Pré-Publication, Document De Travail Année : 2025

HEURISTICS FOR (IR)REDUCIBILITY OF p-RANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES

Résumé

Let $\mathcal{H}_g$ denote the moduli space of smooth hyperelliptic curves of genus $g$ in characteristic $p \geq 3$, and let $\mathcal{H}_g^f$ denote the $p$-rank $f$ stratum of $\mathcal{H}_g$ for $0 \leq f \leq g$. Achter and Pries note 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 $1 \leq f \leq g$. We conclude with a brief discussion on $\mathcal{H}_g^0$ .

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Dates et versions

hal-05102923 , version 1 (07-06-2025)

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Thomas Bouchet, Erik Davis, Zachary Porat, Steven Groen, Benjamin York. HEURISTICS FOR (IR)REDUCIBILITY OF p-RANK STRATA OF THE MODULI SPACE OF HYPERELLIPTIC CURVES. 2025. ⟨hal-05102923⟩
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