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Research News

A Proof of the Full Shimura- Taniyama-Weil Conjecture Is Announced Henri Darmon

On June 23, 1993, unveiled his strat- straight line, finding the third point of intersection egy for proving the Shimura-Taniyama-Weil con- of the line with the curve and reflecting the re- jecture for semistable elliptic curves defined over sulting point about the x-axis. the field Q of rational numbers. Thanks to the After a change of variables is performed to work of Gerhard Frey, Jean-Pierre Serre, and Ken- bring it into the best possible form, the equation neth Ribet, this was known to imply Fermat’s Last defining E can be reduced modulo any prime num- Theorem. Six years later Christophe Breuil, Brian ber p. If the resulting equation is nonsingular over Conrad, , and Richard Taylor have the finite field with p elements Fp = Z/pZ , then E finally announced a proof of the full Shimura- is said to have good reduction at p. All but finitely Taniyama-Weil conjecture for all elliptic curves many primes are primes of good reduction for a over Q. given E. For example, the defined by the cubic equation The Conjecture The Shimura-Taniyama-Weil conjecture relates el- y2 = x3 x2 +1/4 or, equivalently, (1) liptic curves (cubic equations in two variables of the y2 + y = x3 x2, form y2 = x3 + ax + b, where a and b are rational numbers) and modular forms, objects (to be defined has good reduction at all primes except 11. F below) arising as part of an ostensibly different cir- Let Np be the number of solutions (over p) of cle of ideas. the reduced equation, and set ap(E)=p Np. The { } An elliptic curve E can be made into an abelian sequence ap(E) p (indexed by the primes p of group in a natural way after adjoining to it an good reduction) encodes basic arithmetic infor- extra “solution at infinity” that plays the role of mation on E. Some terms in the sequence ap(E) for the identity element. This is what makes elliptic the elliptic curve of equation (1) are given in Table curves worthy of special study, for they alone, 1. among all projective curves (equations in two vari- ables, compactified by the adjunction of suitable p 2 3 5 7 1113 17 19 23 29 31 ... 10007 points at infinity) are endowed with such a natural Np 4 4 4 9 – 9 19 19 24 29 24 ... 9989 group law. If one views solutions geometrically as ap (E) –2 –1 1 –2 – 4 –2 0 –1 0 7 ... 18 points in the (x, y)-plane, the group operation con- sists in connecting two points on the curve by a Table 1. Sequence ap(E) for the elliptic curve (1). Henri Darmon is associate professor of at McGill University and a member of CICMA (Centre In- It has been a long-standing concern of number teruniversitaire en Calcul Mathématique Algébrique) and theory to search for patterns satisfied by sequences the CRM (Centre de Recherches Mathématiques). His e-mail of this sort. For example, in the simpler case of the address is [email protected]. quadratic equation in one variable x2 d =0with

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d an integer, any prime p that does not divide 2d stable elliptic curve). It turns out that the space of is a prime of good reduction, and for such a p the weight two cusp forms of level 11 is one- integer Np is equal to 2 or 0, depending on whether dimensional and is spanned by the function d is a square or not modulo p. Gauss’s quadratic Y∞ reciprocity law implies that this seemingly subtle q (1 qn)2 (1 q11n)2 property of p depends only on the residue class n=1 of p modulo 4d, so that the sequence Np obeys a = q 2q2 q3 +2q4 + q5 +2q6 2q7 simple periodicity law. 9 10 11 12 13 14 In the case of elliptic curves, a similar pattern 2q 2q + q 2q + 4q +4q arises. It is, however, a good deal more subtle—so q15 4q16 2q17 +4q18 +2q20 +2q21 much so that it emerged as a precise conjecture 2q22 q23 4q25 8q26 +5q27 4q28 only in the 1950s through the work of Shimura, 30 31 10007 Taniyama, and Weil. This pattern involves the no- +2q + 7q + + 18q + . tion of a modular form of weight two: an analytic function on the complex upper half-plane The reader will note that the Fourier coefficients {z ∈ C with Im(z) > 0} satisfying suitable growth of this function agree with the numbers computed, conditions at the boundary as well as a transfor- by wholly different methods, in Table 