CONFERINTA
Marti, 27 octombrie 2009, ora 12:15
cladirea Mathematicum, sala π

How to count points on singular surfaces over finite fields

Speaker

Prof. dr. Christopher Deninger (Universität Münster)

Abstract

For a smooth projective variety X over a finite field F the zeta function is a generating series for the numbers of points of X in the finite extension fields of F. If X is given by system of polynomial equations in several variables over F, a point of X in an extension field F’  is a solution of the equations with coordinates in F’. Thus the zeta function encodes the number of solutions of equations over finite fields which is very interesting from a number theoretical point of view. One can think of the condition that X is smooth and projective as being analogous to being a compact manifold. A part of the famous Weil-conjectures which was proved by Grothendieck and Verdier asserts that the zeta function is in fact rational and has a simple functional equation. If the variety is no longer smooth, i.e. if it has singularities the zeta function obtained by counting points naively is still rational but it no longer has a functional equation. The talk explains with what weights one has to count the singular points -at least on surfaces- in order to obtain a new zeta function which is again rational but does have a functional equation. I will try to explain all concepts in the talk and also the required ideas from etale (intersection) cohomology.