Hasse-Weil Zeta Functions Modulo a Prime

Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and $\pi\colon Y\to X$ be a finite $\mathbb{F}_q$-morphism of separated $\mathbb{F}_q$-schemes of finite type. Suppose $\pi$ is generically Galois with group $G$ of prime order $r\neq p$. 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 $\pi$. We give applications to curves and to numerators of hyperelliptic/superelliptic curves.