Published In
Finite Fields and Their Applications
Document Type
Post-Print
Publication Date
1-1-2026
Abstract
In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus g defined over a finite field Fq with a given a-number. In characteristic three this method gives exact probabilities for curves of the form Y 2 = f(X) with f(X) ∈ Fq[X] 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.
Rights
© Copyright the author(s) 2025
DOI
10.1016/j.ffa.2025.102715
Persistent Identifier
https://archives.pdx.edu/ds/psu/44101
Publisher
Elsevier BV
Citation Details
Garton, D., Thunder, J. L., & Weir, C. (2026). The distribution of a-numbers of hyperelliptic curves in characteristic three. Finite Fields and Their Applications, 109, 102715.
Description
This is the author’s accepted version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as: The distribution of a-numbers of hyperelliptic curves in characteristic three. Finite Fields and Their Applications, 109, 102715.