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

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.