{
  "abstract": "This paper studies the Birch-Swinnerton-Dyer (BSD) conjecture for elliptic curves E defined over the rational number field Q. The central focus is the construction of a general structure arising from elliptic curves, followed by transformations that lead to a symmetry comparison between the analytic and algebraic sides. The results are established within the scope of the constructed structure and the specified symmetry-compatibility condition.\n\nDefine the analytic rank and algebraic rank, respectively, by\n\nm = ord_(s=1) L(E,s)\n\nr = rank E(Q).\n\nThe BSD rank conjecture asserts that\n\nFor every elliptic curve E/Q, ord_(s=1) L(E,s) = rank E(Q).\n\nFor analytic ranks zero and one, the paper presents the established results in BSD theory.\n\nCase m = 0:\n\nFor every E/Q, m = 0 implies r = 0.\n\nCase m = 1:\n\nFor every E/Q, m = 1 implies r = 1.\n\nFor the general construction, the paper introduces two complex vector spaces associated with the elliptic curve:\n\nA_E = C{t} / (L(E,1+t))\n\nV_E = E(Q) tensor_Z C.\n\nTheir dimensions are\n\ndim_C A_E = m, dim_C V_E = r.\n\nThe paper examines the symmetry operations\n\ntau([f(t)]) = [f(-t)]\n\niota(v) = -v\n\nand considers whether a complex-linear isomorphism\n\nPhi: A_E -> V_E\n\ncan satisfy the symmetry-compatibility condition\n\nPhi o tau = iota o Phi.\n\nThe properties of these operations show that no such symmetry-compatible isomorphism exists when\n\nm >= 1.\n\nThus, the paper describes the rank cases of the Birch-Swinnerton-Dyer conjecture that have been proved for elliptic curves over Q with analytic rank zero or one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Son Tuyet Tran",
      "orcid": "0009-0007-9871-9871"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/393zpfm3h3ytk.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23491736",
  "fulltext_url": null,
  "id": "393zpfm3h3ytk",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23491736",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23491736/files/Birch_Swinnerton_Dyer.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/393zpfm3h3ytk.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:22Z",
      "native_id": "23491736",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23491736"
    }
  ],
  "subjects": [],
  "title": "The Birch–Swinnerton–Dyer Rank Problem: Reduction of the Analytic Rank Two Case",
  "title_text": "The Birch–Swinnerton–Dyer Rank Problem: Reduction of the Analytic Rank Two Case",
  "type": "preprint",
  "updated_at": "2026-11-03T12:21:22Z",
  "url": "https://research.draconitic.com/paper/393zpfm3h3ytk",
  "venue": "Zenodo",
  "versions": []
}