Zeta functions over finite fields- An exposition of Dwork's methods

TL;DR

This paper explains Dwork's p-adic analytic techniques used in proving the rationality part of the Weil Conjectures, including cohomological methods and an application to hypersurfaces.

Contribution

It provides an exposition of Dwork's original p-adic cohomological methods and demonstrates their application to specific algebraic varieties.

Findings

Proof of rationality of zeta functions over finite fields

Development of p-adic cohomological techniques

Application to Dwork's family of hypersurfaces

Abstract

The paper reviews Dwork's p-adic analytic methods used in the Weil Conjectures. The first two chapters review a version of his proof of the rationality conjecture. The rest of the paper is devoted to Dwork's original cohomological methods, along with some p-adic functional analysis. The last chapter applies what is developed to an example of Dwork's family of hypersurfaces.