On the p-adic Riemann hypothesis for the zeta function of divisors

Abstract

In this paper, we continue the investigation of the zeta function of divisors, as introduced by the first author in Wan (in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications, Springer, Berlin, 2001, pp. 437–461; Manuscripta Math. 74 (1992) 413), for a projective variety over a finite field. Assuming that the set of effective divisors in the divisor class group forms a finitely generated monoid, then there are four conjectures about this zeta function: p-adic meromorphic continuation, rank and pole relation, p-adic Riemann hypothesis, and simplicity of zeros and poles. This paper proves all four conjectures when the Chow the group of divisors is of rank one. Also, an example with higher rank is provided where all four conjectures hold.