ring of integers of E, so that a_{n} ∈ O for all n ≥ 1. 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. 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 in terms of the determinant of the representation ρ_{f,λ}. The latter determinant is a continuous homomorphism Gal(\overline{Q}/Q) → F_{λ}^{∗}. This homomorphism turns out to be the composite of the “cyclotomic character” χ_{ℓ}: Gal(\overline{Q}/Q) → F_{ℓ}^{∗} and the natural inclusion F_{ℓ}^{∗} ⊆ F_{λ}^{∗}. The cyclotomic character may be defined by considering the group μ_{ℓ} of ℓth roots of unity in \overline{Q}. The Galois group Gal(\overline{Q}/Q) acts on μ_{ℓ} by conjugation, giving rise to a continuous homomorphism

Gal(\overline{Q}/Q) → Aut(μ_{ℓ}).

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

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. Thus we have ρ_{f,λ}(c) = −1; one says that ρ_{f,λ} is odd.

This parity observation generalizes to modular forms of weights other than two. Here is a quick synopsis of the situation; for details and further references, one may consult [30]. For integers k ≥ 2, 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. For each non-zero prime ideal λ of O, 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, where χ_{ℓ} again denotes the mod ℓ cyclotomic character. For this, we regard ϵ as a map Gal(\overline{Q}/Q) → F_{λ}^{∗}. To do this, we compose the natural map Gal(\overline{Q}/Q) → (Z/NZ)^{∗} which gives the action of Gal(\overline{Q}/Q) on the Nth roots of unity with the character ϵ, thus obtaining a map Gal(\overline{Q}/Q) → O^{∗}. On reducing this homomorphism mod λ, we obtain the desired variant of ϵ. Evaluating the determinant formula on the “complex conjugation” element of Gal(\overline{Q}/Q), we find

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 set-up.

Serre has conjectured that all two-dimensional odd irreducible representations of Gal(\overline{Q}/Q) over a finite field are essentially of the form ρ_{f,λ} [43]. To give meaning to this statement, we choose a prime number ℓ 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 irreducible representation. We shall say that ρ is modular if one can find: (i) an eigenform f in some space S_{k}(N, ϵ); (ii) a prime λ dividing ℓ in the ring of integers O 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). Serre has conjectured that every two-dimensional odd irreducible representation of Gal(\overline{Q}/Q) is modular. It has emerged that this seemingly innocuous conjecture is equivalent to a much more precise statement (also given in [43]), which immediately implies the Taniyama-Shimura conjecture, Fermat’s Last Theorem, and a host of other assertions!

5. Frey’s construction

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

Frey’s idea runs as follows. Suppose that we have 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. Changing the sign in the value of Z, we obtain a triple (a, b, c) which satisfies the symmetric equation a^{ℓ}+b^{ℓ}+c^{ℓ} = 0. The equation y^{2} = x(x−a^{ℓ})(x+b^{ℓ}) defines an elliptic curve A with unexpected properties.

These properties are catalogued in §4.1 of Serre’s article [43]. After a possible permutation of the three integers a, b and c defining A, one finds that the conductor of A is the product of the prime numbers which divide abc (each occurring to the first power), so that A is semistable. On the other hand, the minimal discriminant Δ of A is the quotient of (abc)^{2ℓ} by the trivial factor 2^{8}. From our point of view, the main “unexpected” property of A is that Δ is the product of a power of 2 and a perfect ℓth power, where ℓ is a prime ≥ 5. (From the point of view of Szpiro’s conjecture and the abc conjecture, the surprising feature of A is that its discriminant is larger than a high power of its conductor.)

This property persuaded Frey that A was unlikely to be 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 to link it to established conjectures. Much of what emerged is off the main topic of these notes we will not discuss the abc conjecture or Szpiro’s conjecture, for instance. Note, however, that these conjectures are treated by such articles as [8], [9], [18], [15], and [28].

