{
  "abstract": "A Bely\\uı function on a smooth projective algebraic curve defined over a number field determines a bipartite graph called a dessin d'enfant. We study the regularity and automorphism groups of dessins with uniform passports. In previous papers, we proved that every passport of the form $[n,n,n]$, $[n,b^{q},n]$, or $[b^{q},b^{q},n]$ of genus at least $2$ admits a dessin with trivial automorphism group. In this paper, we prove the analogous result for passports of the form $[a^{p},b^{q},n]$. The proof is mainly based on a counting argument: we compare a lower bound for the number of permutation representations having the prescribed passport with an upper bound for the number admitting a nontrivial automorphism. Together with our previous results, this shows that every uniform unicellular passport of genus at least $2$ admits a dessin with trivial automorphism group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33602",
  "author_total": 1,
  "authors": [
    {
      "name": "Tatsuya Ohnishi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4k6bfx0haehy9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33602",
  "fulltext_url": null,
  "id": "4k6bfx0haehy9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33602",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214751685",
  "pdf_url": "https://arxiv.org/pdf/2609.33602",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4k6bfx0haehy9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33602",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33602"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups",
  "title_text": "Uniform Unicellular Dessins d'Enfants with Trivial Automorphism Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/4k6bfx0haehy9",
  "venue": "arXiv",
  "versions": []
}