Rationality and meromorphy of zeta functions

The purpose of the article is to explain what these theorems mean, and also to give an outline of the proof of the first one. The intended audience is mathematicians with an interest in finite fields, but no especial expertise on the vast literature which surrounds the topic of equations over finite fields. By way of motivation, we will begin by giving an indication of the historical significance of these two theorems, before giving more formal definitions in Section 2. The basic object of interest to us is a system of polynomial equations over a finite field. Loosely speaking, this is called a variety. Given such a system, one can encode the number of solutions over different finite extension of the base field in a generating function. This is the zeta function of the variety. In the late 1950s Dwork proved that this generating function is always a rational function. Weil had conjectured this some ten years earlier, and conceived a plan for proving it based upon an as yet unknown cohomology theory for varieties over finite fields. Such a theory would associate a vector space with a variety over a finite field, and the rationality of zeta functions would follow from the finite dimensionality of these vector spaces. To everyone’s surprise, Dwork proved rationality without constructing such a theory. He proved instead that the zeta function was meromorphic as a p-adic function, and then deduced that it must be rational. During the next decade Dwork’s work inspired the construction of a true cohomology theory based upon p-adic analysis. Unfortunately though,