Distribution of Points on Cyclic Curves over Finite Fields

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.