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A Proof of the Full Shimura-Taniyama-Weil Conjecture

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A Proof of the Full Shimura-Taniyama-Weil Conjecture comm-darmon.qxp 10/19/99 3:17 PM Page 1397 Research News A Proof of the Full Shimura- Taniyama-Weil Conjecture Is Announced Henri Darmon On June 23, 1993, Andrew Wiles 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, Fred Diamond, 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 elliptic curve 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 mathematics 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 DECEMBER 1999 NOTICES OF THE AMS 1397 comm-darmon.qxp 10/19/99 3:17 PM Page 1398 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 number theory 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, 1398 NOTICES OF THE AMS VOLUME 46, NUMBER 11 comm-darmon.qxp 10/19/99 3:17 PM Page 1399 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.
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