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Proof of the Taniyama-Shimura Conjecture
(From December 1999 Notices of the AMS)
On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, Jean-Pierre Serre, and Kenneth Ribet, this was known to imply Fermat's Last Theorem. Six years later Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally announced a proof of the full Shimura-Taniyama-Weil conjecture for all elliptic curves over Q.
The Shimura-Taniyama-Weil conjecture relates elliptic curves (cubic equations in two variables of the form
An elliptic curve E can be made into an abelian group in a natural way after adjoining to it an extra "solution at infinity" that plays the role of the identity element. This is what makes elliptic curves worthy of special study, for they alone, among all projective curves (equations in two variables, compactified by the adjunction of suitable points at infinity) are endowed with such a natural group law. If one views solutions geometrically as points in the
After a change of variables is performed to bring it into the best possible form, the equation defining E can be reduced modulo any prime number p. If the resulting equation is nonsingular over the finite field with p elements
y2 = x3 - x2 + 1/4
y2 + y = x3 - x2,
has good reduction at all primes except 11.
Let Np be the number of solutions (over Fp) of the reduced equation, and set
It has been a long-standing concern of number theory to search for patterns satisfied by sequences of this sort. For example, in the simpler case of the quadratic equation in one variable
In the case of elliptic curves, a similar pattern arises. It is, however, a good deal more subtle-so much so that it emerged as a precise conjecture only in the 1950s through the work of Shimura, Taniyama, and Weil. This pattern involves the notion of a modular form of weight two: an analytic function on the complex upper half-plane
f((az + b)/(cz + d)) = (cz + d)2f(z),
where is an appropriate "congruence subgroup" of
f(z) = an(f)qn, where q = e2iz.
Of particular interest are the so-called "cusp forms" satisfying a more stringent growth condition at the boundary that implies, in particular, that
ap(E) = ap(f),
for all primes p of good reduction for E. When this is the case, the curve E is said to be modular. The conjecture also predicts the precise value of N: it should be equal to the "conductor" of E, an arithmetically defined quantity that measures the Diophantine complexity of the associated cubic equation. Its prime divisors are precisely the primes of bad reduction of E. If p divides N but p2 does not, then E is said to have semistable reduction at p. In particular, E has semistable reduction at all primes p (i.e., is semistable) precisely when N is square-free.
For instance, the elliptic curve of equation (1) has conductor 11 (and thus is an example of a semistable elliptic curve). It turns out that the space of weight two cusp forms of level 11 is one-dimensional and is spanned by the function
q = (1 - qn)2 · (1 - q11n)2
= q - 2q2 - q3 + 2q4 + q5 + 2q6 - 2q7
- 2q9 - 2q10 + q11 - 2q12 + 4q13 + 4q14
- q15 - 4q16 - 2q17 + 4q18 + 2q20 + 2q21
- 2q22 - q23 - 4q25 - 8q26 + 5q27 - 4q28
+ 2q30 + 7q31 + ... + 18q10007 + ...
The reader will note that the Fourier coefficients of this function agree with the numbers computed, by wholly different methods, in Table 1.
The Shimura-Taniyama-Weil conjecture was widely believed to be unbreachable, until the summer of 1993, when Wiles announced a proof that every semistable elliptic curve is modular. A full proof of this result appeared in 1994 in the two articles [W] and [TW], the second joint with Taylor.Shortly afterwards, Diamond [Di1] was able to remove the semistability assumption in Wiles's argument at all the primes except 3 and 5. Then, in 1998 Conrad, Diamond, and Taylor [CDT] refined the techniques still further, establishing the Shimura-Taniyama-Weil conjecture for all elliptic curves whose conductor is not divisible by 27. This is where matters stood at the start of the summer of 1999, before the announcement of Breuil, Conrad, Diamond, and Taylor.
The Importance of the Conjecture
The Shimura-Taniyama-Weil conjecture and its subsequent, just-completed proof stand as a crowning achievement of number theory in the twentieth century. This statement can be defended on (at least) three levels.
Fermat's Last Theorem
Firstly, the Shimura-Taniyama-Weil conjecture implies Fermat's Last Theorem. This is surprising at first, because the equation
Because the Frey curve is semistable, the original result of [W] and [TW] is enough to imply Fermat's Last Theorem, and the new result of Breuil, Conrad, Diamond, and Taylor yields nothing new on Fermat's equation. It does imply, however, other results of the same nature, such as the statement that a perfect cube cannot be written as a sum of two relatively prime nth powers with
The Arithmetic of Elliptic Curves
Secondly, and more centrally perhaps, the Shimura-Taniyama-Weil conjecture lies at the heart of the theory of elliptic curves.
