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Statistics for $p$-ranks of Artin–Schreier covers

Volume 205 / 2022

Anwesh Ray Acta Arithmetica 205 (2022), 211-226 MSC: Primary 11G20; Secondary 11T06, 11T55, 14G17, 14H25. DOI: 10.4064/aa220315-13-8 Published online: 3 October 2022


Given a prime $p$ and $q$ a power of $p$, we study the statistics of $p$-ranks of Artin–Schreier covers of given genus defined over $\mathbb F_q$, in the large $q$-limit. We refer to this problem as the geometric problem. We also study an arithmetic variation of this problem, and consider Artin–Schreier covers defined over $\mathbb F_p$, letting $p$ go to infinity. Distribution of $p$-ranks has previously been studied for Artin–Schreier covers over a fixed finite field as the genus is allowed to go to infinity. The method requires that we count isomorphism classes of covers that are unramified at $\infty $.


  • Anwesh RayDepartment of Mathematics
    University of British Columbia
    Vancouver, BC, Canada V6T 1Z2

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