A first step toward justifying Frey’s heuristic conclusion was taken in August, 1985 by Serre in a letter to J-F. Mestre [42]. In this letter, Serre formulated two related conjectures about modular forms, which he called C_{1} and C_{2}. He pointed out that Fermat’s Last Theorem is a consequence of the Taniyama-Shimura conjecture together with the two new conjectures. In other words, Serre exhibited the implication

C_{1} + C_{2} + Taniyama-Shimura ⟹ Fermat.

In my article [31], I proved Serre’s conjectures C_{1} and C_{2}. My theorem established the implication

Conjecture of Taniyama-Shimura ⟹ Fermat

which was the goal of Frey’s construction. More generally, let ℓ ≥ 5 be a prime number and suppose that A is a semistable elliptic curve over Q whose discriminant is the product of a power of 2 and a perfect ℓth power. Then the result of [31] shows that A cannot be a modular elliptic curve. (See [29] and [32] for expository accounts of this work.)

After writing [42], Serre presented in [43] a broad group of conjectures linking mod p Galois representations and mod p modular forms. These conjectures constitute the “precise statement” that was mentioned briefly at the end of the last section. As was alluded to above, Serre’s conjectures imply Fermat’s Last Theorem by a simple direct argument, and they imply a host of other statements about Diophantine equations and elliptic curves. There seems no hope of proving Serre’s conjectures in the near term. Nevertheless, certain consequences of the conjectures for the mod p Galois representations associated to modular forms have seemed quite accessible. A group of these consequences have become known as the “weak Serre conjecture”; this conjecture is an intricate generalization of the two statements C_{1} and C_{2} formulated in [42]. Thanks to the work of a large group of mathematicians, the weak Serre conjecture is now a theorem, at least for p ≥ 3; see [35] and [5] for details.

The relationship between the weak Serre conjecture and the full group of conjectures of [43] is that the conjectures of [43] amount to the conjunction of the weak conjecture and the single supplementary statement which was introduced at the end of the previous section. This statement, to the effect that two-dimensional continuous irreducible odd representations of Gal(\overline{Q}/Q) are modular, is the one which appears to be intractable.

6. 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. For example, we might consider 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 shows that an infinite number of these representations are modular (in the broadest possible sense), one can go on to prove that A is modular. The difficulty with this approach, from a philosophical point of view, is that there is no visible program for manufacturing modular forms which can be compared with the ρ_{A,ℓ}.

Alternatively, one may fix a prime ℓ and consider the family of spaces 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 re-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. The natural map Z → Z_{ℓ} is injective, so that Z_{ℓ} contains Z as a subring. 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. Once one finds an eigenform f in S(N) whose Fourier coefficients are related to the traces of ρ_{A,ℓ^{∞}}(Frob_{p}), one has essentially proved that A is modular.

Of course, if ρ_{A,ℓ^{∞}} is modular, then so is ρ_{A,ℓ}. Relating ρ_{A,ℓ^{∞}} to modular forms is therefore at least as hard as the formidable task of relating ρ_{A,ℓ} to modular forms! On the other hand, to prove that A is modular by the ℓ-adic method, we need only work with a single prime ℓ. Wiles’s approach is based on the lucky circumstance that the representations ρ_{A,ℓ} are in fact modular for ℓ ≤ 3, cf. [36, §2.3]. This circumstance arises from the theory of base change à la Saito-Shintani, as developed by Langlands in [19] and then applied by Tunnell in [48]. (For expositions of the results of Langlands, see [10] and [11].)

Wiles’s basic idea is to prove that 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 seeks to ignore A as much as possible the aim is to prove a theorem about ℓ-adic representations with properties generalizing those of ρ_{A,ℓ^{∞}}.

