{
  "abstract": "Let $\\mathcal{X}$ be a projective, geometrically irreducible, nonsingular algebraic curve of genus $g\\ge2$ over an algebraically closed field of characteristic two. We classify cyclic subgroups of $\\Aut(\\mathcal{X})$ of order $N\\ge2g+1$. Besides Kummer extensions of odd degree ramified over three points, precisely two cases occur: $y^2+y=x^m$, with $N=2m=4g+2$, and $y^2+y=c/(x^m+1)$, with $m$ odd, $c\\ne0$, and $N=2m=2g+2$. In particular, $4\\nmid N$. We then classify groups $H$ with a cyclic subgroup of index two and $|H|>4g+4$. They are precisely the groups $C_k\\times D_{2M}$ on the curves $y^k=x^M+x^{-M}$, where $k,M\\ge3$ are odd and coprime, and $2g=M(k-1)$. Their possible orders are $4g+2M$, where $M$ runs over certain odd divisors of $g$, and $|H|\\le6g$. No curve in this family is ordinary. Hermitian curves occur in the family, and we give two sufficient conditions for maximality over finite fields. For dihedral groups the sharp bound is $4g+4$ in even genus and $4g$ in odd genus. The equality cases are given by explicit Artin--Schreier equations, and in each case the dihedral group is the full automorphism group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34709",
  "author_total": 1,
  "authors": [
    {
      "name": "Marco Timpanella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0139knj6d5p7g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34709",
  "fulltext_url": null,
  "id": "0139knj6d5p7g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34709",
  "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": "W7214748793",
  "pdf_url": "https://arxiv.org/pdf/2609.34709",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0139knj6d5p7g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34709",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34709"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two",
  "title_text": "Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0139knj6d5p7g",
  "venue": "arXiv",
  "versions": []
}