Harmonic analysis on discrete groups and analytic properties of zeta-functions of algebraic varieties

Let X be an algebraic variety defined over a finite field and let us consider its zeta-function defined by the Euler product. The Grothendieck cohomological method solves two main problems: analytic continuation of zeta-function of X to the whole s-plane and existence of a functional equation. If X is an algebraic curve then the same problems can be solved by the adelic method developed by Tate and Iwasawa. In general, zeta-function of X can be written as a sum over the discrete group of 0-cycles on X. In the talk, we show how to develop a harmonic analysis on this group for curves and apply it to study the zeta-functions. Next, we describe what can be done for algebraic surfaces along these lines.