Statistics for ordinary Artin-Schreier covers and other 𝑝-rank strata

Bucur, Alina; David, Chantal; Feigon, Brooke; Lalín, Matilde

doi:10.1090/tran/6410

Statistics for ordinary Artin-Schreier covers and other $p$-rank strata
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by Alina Bucur, Chantal David, Brooke Feigon and Matilde Lalín PDF

Trans. Amer. Math. Soc. 368 (2016), 2371-2413 Request permission

Abstract:

We study the distribution of the number of points and of the zeroes of the zeta function in different $p$-rank strata of Artin-Schreier covers over $\mathbb {F}_q$ when $q$ is fixed and the genus goes to infinity. The $p$-rank strata considered include the ordinary family, the whole family, and the family of covers with $p$-rank equal to $p-1.$ While the zeta zeroes always approach the standard Gaussian distribution, the number of points over $\mathbb {F}_q$ has a distribution that varies with the specific family.

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