Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field

Abstract

For a function field $k$ over a finite field with $\mathbb{F}_q$ as the field of constant, and a finite abelian group $G$ whose exponent is divisible by $q-1$, we study the distribution of zeta zeroes for a random $G$-extension of $k$, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.