The distribution of a-numbers of hyperelliptic curves in characteristic three

In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus g defined over a finite field $F_q$ with a given a-number. In characteristic three this method gives exact probabilities for curves of the form $Y^2=f\left(X\right)$ with $f\left(X\right)\in F_q\left[X\right]$ monic and cubefree—probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of q) for these probabilities when restricting to squarefree f. As a consequence, for positive integers a and g we show that the nonempty strata of the moduli space of hyperelliptic curves of genus g consisting of those curves with a-number a are of codimension $2a-1$. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are $a(a+1)/2$. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.—one that matches the available data.