Recent progress on the Langlands conjectures
by
Guy Henniart
Univ. of Paris XI, France

It was recently proved that every elliptic curve over Q is modular. That result can be seen as a special case of an outstanding set of conjectures, due to R.P. Langlands. The most elementary result of this kind is the reciprocity law of Gauss. In general, the conjectures say roughly the following : if you consider a system of polynomial equations (in several variables) with integer coefficients, and you compute the number of solutions modulo a prime number p, then when p varies that number exhibits a regularity which can be expressed by harmonic analysis via devices named L-functions, which generalize Riemann's zeta function.

Gauss' reciprocity law concerns the equation X²-n where n is an integer. Modulo a prime number p not dividing n, that equation has 0 or 2 solutions. The reciprocity law implies that the number of solutions only depends on congruences on p modulo 4n, which is rather surprising. Since Gauss such a reciprocity result has been greatly generalized. If for example we consider a polynomial P irreducible in Q[X], and we compute the number of solutions of P(X) \equiv 0 mod p for a large enough prime number p, then that number depends only on congruences on p, provided that the Galois group of P is abelian : this is a consequence of the celebrated class field theory (for the number field Q).

What Langlands proposes is a generalization of class field theory, which would deal with equations in one variable with non abelian Galois group, or even with system of equations in several variables. For example, we can consider an elliptic curve E with equation y²=x³+ax+b, where x³+ax+b in Q[x] has no multiple roots in C. For a prime number p not dividing the denominator of a or b, we can consider the number of solutions of y² \equiv x³+ax+b modulo p ; write it p-a_p (to compare it to the affine line over Z/pZ). Then we can form an L-function L(E, s) which is a product of L-factors L_p(s) where for p large L_p(s) is the inverse of (1-a_pp^-s+p^1-2s). Write L(E, s) in the form \sum_{n >= 1}a_nn^-s, a_n in Z. The Shimura-Taniyama-Weil conjecture, which can be seen as a special case of the Langlands conjectures, says that the function f(z)= \sum_{n >= 0}a_ne^2\piinz, as a function of a complex number z with Im(z) > 0, is a (classical) modular form of weight 2 for a specific subgroup of SL₂(Z). In particular, there is a functional equation f(-1/z)= +/- f(z). Such modular forms can often be constructed by other means, for example using theta-series, and that yields some regularity for the number a_p as a function of p (but in general it is much more subtle than saying that a_p depends only on congruences on p!).

The Shimura-Taniyama-Weil conjecture has now been proved completely by Breuil, Conrad, Diamond and Taylor. Previously, Wiles had settled the conjecture for many elliptic curves over Q, thus proving the celebrated Fermat's ``theorem''.

In the general case, to a system of equations in several variables over Q, or even over a number field F, one would attach, essentially by counting points modulo prime numbers p, L-functions L(s) as above. It is expected (by Langlands) that such L-functions also appear in harmonic analysis, as L-functions attached to automorphic forms, which are representation - theoretic generalizations of classical modular forms : automorphic forms exist for GL_n over F, whereas classical modular forms are attached to GL₂ over Q only. Harris and Taylor have recently shown that some - rather special, but already numerous and very interesting - L-functions coming from automorphic forms are also L-functions coming from systems of equations. This has led to a complete proof of the counterpart of the Langlands conjectures for the p-adic field Q_p and its extensions.

Hopefully there is some more progress to come in the near future.

Date received: May 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-75.