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

TL;DR

This paper investigates the distribution of the number of points on cyclic covers of the projective line over finite fields, showing it converges to a sum of independent random variables as genus increases.

Contribution

It extends previous results by analyzing the distribution over the entire moduli space of cyclic covers, using cyclic function field extensions and ramification counting techniques.

Findings

Distribution converges to a sum of i.i.d. variables

Provides statistical description for all cyclic covers of fixed genus

Relates cyclic covers to cyclic function field extensions

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.