Zeta functions and Dwork Modules

Abstract

Until 1949, studying the number of points on an algebraic variety over a finite field did not seem as worthwhile as it is today, but when André Weil proposed his Weil Conjectures, linking the number of points over a finite field to certain topological properties of an algebraic variety over C via the zeta function, mathematicians have not only occupied themselves with proving these conjectures, but also with finding feasible methods to compute the zeta function and consequently derive the associated topological properties. In this thesis, we will focus on the methods of the proof of rationality of the zeta function, found by Bernard Dwork in 1960. We will study these methods -along with other theories by Dwork- which we can use to derive new ways of computing the zeta function, which allows us to derive a new method to compute the zeta function of a non-degenerate hypersurface.