Hasse-Weil Zeta Functions Modulo a Prime

Abstract

Let ${\mathbb{F}}_q$ be a finite field of characteristic $p$ and $\ddot{I}€:Y\to X$ be a finite ${\mathbb{F}}_q$-morphism of separated ${\mathbb{F}}_q$-schemes of finite type. Suppose $\ddot{I}€$ is generically Galois with group $G$ of prime order $r\ne p$. We determine the mod-$r$ reduction of the zeta function of $Y$ in terms of the zeta function of $X$ and the branch locus $Z\subset X$ of $\ddot{I}€$. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.

Comments welcome