Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves

Abstract

Let ${\mathcal{H}}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p\ge 3$, and let ${\mathcal{H}}_g^f$ denote the $p$-rank $f$ stratum of ${\mathcal{H}}_g$ for $0\le f\le g$. Achter and Pries note in their 2011 work that determining the number of irreducible components of ${\mathcal{H}}_g^f$ would lead to several intriguing corollaries. In this paper, we present a computational approach for estimating the number of irreducible components in various $p$-rank strata. Our strategy involves sampling curves over finite fields and calculating their $p$-ranks. From the data gathered, we conjecture that the non-ordinary locus is geometrically irreducible for all genera $g>1$. The data also leads us to conjecture that the moduli space ${\mathcal{H}}_g^{g-2}$ is irreducible and suggests that ${\mathcal{H}}_g^f$ is irreducible for all $1\le f\le g$. We conclude with a brief discussion on ${\mathcal{H}}_g^0$.

19 pages, 3 figures, 5 tables, comments welcome