Number of Points on the Full Moduli Space of Curves over Finite Fields

TL;DR

This paper advances understanding of the distribution of the number of points on algebraic curves over finite fields by extending previous results to more general abelian Galois groups, specifically prime power cyclic groups.

Contribution

It generalizes prior work by determining the distribution of points on curves with abelian Galois groups that are prime power cyclic, broadening the scope of the moduli space analysis.

Findings

Distribution determined for curves with Galois group as a prime power cyclic group.

Extended previous results from prime cyclic groups to prime power cyclic groups.

Provides a step towards understanding distributions for more general abelian Galois groups.

Abstract

The distribution of the number of points on abelian covers of $\mathbb{P}1(\mathbb{F}_q)$ ranging over an irreducible moduli space has been answered in recent work by the author. Bucur, et al. determined the distribution over the whole moduli space for curves with Gal$(K(C)/K)$ a prime cyclic. In this paper, we prove a result towards determining the distribution over the whole moduli space of curves with Gal$(K(C)/K)$ any abelian group. We successfully determine the distribution in the case Gal$(K(C)/K)$ is a power of a prime cyclic.