{
  "abstract": "Let $X$ be a nonsingular projective curve of genus $g\\ge2$ over an algebraically closed field of odd characteristic $p$. Giulietti and Korchmáros established a quadratic genus bound with constant $900$ forcing zero $p$-rank for curves of even genus. We extend this bound to odd genus: for every $g\\ge2$, positive $p$-rank implies $|\\Aut(X)|<900g^2$. Starting from Montanucci's two-short-orbit reduction, we combine ramification estimates and a minimum-genus argument to reduce a counterexample to an almost-simple group acting on a cover of the projective line with two branch points. The main ingredients are an intrinsic description of the second ramification group via Hasse--Arf and genus bounds for intermediate quotients associated with parabolic subgroups. These, together with order and local-structure estimates, exclude every possible simple socle. The proof uses the classification of finite simple groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05562",
  "author_total": 1,
  "authors": [
    {
      "name": "Saeed Tafazolian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ar990va809fjg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05562",
  "fulltext_url": null,
  "id": "ar990va809fjg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05562",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220731139",
  "pdf_url": "https://arxiv.org/pdf/2610.05562",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ar990va809fjg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05562",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05562"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Quadratic Bound for Automorphism Groups of Curves of Positive $p$-Rank in Odd Characteristic",
  "title_text": "A Quadratic Bound for Automorphism Groups of Curves of Positive p-Rank in Odd Characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/ar990va809fjg",
  "venue": "arXiv",
  "versions": []
}