Two theorems in this direction are stated in [49]. In each theorem, the prime ℓ is taken to be odd, and the representation ρ_{A,ℓ} is required to be irreducible. One theorem pertains to the case where A is semistable; this is the result which would apply to Frey curves arising in connection with Fermat’s Last Theorem, and also the theorem which is in difficulty. That there is difficulty means that the theorem is not known to be true for all semistable elliptic curves over Q. However, it is apparently possible to use the technique of the proof to exhibit a genuinely infinite class of semistable elliptic curves over Q which are modular. As I mentioned at the beginning of these notes, no such class was known before Wiles’s methods were introduced. The second theorem treats the case where the representation ρ_{A,ℓ} arises from a complex multiplication elliptic curve A′ over Q. Here, there is no “gap” in the proof, and one can again use the theorem to exhibit an explicit infinite collection of elliptic curves over Q which are modular [37].

Wiles seeks to use the first theorem to prove that all semistable elliptic curves are modular. His ingenious argument for doing this is sketched by Rubin and Silverberg in [36, Prop. 2.4]. Suppose that A is a semistable elliptic curve over Q. If ρ_{A,3} happens to be irreducible, then the results of Langlands and Tunnell, plus the assertion of Wiles’s first theorem, give the desired result that A is modular. The hard case is that where ρ_{A,3} is reducible. If ρ_{A,5} is reducible as well, then it is not hard to show directly that A is modular. So we might as well assume that ρ_{A,5} is irreducible. Wiles shows then that one can find a second elliptic curve A′ whose mod 5 representation is isomorphic to that of A and whose mod 3 representation is irreducible [36, Appendix B]. Two applications of the “first theorem” then suffice to show that A is modular.

7. The language of deformations

Suppose that A is a semistable elliptic curve over Q and that ℓ ≥ 3 is 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 GL(2, F_{ℓ}). We seek to show that ρ_{A,ℓ^{∞}} is modular by showing that all suitable lifts of ρ_{A,ℓ} are modular. The idea of looking at all lifts at once stems from a fundamental article of Mazur [23]. For historical reasons arising from algebraic geometry, isomorphism classes of lifts are called “deformations.”

The meaning of “suitable” depends on an auxiliary choice: a finite set of prime numbers Σ which contains Σ_{0}, where Σ_{0} is the set of primes at which ρ_{A,ℓ} is ramified. (Thus, by definition, Σ_{0} is the set of primes dividing the discriminant of the number field obtaining by adjoining to Q the coordinates of all points in A[ℓ].) A priori, 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 [44].

That Σ_{0} might be strictly smaller than the union should be regarded as a complicating factor. The reason for this is that the so-called “minimal” case Σ = Σ_{0} is expected to be the easiest to deal with, while the choice Σ = Σ^{∗} is the one which is relevant to the representation ρ_{A,ℓ^{∞}}.

The following description is derived from §3.4 of [36]. Our discussion differs from that of [36] mainly in our writing ℓ for the prime called p in [36] and in our focus on the auxiliary choice Σ.

Having fixed this choice for the moment, one considers continuous homomorphisms ρ̃: Gal(\overline{Q}/Q) → GL(2, A), where A is a complete Noetherian Z_{ℓ}-algebra with residue field F_{ℓ}. (For technical reasons, it might be useful to replace Z_{ℓ} by the integer ring of a finite extension of the ℓ-adic field Q_{ℓ}, but we will ignore this complication.) One demands a number of properties of ρ̃, which can be summarized as follows: First, note that the residue map A → F_{ℓ} induces a homomorphism GL(2, A) → GL(2, F_{ℓ}); we require that ρ_{A,ℓ} coincide with the composite of ρ̃ and this residue homomorphism. Second, we demand that ρ̃ be unramified outside Σ. Third, we ask that ρ̃ have the same qualitative behavior at the primes in Σ_{0} as the representation ρ_{A,ℓ}. (In the language of [36], we ask that ρ̃ be ordinary at ℓ if A has ordinary or multiplicative reduction at ℓ and that ρ̃ be flat at ℓ if A has supersingular reduction at ℓ.) Finally, we insist 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 latter condition can be removed or modified if one is willing to consider spaces of modular forms which are more general than the spaces S(N).

