{
  "abstract": "Let $\\cX$ be a curve of genus $g\\ge2$ and $p$-rank zero over an algebraically closed field of odd characteristic $p$, and let $G\\le\\Aut(\\cX)$ satisfy $|G|>24g(g-1)$. We prove that $G$ fixes a point unless $(\\cX,G)$ belongs to an explicit list. For $g\\ge4$, the curve is a cyclic quotient of a Hermitian curve, of a Giulietti--Korchmáros curve, or of Skabelund's cover of a Ree curve. We determine the equations and all the groups $G$ that occur, including their central kernels. The exceptional $A_7$ action in characteristic $5$ is on the Hermitian curve of degree $6$; genera $2$ and $3$ are treated separately. The equations are recovered from the group action by two rigidity lemmas for equivariant one-point $p$-group covers of the affine line. If $G$ has no common fixed point and its Sylow $p$-subgroups are noncyclic, then, unless $G$ induces ${}^2G_2(3)$ on the orbit of Sylow fixed points, the bound $|G|>\\max\\{84(g-1),2g^2\\}$ gives a rational quotient with exactly one wild and one tame short orbit, and a rational Sylow quotient.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33185",
  "author_total": 1,
  "authors": [
    {
      "name": "Saeed Tafazolian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0dc5e22d15a75.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33185",
  "fulltext_url": null,
  "id": "0dc5e22d15a75",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33185",
  "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": "W7214806610",
  "pdf_url": "https://arxiv.org/pdf/2609.33185",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0dc5e22d15a75.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.33185",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33185"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A classification of zero $p$-rank curves with large automorphism groups in odd characteristic",
  "title_text": "A classification of zero p-rank curves with large automorphism groups in odd characteristic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0dc5e22d15a75",
  "venue": "arXiv",
  "versions": []
}