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Given a projective variety X defined over a finite field, the zeta functionof divisors attempts to count all irreducible, codimension one subvarieties ofX, each measured by their projective degree. When the dimension of X is greaterthan one, this is a purely p-adic function, convergent on the open unit disk.Four conjectures are expected to hold, the first of which is p-adic meromorphiccontinuation to all of C_p.  When the divisor class group (divisors modulo linear equivalence) of X hasrank one, then all four conjectures are known to be true. In this paper, wediscuss the higher rank case. In particular, we prove a p-adic meromorphiccontinuation theorem which applies to a large class of varieties. Examples ofsuch varieties are projective nonsingular surfaces defined over a finite field(whose effective monoid is finitely generated) and all projective toricvarieties (smooth or singular).

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