Distribution of Points on Cyclic Curves over Finite Fields

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}$-powerfreewhen$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.