On the zeta function of divisors for projective varieties with higher rank divisor class group

Given a projective variety X defined over a finite field, the zeta function of divisors counts all irreducible, codimension one subvarieties of X, each measured by their projective degree. For dim(X) ≥ 2, this is a purely p-adic function. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to the all of Cp. When the divisor class group of X has rank one, then all four conjectures are known to be true. In this paper, we prove a p-adic meromorphic continuation theorem for the zeta function of divisors which applies to a large class of varieties with higher rank divisor class group; for instance, all projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular) lie in this class.