1. mation rule of the form The Shimura-Taniyama-Weil conjecture was widely believed to be unbreachable, until the sum- az + b f =(cz + d)2f (z), mer of 1993, when Wiles announced a proof that cz + d ! every semistable elliptic curve is modular. A full ab proof of this result appeared in 1994 in the two for all ∈ , cd articles [W] and [TW], the second joint with Tay- lor. Shortly afterwards, Diamond [Di1] was able to where is an appropriate “congruence subgroup” remove the semistability assumption in Wiles’s ar- Z of SL2( ). The main example of a congruence sub- gument at all the primes except 3 and 5. Then, in group, sufficient for the formulation of the 1998 Conrad, Diamond, and Taylor [CDT] refined Shimura-Taniyama-Weil conjecture, is the group the techniques still further, establishing the Z 0(N) of matrices in SL2( ) whose lower-left entries Shimura-Taniyama-Weil conjecture for all elliptic are divisible by N. A modular form of weight two curves whose conductor is not divisible by 27. on 0(N) (also said to be of level N) is, in particu- This is where matters stood at the start of the lar, invariant under translation by 1, and it can be summer of 1999, before the announcement of expressed as a Fourier series Breuil, Conrad, Diamond, and Taylor. X∞ n 2iz f (z)= an(f )q , where q = e . The Importance of the Conjecture n=0 The Shimura-Taniyama-Weil conjecture and its subsequent, just-completed proof stand as a crown- Of particular interest are the so-called “cusp ing achievement of in the twenti- forms” satisfying a more stringent growth condi- eth century. This statement can be defended on (at tion at the boundary that implies, in particular, that least) three levels. a0(f )=0. The Shimura-Taniyama-Weil conjecture asserts that if E is an elliptic curve over Q, then Fermat’s Last Theorem there is an integer N 1 and a weight-two cusp Firstly, the Shimura-Taniyama-Weil conjecture im- form f of level N, normalized so that a1(f )=1, plies Fermat’s Last Theorem. This is surprising at n n n such that first, because the equation x + y = z is not a ap(E)=ap(f ), cubic and bears, on the face of it, no relation with elliptic curves. But to a nontrivial solution for all primes p of good reduction for E. When this ap + bp = cp of Fermat’s equation with prime ex- is the case, the curve E is said to be modular. The ponent p>5, Frey associated the elliptic curve conjecture also predicts the precise value of N: it (now known as a “Frey curve”) given by the equa- should be equal to the “conductor” of E, an arith- tion y2 = x(x ap)(x + bp) . The conductor of E metically defined quantity that measures the Dio- when a, b, and c are relatively prime is the prod- phantine complexity of the associated cubic equa- uct of the primes dividing abc (so that, in partic- tion. Its prime divisors are precisely the primes of ular, E is semistable). Ribet, guided by conjectures bad reduction of E. If p divides N but p2 does not, of Serre, proved that such an elliptic curve could then E is said to have semistable reduction at p. In not possibly correspond to a modular form in the particular, E has semistable reduction at all primes way predicted by the Shimura-Taniyama-Weil con- p (i.e., is semistable) precisely when N is square- jecture. free. Because the Frey curve is semistable, the origi- For instance, the elliptic curve of equation (1) nal result of [W] and [TW] is enough to imply Fer- has conductor 11 (and thus is an example of a semi- mat’s Last Theorem, and the new result of Breuil,

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Conrad, Diamond, and Taylor yields nothing new This conjecture is of fundamental importance for on Fermat’s equation. It does imply, however, other the arithmetic of elliptic curves and is still far results of the same nature, such as the statement from being settled, although the work of Gross- that a perfect cube cannot be written as a sum of Zagier and Kolyvagin shows that it is true when two relatively prime nth powers with n 3, gen- ords=1(L(E,s)) 1. eralizing Euler’s result for n =3. As in the case of Knowing that E is modular also gives control on Fermat’s Last Theorem, a solution to the equation the arithmetic of E in other ways, by allowing the ap + bp = c3 is used to construct an elliptic curve construction of certain global points on E defined whose