Distribution of Points on Abelian Covers Over Finite Fields

Abstract

We determine in this paper the distribution of the number of points on the covers of ${\mathbb{P}}^1({\mathbb{F}}_q)$ such that $K(C)$ is a Galois extension and $\mbox{Gal}(K(C)/K)$ is abelian 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 ${\mathbb{F}}_q$. In all cases, the distribution is given by a sum of $q+1$ random variables.

References in corpus (1)

• Statistics for traces of cyclic trigonal curves over finite fields