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

TL;DR

This paper investigates the distribution of zeta zeroes in abelian covers of algebraic curves over finite fields, demonstrating uniform distribution and Gaussian variance as the degree of conductors increases.

Contribution

It establishes the asymptotic uniform distribution of zeta zeroes and Gaussian variance behavior for large-degree abelian covers of algebraic curves over finite fields.

Findings

Zeta zeroes are uniformly distributed in the limit.

Variance of zeroes count follows a Gaussian distribution.

Results apply to G-extensions with exponent divisible by q-1.

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.