existence is shown in [DM] to contradict the over abelian extensions of quadratic imaginary Shimura-Taniyama-Weil conjecture. In many fields via the theory of complex multiplication. cases—whenever 3 does not divide ab—the con- Such analytic constructions of global points on E ductor of this curve is divisible by 27, so that the actually play an important role in studying the full strength of the result of Breuil, Conrad, Dia- Birch and Swinnerton–Dyer conjecture through mond, and Taylor is needed to conclude the ar- the work of Gross-Zagier and of Kolyvagin. gument. The Langlands Program The Arithmetic of Elliptic Curves A Galois representation is a (finite-dimensional) Secondly, and more centrally perhaps, the Shimura- representation Taniyama-Weil conjecture lies at the heart of the : GQ → GLn(F), theory of elliptic curves. A theorem of Mordell asserts that the abelian where GQ := Gal(Q¯/Q) is the absolute Galois group group, denoted E(Q), of points of E with rational of Q and F is any field. (Of special interest are the C Q¯ F¯ coordinates is finitely generated, so that it is iso- cases where F = , `, or `.) morphic as an abstract group to Zr T, where T Wiles’s work can be viewed in the broader per- is finite. It is known how to determine T explicitly. spective of establishing connections between au- The integer r 0, called the rank of E over Q, is tomorphic forms—objects arising in the (infinite- a more subtle invariant: no algorithm is known at dimensional) representation theory of adelic present to calculate r as a function of E. groups—and Galois representations. Viewed in It has been a long-standing feeling that much this light, it becomes part of a vast conjectural ed- information on the arithmetic of E (such as the in- ifice put together by Langlands, based on earlier insights of Tate, Shimura, Taniyama, and many variant r) can be gleaned from the sequence Np(E), others. In this setting, Wiles’s discoveries have en- or equivalently ap(E) as p varies. A convenient way to package the information contained in this riched the theory with a powerful new method sequence is to form the L-series of E, a function that should keep the experts occupied well into the of the complex variable s defined initially by the new millennium. Indeed, the impact of Wiles’s Euler product ideas has only started being felt in many diverse Y aspects of the Langlands program: s 12s 1 L(E,s):= (1 ap(E)p + p ) . Two-dimensional complex representations of GQ : p-N Emil Artin associated to a Galois representation (In the later parts of the theory, elementary factors : GQ → GLn(C) an L-function L(,s) and con- are included in the product for the finitely many jectured that it has an analytic continuation to the primes p dividing N.) This product converges when whole complex plane. Via work of Deligne and Re(s) > 3/2. But the Shimura-Taniyama-Weil con- Serre, the Langlands program relates such repre- jecture gives a strong control on the (arithmetically sentations, when n =2, to certain “cusp forms of defined) sequence ap(E) and implies through the weight one” on a group slightly different from work of Hecke that L(E,s) extends to an analytic 0(N). This relation implies the analytic continua- function on the entire complex plane. In particu- tion of L(,s), just as the modularity of an ellip- lar, it makes sense to study the behavior of L(E,s) tic curve implies the analytic continuation of its L- in a neighborhood of s =1. Note that formally series through the work of Hecke. Before Wiles Y the only cases that could be attacked with any p L(E,1) = . generality were the case where is reducible, by p Np +1 work of Hecke, and where the image of is solv- It is believed that the size of r might affect the size able, thanks to the work of Langlands and Tunnell. (It should be noted that for a specific , the mod- of Np on average, which may in turn be reflected in the analytic behavior of L(E,s) near s =1. Indeed, ular form attached to it can in principle be found after a finite amount of computation, so that the Birch and Swinnerton–Dyer in the 1960s conjec- Langlands conjecture could be checked for a finite tured that the order of vanishing of L(E,s) at s =1 number of with nonsolvable image; the first such is equal to r: example was produced by Joe Buhler in his Har- ords=1L(E,s)=r. vard Ph.D. thesis.)

