I will be presenting Dwork’s proof of rationality of zeta functions.

Abstract: Bernard Dwork surprised the mathematical community by proving the first of the famous Weil conjectures in 1960. His proof is an ingenious application of p-adic analysis. The proof naturally breaks into two parts. He first expresses the number of points of an algebraic variety over finite fields in terms of traces of certain infinite matrices with coefficients in a field extension of the p-adic numbers. From this analysis, he concludes that the zeta function is a ratio of p-adic two entire functions. A p-adic version of a classical theorem by Borel then implies that the zeta function is rational.