On the Rationality of Zeta Functions and L-Series

Abstract

Let V be an algebraic variety, defined over GF[q]. We recall the definition of the zeta function of V

$$ Z(V,t) = \exp (\sum\limits_{s = 1}^\infty {{N_s}{t^s}/s} ) $$

where N _s is the number of points of V which are rational over GF[q _s ]. (For definition of L-series, see chapter II). It has been known for some time [1] that the zeta function is rational and a second exposition of the same proof has been given by Serre [2]. It is rather questionable as to whether there is any need to repeat such well known material. However because of its connection with p-adic analysis it may be in accord with the purpose of this conference to outline the old proof. This will be done in chapter I but we will use results and points of view which were not available in 1959.