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Using Wiles’s method, Taylor has formulated a the Eichler-Shimura construction, f gives rise to an novel strategy [Ta] for proving the Artin conjecture `-adic Galois representation in the remaining (most interesting) case where the Q¯ Q → Q¯ image of in PGL2(C) is isomorphic to A5—the so- f : Gal( / ) GL2( `) called icosahedral case. Enough of Taylor’s program has now been carried out in joint work with Kevin satisfying trace(f (Frobp)) = (ap(f )), for all primes N` Buzzard, Mark Dickinson, and Nicholas Shep- p not dividing . Here Frobp is the “Frobenius el- p herd–Barron to establish the truth of the Artin ement” at . A notion of conductor can be defined for an `-adic Galois representation, and it follows conjecture for infinitely many icosahedral Galois from the work of Carayol, Deligne, Igusa, Lang- representations. lands, and Shimura that the conductor of f is Generalizations to other number fields. A number equal to the level of f. of ingredients in Wiles’s method have been sig- When f is an eigenform with rational Fourier co- nificantly simplified, by Diamond and Fujiwara efficients corresponding to an elliptic curve Ef among others. Fujiwara, Skinner, and Wiles have under the original Eichler-Shimura construction, been able to extend Wiles’s results to the case then f is simply obtained by piecing together the n where the field Q is replaced by a totally real num- natural action of GQ on the space of ` -torsion ber field K. In particular, this yields analogues of points of Ef (Q¯) as n varies. the Shimura-Taniyama-Weil conjecture for a large It becomes natural to formulate a more general class of elliptic curves defined over such a field. version of the Shimura-Taniyama-Weil conjecture, replacing elliptic curves with two-dimensional rep- n-dimensional generalizations. Michael Harris and resentations of GQ with coefficients in Q¯ . This Richard Taylor have explored generalizations of the ` more general version would have the virtue of main results of [W] and especially [TW] to the con- avoiding the subtleties associated with fields of de- text of n-dimensional representations of GQ . (This finition of Fourier coefficients of eigenforms. work, as well as the proof of the local Langlands An important insight that emerged over the last conjecture for GL by Harris and Taylor, is ex- n decades through the work of Alexander Grothen- pected to be covered in a future Notices article.) dieck, Pierre Cartier, Jean Dieudonné, and finally Jean-Marc Fontaine and his school is that it should The Work of Breuil, Conrad, Diamond, and be possible to characterize the `-adic representa- Taylor tions arising from modular forms entirely in Ga- The space S2(N) of weight-two cusp forms on 0(N) lois-theoretic terms—or, more precisely, in terms is a finite-dimensional complex vector space of their restriction to a “decomposition group” Q¯ Q equipped with an action of a natural family of Gal( `/ `) at ` or even an “inertia group” at this commuting self-adjoint operators, the so-called prime. Such representations are called “potentially “Hecke operators”. A normalized newform on 0(N) semistable”, and this notion is a key ingredient for is a simultaneous eigenvector for these operators, generalizing the Shimura-Taniyama-Weil conjecture normalized so that its first Fourier coefficient is to `-adic Galois representations. Around 1990 equal to 1, and not already arising in the space of Fontaine and Mazur conjectured that the `-adic cusp forms on 0(D) for any D dividing N. A con- Eichler-Shimura construction yields a bijection struction of Eichler and Shimura associates to a nor- from the set mod(N) of normalized eigenforms malized newform of level N with rational Fourier of level N to the set gal(N) of `-adic Galois rep- coefficients an elliptic curve over Q of conductor resentations of conductor N that are potentially N. The original Shimura-Taniyama-Weil conjec- semistable at `. Wiles’s proof in essence amounts ture states that this construction