{
  "abstract": "Belyi's theorem asserts that a smooth projective curve over $\\mathbb{C}$ is defined over a number field if and only if it admits a non-constant morphism to $\\mathbb{P}^1$ ramified over at most three points. This article gives a complete, self-contained account of the theorem -- both implications, with all descent machinery proved rather than quoted -- together with a study of the objects the proof produces. Beyond the exposition it contains original results, proved in full. The principal one is a structure theorem for the Belyi polynomials $P_{m,n}(z)=\\frac{(m+n)^{m+n}}{m^{m}n^{n}}z^{m}(1-z)^{n}$: their dessins are double stars, they satisfy the exact identity $P_{m,n}=π_{k}\\circ P_{m/k,n/k}$ with $k=\\gcd(m,n)$ and $π_{k}(w)=w^{k}$, and their monodromy is the wreath product $\\mathfrak{S}_{(m+n)/k}\\wr\\mathbb{Z}/k$, the full symmetric group precisely when $\\gcd(m,n)=1$. Around the sharp lower bounds $d\\geq 2g+1$ (all Belyi maps) and $d\\geq 4g$ (clean maps) we study the extremal maps attaining $d=2g+1$: their monodromy lies in the alternating group, they need not be cyclic -- the smallest non-cyclic one has degree $5$, genus $2$, monodromy $A_{5}$, and is defined over $\\mathbb{Q}$ -- and they satisfy the exact mass formula $\\sum 1/|\\mathrm{Aut}|=2(d-1)!/d(d+1)$, obtained from Boccara's cycle-factorization count -- reproved by a self-contained Frobenius computation -- and verified by complete enumeration for $d\\leq7$. We also bound the degree of the rationalization step of Belyi's algorithm by $N!$ in the number $N$ of irrational branch values. The theory is illustrated by fully computed examples, and by an explicit $G_{\\mathbb{Q}}$-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field $\\mathbb{Q}(\\sqrt[3]{2})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19053",
  "author_total": 1,
  "authors": [
    {
      "name": "Djounvouna Dinamo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3qaybwxvwtkvg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19053",
  "fulltext_url": null,
  "id": "3qaybwxvwtkvg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19053",
  "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": "W7213496749",
  "pdf_url": "https://arxiv.org/pdf/2609.19053",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3qaybwxvwtkvg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.19053",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19053"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Belyi's theorem: coverings, dessins, and fields of definition",
  "title_text": "Belyi's theorem: coverings, dessins, and fields of definition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/3qaybwxvwtkvg",
  "venue": "arXiv",
  "versions": []
}