{
  "abstract": "Motivated by Schoof-Pila-type point-counting algorithms, we study the fields of definition and factorization patterns of Galois orbits of $\\ell$-torsion points on the Jacobians of genus-3 hyperelliptic curves over finite fields. We show that the degree of the field of definition of the $\\ell$-torsion points can be bounded by $O(\\ell^4)$, improving the previously expected $O(\\ell^6)$ bound (which reflects the size of the $\\ell$-torsion subgroup, of order $\\ell^6$ for a genus 3 Jacobian). Moreover, we establish a precise correspondence between the rank of the $\\ell$-torsion subgroup and the Galois orbits of $\\ell$-torsion divisors. Our approach relies on a detailed analysis of the Jordan decomposition of the Frobenius action on $J$ and its nilpotent part.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03171",
  "author_total": 2,
  "authors": [
    {
      "name": "Amalia Pizarro-Madariaga"
    },
    {
      "name": "Edgardo Riquelme"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2fh546qpdq5g4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03171",
  "fulltext_url": null,
  "id": "2fh546qpdq5g4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03171",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7208742461",
  "pdf_url": "https://arxiv.org/pdf/2609.03171",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2fh546qpdq5g4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03171",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03171"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Factorization patterns and fields of definition of $\\ell$-torsion points on the Jacobians of genus 3 hyperelliptic curves",
  "title_text": "Factorization patterns and fields of definition of ℓ-torsion points on the Jacobians of genus 3 hyperelliptic curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/2fh546qpdq5g4",
  "venue": "arXiv",
  "versions": []
}