{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/heuristics-for-ir-reducibility-of-p-rank","title":"Heuristics for (ir)reducibility of $p$-rank strata of the moduli space of hyperelliptic curves","arxiv_id":"2506.06457","date":"2025-06-06","proceeding":null,"authors":["Thomas Bouchet","Erik Davis","Steven R. Groen","Zachary Porat","Benjamin York"],"abstract":"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 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 $\\mathcal{H}^{g-2}_g$ is irreducible and suggests that $\\mathcal{H}^f_g$ is irreducible for all $1 \\leq f \\leq g$. We conclude with a brief discussion on $\\mathcal{H}^0_g$.","url_abs":"https://arxiv.org/abs/2506.06457v1","url_pdf":"https://arxiv.org/pdf/2506.06457v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"heuristics-for-ir-reducibility-of-p-rank","repo_url":"https://github.com/thittho/non-ordinary-locus","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}