Aleksei Nikolaevich Parshin

Harmonic analysis on local fields and adelic spaces. II

We develop harmonic analysis in certain categories of filtered Abelian groups and vector spaces. The objects of these categories include local fields and adelic spaces arising from arithmetic surfaces. We prove some structure theorems for quotients of the adele groups of algebraic and arithmetic surfaces.

Questions and remarks to the Langlands programme

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 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 Lafforgues 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.