A theorem of Mordell asserts that the abelian group, denoted
It has been a long-standing feeling that much information on the arithmetic of E (such as the invariant r) can be gleaned from the sequence
L(E,s) := (1-ap(E)p-s + p1-2s)-1.
(In the later parts of the theory, elementary factors are included in the product for the finitely many primes p dividing N.) This product converges when
L(E,1) = .
It is believed that the size of r might affect the size of Np on average, which may in turn be reflected in the analytic behavior of
ords=1L(E,s) = r
This conjecture is of fundamental importance for the arithmetic of elliptic curves and is still far from being settled, although the work of Gross-Zagier and Kolyvagin shows that it is true when
Knowing that E is modular also gives control on the arithmetic of E in other ways, by allowing the construction of certain global points on E defined over abelian extensions of quadratic imaginary fields via the theory of complex multiplication. Such analytic constructions of global points on E actually play an important role in studying the Birch and Swinnerton-Dyer conjecture through the work of Gross-Zagier and of Kolyvagin.
The Langlands Program
A Galois representation is a (finite-dimensional) representation
Wiles's work can be viewed in the broader perspective of establishing connections between automorphic forms-objects arising in the (infinite-dimensional) representation theory of adelic groups-and Galois representations. Viewed in this light, it becomes part of a vast conjectural edifice put together by Langlands, based on earlier insights of Tate, Shimura, Taniyama, and many others. In this setting, Wiles's discoveries have enriched the theory with a powerful new method that should keep the experts occupied well into the new millennium. Indeed, the impact of Wiles's ideas has only started being felt in many diverse aspects of the Langlands program:
Using Wiles's method, Taylor has formulated a novel strategy [Ta] for proving the Artin conjecture in the remaining (most interesting) case where the image of in
Generalizations to other number fields. A number of ingredients in Wiles's method have been significantly simplified, by Diamond and Fujiwara among others. Fujiwara, Skinner, and Wiles have been able to extend Wiles's results to the case where the field Q is replaced by a totally real number field K. In particular, this yields analogues of the Shimura-Taniyama-Weil conjecture for a large class of elliptic curves defined over such a field.
n-dimensional generalizations. Michael Harris and Richard Taylor have explored generalizations of the main results of [W] and especially [TW] to the context of n-dimensional representations of GQ. (This work, as well as the proof of the local Langlands conjecture for GLn by Harris and Taylor, is expected to be covered in a future Notices article.)
The Work of Breuil, Conrad, Diamond, and Taylor
In general, the Fourier coefficients of a normalized eigenform f are algebraic numbers defined over a finite extension
f: Gal(Q/Q) GL2(Q)
When f is an eigenform with rational Fourier coefficients corresponding to an elliptic curve Ef under the original Eichler-Shimura construction, then f is simply obtained by piecing together the natural action of GQ on the space of n-torsion points of
It becomes natural to formulate a more general version of the Shimura-Taniyama-Weil conjecture, replacing elliptic curves with two-dimensional representations of GQ with coefficients in Q. This more general version would have the virtue of avoiding the subtleties associated with fields of definition of Fourier coefficients of eigenforms.
An important insight that emerged over the last decades through the work of Alexander Grothendieck, Pierre Cartier, Jean Dieudonné, and finally Jean-Marc Fontaine and his school is that it should be possible to characterize the -adic representations arising from modular forms entirely in Galois-theoretic terms-or, more precisely, in terms of their restriction to a "decomposition group"
The main tools in controlling the size of mod are supplied by the theory of "Hecke rings" and congruences between modular forms, a rich body of techniques developed by Mazur, Hida, and Ribet and used to great effect by Ribet to derive Fermat's Last Theorem from the Shimura-Taniyama-Weil conjecture.
o The theory of "base change" and in particular the work of Langlands and Tunnell on solvable base change.
o The theory of deformations of Galois representations 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 nonsolvable 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 accident of group theory: the group
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 restricted to the inertia group at 3-and a precise description
and understanding of this behavior are required to control the set
[CDT]BRIAN CONRAD, FRED DIAMOND, and RICHARD TAYLOR, Modularity of certain potentially Barsotti-Tate Galois 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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