Rationality of the zeta-function over a finite field

Abstract:

Suppose that we have a multivariate polynomial f defined over a finite field with q elements. We can count the number of roots of f over finite field extensions containing q2 elements, q3 elements and onwards. Denote by Ns the number of roots of f in the field with qs elements. The zeta-function is a power series which expresses Ns in terms of its coefficients.

Weil conjectured interesting properties about the zeta-function which have since been proven. In particular, Dwork showed that the zeta-function is a rational function, the quotient of two polynomials, using p-adic analysis. This tells us that we need only to find the values Ns up to some bound for s, in order to calculate the zeta-function. From this we can find Ns for any s.

We will discuss the ideas behind Dworks proof and the role that p-adic analysis plays in his argument.