{
  "abstract": "The factorizations over $\\mathbb{F}_p$ of two supersingular polynomials $h_p(x)$ and $g_p(x)$ for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo $p$ to the Jacobi polynomials $P_n^{(α,0)}(1-2x)$, for $α= \\pm 1/4, \\pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\\mathbb{Q}(\\sqrt{-dp})$, where $d \\in \\{1,2,3\\}$. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials $P_n(x)$; previous results on linear and binomial quadratic factors of $P_{(p-e)/4}(x)$ and $P_{(p-\\bar e)/3}(x)$ proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations $H_{-3p}(X)$ and $H_{-12p}(X)$ (mod $p$). The quadratic transformation and linear factor results for $h_p(x) \\equiv P_n^{(\\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.21198",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrew Karl"
    },
    {
      "name": "Patrick Morton"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cbexbdtqfbfzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.21198",
  "fulltext_url": null,
  "id": "cbexbdtqfbfzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.21198",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204143679",
  "pdf_url": "https://arxiv.org/pdf/2608.21198",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cbexbdtqfbfzv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.21198",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.21198"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials",
  "title_text": "Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/cbexbdtqfbfzv",
  "venue": "arXiv",
  "versions": []
}