The Distribution of 𝔽q-Points on Cyclic ℓ-Covers of Genusg

DOI10.1093/IMRN/RNV279zbMATH Open1404.11088arXiv1505.07136OpenAlexW2177021943MaRDI QIDQ4560474

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Publication date: 12 December 2018

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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 $qequiv1modell$ 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.

Full work available at URL: https://arxiv.org/abs/1505.07136

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