{
  "abstract": "Let $p\\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y^2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2^\\infty$-Selmer corank one for $p\\equiv 3,5\\pmod{8}$ and corank zero for $p\\equiv 7\\pmod{8}$. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class $p\\equiv 1\\pmod{8}$. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09370",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruihan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ehwq8c84q8z6q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09370",
  "fulltext_url": null,
  "id": "ehwq8c84q8z6q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09370",
  "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": "W7220963440",
  "pdf_url": "https://arxiv.org/pdf/2610.09370",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ehwq8c84q8z6q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09370",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09370"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$",
  "title_text": "On the algebraic and analytic ranks of the twin-prime elliptic curve y^2=x(x−2)(x−p)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/ehwq8c84q8z6q",
  "venue": "arXiv",
  "versions": []
}