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

TL;DR

This paper investigates the zeta function of divisors on higher rank divisor class group projective varieties over finite fields, proving a p-adic meromorphic continuation for a broad class of such varieties, including certain surfaces and toric varieties.

Contribution

It extends known results from rank-one divisor class groups to higher ranks by proving a p-adic meromorphic continuation theorem for a large class of varieties.

Findings

Proved p-adic meromorphic continuation for higher rank divisor class groups.

Established results for projective nonsingular surfaces over finite fields.

Extended the theory to all projective toric varieties, smooth or singular.

Abstract

Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than 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 meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case. In particular, we prove a p-adic meromorphic continuation theorem which applies to a large class of varieties. Examples of such varieties are projective nonsingular surfaces defined over a finite field (whose effective monoid is finitely generated) and all projective toric varieties (smooth or singular).