yields a bijection to a sophisticated counting argument in which from the set of normalized newforms on 0(N) these two sets are compared and found to be of with rational Fourier coefficients to the set of el- the same size. liptic curves of conductor N, taken modulo an The main tools in controlling the size of mod equivalence, weaker than isomorphism, known as are supplied by the theory of “Hecke rings” and con- isogeny. It appears difficult even now to give an a gruences between modular forms, a rich body of priori estimate for the size of either set as a func- techniques developed by Mazur, Hida, and Ribet tion of N; in fact, the question of the rationality and used to great effect by Ribet to derive Fermat’s of the Fourier coefficients of an eigenform is a sub- Last Theorem from the Shimura-Taniyama-Weil tle one that seems hard to come to terms with. conjecture. In general, the Fourier coefficients of a nor- The set gal(N) is in many ways the more sub- malized eigenform f are algebraic numbers de- tle object of the two, about which there is a priori fined over a finite extension Kf Q¯ of Q. Fix a the least explicit information. There are two major Q¯ → Q¯ Q prime ` and an inclusion : `, where ` is ingredients used to estimate the size of gal(N) the field of `-adic numbers. By a generalization of and relate it to mod(N).

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• The theory of “base change” and in particular the work of Langlands and Tunnell on solvable base change. • The theory of deformations of Galois repre- sentations pioneered by Mazur and Hida. The second ingredient is extremely general and flexible and is being honed into a powerful tool in the arithmetic study of automorphic forms. The first ingredient, by contrast, is available only when the image of f is a (pro)-solvable group. Number theory has, over the last hundred years, developed an arsenal of techniques for understanding abelian and solvable extensions, as evidenced in class field theory, which gives a precise description of all the abelian extensions of a given number field as well as the behavior of the Frobenius elements in these extensions. Arriving at an understanding of non- solvable extensions on the same terms has proved far more elusive. Unfortunately, the image of f is rarely solvable. But it is when the prime ` is 3 by a fortuitous ac- cident of group theory: the group GL2(F3), and hence GL2(Z3), is solvable, a fact that ceases to be true as soon as 3 is replaced by any larger prime. It is for this reason that in the application to the Shimura-Taniyama-Weil conjecture it is indis- pensable in Wiles’s strategy to work with the prime ` =3. The last obstacle to carrying out Wiles’s program to a complete proof of the Shimura-Taniyama-Weil conjecture arose from a technical difficulty: the 3- adic Galois representations of conductor N, when 27 divides N, have an intricate behavior when re- stricted to the inertia group at 3—and a precise de- scription and understanding of this behavior are required to control the set gal(N) when ` =3. Overcoming this difficulty required some new in- sights into the structure of 3-adic representations of GQ that are “highly ramified” at the prime 3. A number of these key insights were provided by the work of Breuil strengthening Fontaine’s theory.

References [CDT], FRED DIAMOND, and RICHARD TAYLOR, Modularity of certain potentially Barsotti-Tate Ga- lois representations, J. Amer. Math. Soc. 12 (1999), 521–567. [DM]HENRI DARMON and LOIC MEREL, Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math. 490 (1997), 81–100. [Di1]FRED DIAMOND, On deformation rings and Hecke rings, Ann. of Math. (2) 144 (1996), 137–166. [Ta] RICHARD TAYLOR, Icosahedral Galois representations, Olga Taussky-Todd: In memoriam, Pacific J. Math., Special Issue (1997), 337–347. [TW]RICHARD TAYLOR and ANDREW WILES, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572. [W] ANDREW WILES, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.

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