Mathematical Research Letters

Volume 19 (2012)

Number 6

Distribution of zeta zeroes of Artin–Schreier covers

Pages: 1329 – 1356



Alina Bucur (Department of Mathematics, University of California at San Diego)

Chantal David (Department of Mathematics and Statistics, Concordia University, Montreal, Canada)

Brooke Feigon (Department of Mathematics, The City College of New York, CUNY, New York, U.S.A.)

Matilde Lalín (Département de mathématiques et de statistique, Université de Montréal, Montreal, Canada)

Kaneenika Sinha (Department of Mathematics, Indian Institute of Science Education and Research (IISER), Pune, Maharashtra, India)


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 $[−π, π)$ has a standard Gaussian distribution (when properly normalized).

2010 Mathematics Subject Classification

Primary 11G20. Secondary 11M50, 14G15.

Published 18 July 2013