Distribution of Points on Abelian Covers Over Finite Fields

TL;DR

This paper analyzes the distribution of the number of points on abelian covers of the projective line over finite fields, revealing a sum of $q+1$ random variables as the distribution pattern.

Contribution

It generalizes previous work by explicitly determining the point distribution on abelian covers over finite fields as genus tends to infinity.

Findings

Distribution given by a sum of $q+1$ random variables

Extends prior results to abelian Galois covers

Provides explicit distribution formulas

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.