{
  "abstract": "Let $E\\colon y^2=x^3-Dx$ have rank two and let $p\\equiv1\\pmod4$ be a prime of good reduction. We study the conjecture that the $p$-adic quotient $q_p$, the leading coefficient of the cyclotomic $p$-adic $L$-function divided by the Euler factor and the $p$-adic regulator, equals the complex quotient $q_\\infty=L''(E,1)/(2Ω_E\\mathrm{Reg}_\\infty)$, and that the latter is rational. In rank one the analogous statement is a theorem of Perrin-Riou and Rubin, because a Heegner point computes both sides; in rank two there is no such point. We show that $q_p$ is the constant of the quadratic term of the two-variable Katz measure; this is how the conjecture arises, and it computes $q_p$ modulo any power of $p$ from critical Hecke $L$-values. We then show that $q_p$ is the coordinate of two global classes: Rubin's rank-two elliptic-unit class, whose position in $E(\\mathbb{Q})\\otimes\\mathbb{Q}_p$ is $q_p$ times an explicit vector, and the derived Heegner class over an auxiliary imaginary quadratic field $F$, for which we prove an exact identity. As consequences, whenever such a field $F$ satisfying our hypotheses exists, $q_p$ is a square in $\\mathbb{Q}_p$ times the product of the Tamagawa numbers of $E$, so that the unit $q_p/A_E$ of the $p$-adic Birch and Swinnerton-Dyer formula is a square when Sha is finite; if moreover the derived Heegner class has non-zero formal logarithm, the conjecture of Agboola and Castella for $E/F$ holds up to a $p$-adic unit, and with a theorem of Burungale, Castella and Skinner the anticyclotomic main conjecture holds integrally; and our conjecture is equivalent to the analytic form of theirs, with the products of anticyclotomic heights read sesquilinearly, which differs by a sign from its literal formulation. The conjecture is verified modulo $p^3$ for 5520 pairs $(E,p)$ with conductors up to $3.9\\cdot10^{12}$, including curves with analytic Sha of order 4 and 9.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.09192",
  "author_total": 1,
  "authors": [
    {
      "name": "Miquel Calonge Fabregat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b9c24kbkffbzv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.09192",
  "fulltext_url": null,
  "id": "b9c24kbkffbzv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.09192",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.09192",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b9c24kbkffbzv.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.09192",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.09192"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two",
  "title_text": "Elliptic units, derived Heegner classes and the p-adic Birch and Swinnerton-Dyer quotient of CM elliptic curves of rank two",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/b9c24kbkffbzv",
  "venue": "arXiv",
  "versions": []
}