Computing zeta functions over finite fields

Abstract

In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a hypersurface, where $p$ is the characteristic of the finite field. In particular, this applies to the problem of counting rational points of an algebraic variety over a finite field.