Distribution of zeta zeroes of Artin–Schreier covers

Abstract

We study the distribution of the zeroes of the zeta functions of the family of Artin–Schreier covers of the projective line over Fq 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 [−π,π) has a standard Gaussian distribution (when properly normalized).