{
  "abstract": "We prove the André--Oort conjecture for hypersurfaces $V \\subset Y(1)^n \\cong \\mathbb{A}^n_\\mathbb{C}$ defined by an equation $a_1 x_1^m + \\ldots + a_n x_n^m = b$, where $a_1, \\ldots, a_n, b \\in \\overline{\\mathbb{Q}}$ and $m \\in \\mathbb{Z}_{>0}$. Unlike previous proofs, our result is both effective and uniform in the height of the coefficients $a_1, \\ldots, a_n, b$. This is the first effective proof of a uniform André--Oort statement for a class of subvarieties with arbitrary dimension and non-empty special locus. We also prove an analogous result for hypersurfaces $V \\subset Y(1)^n \\times \\mathbb{G}_m^l$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18934",
  "author_total": 1,
  "authors": [
    {
      "name": "Guy Fowler"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fecwj75hb7jq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18934",
  "fulltext_url": null,
  "id": "0fecwj75hb7jq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18934",
  "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": "W7213507225",
  "pdf_url": "https://arxiv.org/pdf/2609.18934",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fecwj75hb7jq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18934",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18934"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A uniform effective André--Oort result",
  "title_text": "A uniform effective André--Oort result",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/0fecwj75hb7jq",
  "venue": "arXiv",
  "versions": []
}