Rationality and meromorphy of zeta functions

This article is all about two theorems on equations over finite fields which have been proved in the past decade. First, the finiteness of the rigid cohomology of a variety over a finite field. Second, the p-adic meromorphy of the unit root zeta function of a family of varieties over a finite field of characteristic p. 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 field, but no especial expertise on the vast literature which surrounds the topic of equations over finite filelds.