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# Mathematics > Number Theory

# Title: Statistics for p-ranks of Artin-Schreier covers

(Submitted on 17 Oct 2021 (v1), last revised 21 Oct 2021 (this version, v2))

Abstract: 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 been previously studied for a 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 $\infty$.

## Submission history

From: Anwesh Ray [view email]**[v1]**Sun, 17 Oct 2021 04:13:20 GMT (12kb)

**[v2]**Thu, 21 Oct 2021 19:56:29 GMT (13kb)

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