Distribution of zeta zeroes of Artin--Schreier curves

TL;DR

This paper investigates the distribution of zeros of zeta functions for Artin-Schreier curves over finite fields, showing they follow a Gaussian distribution in large genus limits across different regimes.

Contribution

It provides a rigorous analysis of zeroes distribution for Artin-Schreier zeta functions, establishing Gaussian behavior in both global and mesoscopic regimes as genus increases.

Findings

Zeroes distribution converges to Gaussian in large genus limit

Results hold for both global and mesoscopic regimes

Distribution of zeroes angles becomes normally distributed

Abstract

We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-\pi,\pi)$ has a standard Gaussian distribution (when properly normalized).