Statistics for $p$-ranks of Artin–Schreier covers
Tom 205 / 2022
Acta Arithmetica 205 (2022), 211-226
MSC: Primary 11G20; Secondary 11T06, 11T55, 14G17, 14H25.
DOI: 10.4064/aa220315-13-8
Opublikowany online: 3 October 2022
Streszczenie
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 $.
