Questions and Remarks to the Langlands Program

Abstract

A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the Langlands program to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. We formulate a conjecture on a direct image (=automorphic induction) of automorphic forms which links the Langlands correspondences in dimension 2 and 1. The direct image conjecture implies the classical Hasse-Weil conjecture on the analytical behaviour of the L-functions of curves defined over global fields of dimension 1.