{
  "abstract": "Let $X$ be a curve of genus $g\\ge2$ over an algebraically closed field of odd characteristic $p$, and let $G=\\mathrm{Aut}(X)$. A theorem of Giulietti and Korchmaros shows that if $G$ fixes no point and a $p$-subgroup $S\\le G_P$ satisfies $|S|>\\frac{p}{p-1}g$, then $X$ is one of the classical hyperelliptic, Hermitian, or Ree curves. We determine the sharp boundary and the first gap below it. At equality, besides the classical curves, exactly three types occur: the ordinary curves $y^p-y=x+c/x$, the zero-$p$-rank curves $y^3=x^p-x$ for $p\\equiv2\\pmod3$, and, in characteristic $3$, the ordinary genus-six family $y^3-y=c\\left(x+\\frac1{x^3-x}\\right)$. As a key ingredient, we classify zero-$p$-rank curves whose full automorphism group fixes no point and which admit a $p$-subgroup of order larger than the genus. We then prove that, when $|S|>p$, the boundary classification already follows from the weaker inequality $|S|>p(g-p+1)/(p-1)$. For $|S|\\ge p^3$, the forbidden interval above $m=(p-1)|S|/p$ has width at least $p(p-1)/2$, and below $m+(p^2-1)/2$ at most one further genus can occur, with a rigid ramification structure. Finally, we classify the first admissible layer: it occurs only for $|S|=p^2$, and the resulting ordinary curves admit a complete description in terms of two-dimensional Artin--Schreier spaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05539",
  "author_total": 1,
  "authors": [
    {
      "name": "Saeed Tafazolian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwf6vjhzkn1a5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05539",
  "fulltext_url": null,
  "id": "fwf6vjhzkn1a5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05539",
  "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": "W7220702330",
  "pdf_url": "https://arxiv.org/pdf/2610.05539",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwf6vjhzkn1a5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05539",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05539"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Sharp bounds and gap phenomena for large $p$-subgroups of automorphism groups of curves",
  "title_text": "Sharp bounds and gap phenomena for large p-subgroups of automorphism groups of curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/fwf6vjhzkn1a5",
  "venue": "arXiv",
  "versions": []
}