Distribution of Points on Cyclic Curves over Finite Fields

TL;DR

This paper analyzes the distribution of the number of points on cyclic covers of the projective line over finite fields, generalizing previous work and showing the distribution as a sum of random variables.

Contribution

It extends existing results by determining the point distribution on cyclic curves with affine models over finite fields as genus tends to infinity.

Findings

Distribution described by a sum of random variables

Generalizes previous results on different families of curves

Applicable to cyclic covers with affine models

Abstract

We determine in this paper the distribution of the number of points on the cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r = F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is fixed and the genus, $g$, tends to infinity. This generalize the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over $\mathbb{F}_q$. In all cases, the distribution is given by a sum of random variables.