Proportion of ordinarity in some families of curves over finite fields

Abstract

A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an ${\mathbb{F}}_p$ vector space of dimension equal to its genus. In this paper we consider the following question: fix a finite field ${\mathbb{F}}_q$ and a family $\mathcal{F}$ of curves over ${\mathbb{F}}_q$. Then, what is the probability that a curve in this family is ordinary? We answer this question when $\mathcal{F}$ is either the Artin-Schreier family in any characteristic or a superelliptic family in characteristic 2.