Questions and remarks to the Langlands programme

A. N. Parshin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 67 (2012) no. 3, pp. 509-539

Résumé

A brief survey is given of the classical Langlands programme to construct a correspondence between $n$-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups $\operatorname{GL}(n)$ connected with these fields and their adelic rings. A generalization of the Langlands programme to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. A conjecture on the direct image of automorphic forms is stated which links the Langlands correspondences in dimensions 2 and 1. In the geometric case of surfaces over a finite field the conjecture is shown to follow from Lafforgue's theorem on the existence of a global Langlands correspondence for curves. The direct image conjecture also implies the classical Hasse–Weil conjecture on the analytic behaviour of the zeta- and $L$-functions of curves defined over global fields of dimension 1. Bibliography: 57 titles.

Détail

Keywords: Langlands correspondence, automorphic forms, $L$-functions, two-dimensional local fields, $K$-groups, class field theory
Mots-clés : adeles, direct images.