{
  "abstract": "FINDING: The Birch and Swinnerton-Dyer (BSD) conjecture links the algebraic rank of an elliptic curve (number of independent rational points) to the order of vanishing of its L-function at s=1, with recent progress on the 2-part for quadratic twists. | MATH: BSD conjecture: \\( \\text{ord}_{s=1} L(E,s) = \\text{rank}(E(\\mathbb{Q})) \\). The refined BSD formula: \\( \\frac{L^{(r)}(E,1)}{r! \\Omega_E} = \\frac{|\\text{Ш}(E)|\\cdot \\prod_p c_p \\cdot \\text{Tamagawa}}{\\#E(\\mathbb{Q})_{\\text{tors}}^2} \\). Key constants: \\( \\Omega_E \\) (real period), \\( c_p \\) (Tamagawa numbers), \\( \\text{Ш}(E) \\) (Tate-Shafarevich group). Recent result (arXiv:1712.01271): for a large class of elliptic curves, explicit infinite families of quadratic twists with analytic rank 0 exist, and the 2-part of BSD holds for many — i.e., \\( \\text{ord}_{s=1} L(E^d,s)=0 \\) implies \\( \\text{Ш}(E^d)[2] \\) is finite and its size matches the predicted value. | CONNECTION: Elliptic curves are 1-dimensional abelian varieties — their rat\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/ejtz7txx9fsnk.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23446956",
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  "id": "ejtz7txx9fsnk",
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  "landing_url": "https://zenodo.org/records/23446956",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
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  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ejtz7txx9fsnk.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T12:21:49Z",
      "native_id": "23446956",
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  "subjects": [],
  "title": "Refined BSD Conjecture: Linking Elliptic Curve Rank to L-Function Vanishing — E8 Intelligence Research",
  "title_text": "Refined BSD Conjecture: Linking Elliptic Curve Rank to L-Function Vanishing — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-01T12:21:49Z",
  "url": "https://research.draconitic.com/paper/ejtz7txx9fsnk",
  "venue": "Zenodo",
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}