{
  "abstract": "We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period $Ω_E =\\frac{Γ(1/4)^2}{2\\sqrtπ\\,17^{1/4}}$ and the Tate--Shafarevich group $ \\Sha(E/\\mathbb{Q})\\cong \\mathbb{Z}/2\\mathbb{Z}\\times \\mathbb{Z}/2\\mathbb{Z}$ remember the elliptic curve $E: y^2=x^3+17x$. Let $K$ be a number field admitting a real embedding, and fix one such embedding $σ\\colon K\\hookrightarrow\\mathbb{R}$. We first prove that, for an elliptic curve $E$ over $K$, the real period of $E$ attached to $σ$ determines the $K$-isogeny class of $E$. We then prove that an elliptic curve $E$ over $\\mathbb{Q}$ is determined up to isomorphism over $\\mathbb{Q}$ by its real period, the number of connected components of $E(\\mathbb{R})$, and the real period of the quadratic twist $E^{D}$ by a negative square-free integer $D\\equiv 1 \\bmod 4$ coprime to the minimal discriminant of $E$. Moreover, over $\\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06015",
  "author_total": 1,
  "authors": [
    {
      "name": "Asuka Shiga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2ysrny938mhmp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06015",
  "fulltext_url": null,
  "id": "2ysrny938mhmp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06015",
  "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": "W7220745832",
  "pdf_url": "https://arxiv.org/pdf/2610.06015",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2ysrny938mhmp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06015",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06015"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "When the BSD Real Period Remembers Elliptic Curves over $\\mathbb{Q}$",
  "title_text": "When the BSD Real Period Remembers Elliptic Curves over ℚ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/2ysrny938mhmp",
  "venue": "arXiv",
  "versions": []
}