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arXiv · 1505.07136

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

Alina Bucur·Chantal David·Brooke Feigon·Nathan Kaplan·Matilde Lalín·Ekin Ozman·Melanie Matchett Wood
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Abstract

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.

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Alina Bucur, Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalín, Ekin Ozman, Melanie Matchett Wood. 2015-05-26. The distribution of $\mathbb{F}_q$-points on cyclic $\ell$-covers of genus $g$. https://arxiv.org/abs/1505.07136

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