Recent Advances in the Langlands Program

The Langlands Program has emerged in the late 60’s in the form of a series of far-reaching conjectures tying together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic forms [L1]. To motivate it, consider the old question from number theory: what is the structure of the Galois group Gal(Q/Q) of the field Q of rational numbers, i.e., the group of automorphisms of an algebraic closure Q of Q which fix Q ⊂ Q? The answer is still unknown, but the classical Kronecker-Weber theorem gives a complete description of the maximal abelian quotient of Gal(Q/Q). This theorem states that the maximal extension of Q, whose Galois group is abelian, is obtained by adjoining to Q all roots of unity. Therefore the maximal abelian quotient of Gal(Q/Q) is isomorphic to the projective limit of the groups of units of the rings Z/NZ (which are the Galois groups of the cyclotomic fields Q(ζN ), where ζN is a primitive Nth root of 1). This projective limit is nothing but the direct product of the multiplicative groups of p–adic integers, Zp , where p runs over the set of all primes. The abelian class field theory gives a similar description for the maximal abelian quotient Gal(F ab/F ) of the Galois group Gal(F/F ), where F is an arbitrary global field, i.e., a finite extension of Q (number field), or the field of rational functions on a smooth projective curve defined over a finite field (function field). Namely, Gal(F ab/F ) is isomorphic to the profinite completion of the quotient F\A×F , where AF is the ring of adèls of F , a subring in the direct product of all completions of F (see Section 1). Thus, when F = Q, the ring AQ is a subring of the direct product of the fields Qp of p–adic numbers and the field R of real numbers, so that the profinite completion of the quotient F\AF is indeed isomorphic to ∏