Representations ρ̃ with these properties define deformations of ρ_{A,ℓ} of type Σ. The deformations are equivalence classes of representations with respect to a natural identification: Two representations give the same deformation if they are conjugate by an element of GL(2, A) which maps to the identity matrix in GL(2, F_{ℓ}).

According to a theorem of Mazur and Ramakrishna, there is a universal deformation. This is a deformation

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

of type Σ which is distinguished by the property that an arbitrary deformation ρ̃: Gal(\overline{Q}/Q) → GL(2, A) of type Σ may be obtained from ρ_{Σ} by a unique homomorphism R_{Σ} → A. Here R_{Σ} should be regarded as a somewhat mysterious ring, which is not given explicitly, at least at first. Its construction depends on an abstract representability theorem, which one applies to the given deformation property.

Somewhat analogously, one can define a universal modular deformation of type Σ:

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

The problem is then to prove that ρ_{Σ} and ρ′_{Σ} coincide. In contrast to R_{Σ}, the ring 𝕋_{Σ} may be constructed quite directly as an essentially classical Hecke algebra. It is free of finite rank over Z_{ℓ}. Following a line of inquiry initiated by Mazur [22, Ch. II, §15] (and taken up by a fair number of other mathematicians, including the author), Wiles proves that 𝕋_{Σ} is a Gorenstein ring.

The Gorenstein property leads to further speculation as to the precise nature of 𝕋_{Σ} as an object of commutative ring theory. Wiles realized several years ago that the key to proving that ρ_{Σ} and ρ′_{Σ} coincide is to show that 𝕋_{Σ} is a complete intersection ring in the sense of [21]. In fact, Wiles shows that if 𝕋_{Σ} is a complete intersection ring, then the universal deformation of type Σ′ is modular for all Σ′ ⊇ Σ. (For another perspective on questions of this type, see [25].)

In comparing ρ_{Σ} and ρ′_{Σ}, one begins with the canonical homomorphism

φ_{Σ} : R_{Σ} → 𝕋_{Σ}

which results from the universality of ρ_{Σ}. It is known in most cases, and should be possible to prove in general, that this homomorphism is surjective. To show that all deformations of type Σ are modular is to prove that φ_{Σ} is an isomorphism. (Analogous conjectures had been made previously by Mazur, cf. [26, p. 85]. For a bit more detail see [36, §4.2])

Using techniques from commutative algebra, Wiles proves a proposition to the effect that φ_{Σ} is an isomorphism whenever a specific numerical inequality is satisfied [36, Th. 5.2]. Although the statement of the proposition seems to require that 𝕋_{Σ} be Gorenstein, Lenstra has shown how to re-state the proposition so that no Gorenstein hypothesis appears [20]. Furthermore, Wiles interprets the numerical condition as an upper bound for the order of a Selmer-like group S_{Σ}, whose finiteness is not known a priori [36, §5.2]. The theory of congruences between modular forms of different levels shows that the desired bound on S_{Σ}, if valid for a given set Σ, is automatically verified for all supersets Σ′ ⊇ Σ. This means that it will be sufficient to verify the sought-after inequality in the minimal case Σ = Σ_{0}. Incidentally, I do not know whether φ_{Σ} is an isomorphism for a given set Σ if and only if the inequality is true for Σ. Given simply that φ_{Σ} is an isomorphism, can one conclude that φ_{Σ′} is an isomorphism for all Σ′ ⊇ Σ?

As the authors of [36] explain, Wiles has sought to verify his inequality for Σ = Σ_{0} by constructing a “geometric Euler system,” thereby generalizing work of M. Flach [7]. At this juncture, no satisfactory construction has been exhibited. Nevertheless, this area of inquiry is likely to become extremely active, and one feels that some progress has been made. For example, Mazur’s 1993–94 Harvard graduate course [25] suggests that new arguments in commutative algebra may uncover further information which is implicit in Flach’s construction.

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UC Mathematics Department, Berkeley, CA 94720-3840 USA E-mail address: ribet@math.berkeley.edu