Partial Zeta Functions of Algebraic Varieties over Finite Fields

Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety defined over a finite field. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil type conjectures. The first approach, using an inductive fibred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork’s rationality theorem. The second approach, due to Faltings, shows that the partial zeta function is always nearly rational.