Distribution of zeta zeroes of Artin--Schreier curves

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).

22 pages

References in corpus (1)

• The distribution of the number of points on trigonal curves over $\F_q$

Cited by in corpus (3)

• Distributions of traces of Frobenius for smooth plane curves over finite fields
• Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field
• Statistics for ordinary Artin-Schreier covers and other $p$-rank strata