The distribution of \(\mathbb{F}_q\)-points on cyclic \(\ell\)-covers of genus \(g\)

We study fluctuations in the number of points of \(\ell\)-cyclic covers of the projective line over the finite field \(\mathbb{F}_q\) when \(q \equiv 1 \mod \ell\) is fixed and the genus tends to infinity. The distribution is given as a sum of \(q+1\) i.i.d. random variables. This was settled for hy...

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Bibliographic Details
Published in:arXiv.org 2015-05
Main Authors: Bucur, Alina, David, Chantal, Feigon, Brooke, Kaplan, Nathan, Lalín, Matilde, Ozman, Ekin, Melanie Matchett Wood
Format: Article
Language:English
Subjects:
Fields (mathematics)
Random variables
Variations
Online Access:Get full text
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Summary:We study fluctuations in the number of points of \(\ell\)-cyclic covers of the projective line over the finite field \(\mathbb{F}_q\) when \(q \equiv 1 \mod \ell\) is fixed and the genus tends to infinity. The distribution is given as a sum of \(q+1\) i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of \(\ell\)-cyclic covers by Bucur, David, Feigon and Lal\'{i}n. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of \(\ell\)-cyclic covers of genus \(g\). This is achieved by relating \(\ell\)-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes.
ISSN:2331-8422