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 P1(Fq) with affine models C:Yr=F(X), where F(X)∈Fq[X] and rth-power free when q is fixed and the genus, g, tends to infinity. This generalizes the work of Kurlberg and Rudnick and Bucur, David, Feigon and Lalin who considered different families of curves over Fq. In all cases, the distribution is given by a sum of random variables.