{
  "abstract": "FINDING: Torsion subgroups of elliptic curves over number fields are finite and highly constrained by the Nagell-Lutz theorem, with deep links to Galois representations and crystallographic-type group actions. | MATH: For an elliptic curve \\(E: y^2 = x^3 + ax + b\\) with \\(a,b \\in \\mathbb{Z}\\), the Nagell-Lutz theorem states: if \\(P=(x,y)\\) is a torsion point of order \\(n>1\\), then \\(x,y \\in \\mathbb{Z}\\) and either \\(y=0\\) (order 2) or \\(y^2 \\mid 4a^3 + 27b^2\\) (the discriminant \\(\\Delta\\)). The torsion subgroup \\(E(\\mathbb{Q})_{\\text{tors}}\\) is finite (Mordell-Weil) and, by Mazur's theorem, is one of 15 possible groups: \\(\\mathbb{Z}/m\\mathbb{Z}\\) for \\(m=1,\\dots,10,12\\), or \\(\\mathbb{Z}/2\\mathbb{Z} \\times \\mathbb{Z}/2m\\mathbb{Z}\\) for \\(m=1,\\dots,4\\). The formal group \\(E(p\\mathbb{Z}_p)\\) has torsion isomorphic to \\(\\mathbb{Z}/p^n\\mathbb{Z}\\) or trivial, depending on the reduction type. | CONNECTION: The finite list of torsion subgroups (orders 1–10, 12) includes the numbers 2, 3, 4,\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher, United Kingdom"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrew Stewart Caldin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7xje58efgjt1z.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23635515",
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  "id": "7xje58efgjt1z",
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  "landing_url": "https://zenodo.org/records/23635515",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
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  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7xje58efgjt1z.ris",
  "sources": [
    {
      "first_seen": "2026-11-08T12:21:54Z",
      "native_id": "23635515",
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      "url": "https://zenodo.org/records/23635515"
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  "title": "Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research",
  "title_text": "Torsion Points on Elliptic Curves: Nagell-Lutz, Galois Actions, and Finite Constraints — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:54Z",
  "url": "https://research.draconitic.com/paper/7xje58efgjt1z",
  "venue": "Zenodo",
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}