{
  "id": "1512.06894",
  "version": "v1",
  "published": "2015-12-21T22:26:35.000Z",
  "updated": "2015-12-21T22:26:35.000Z",
  "title": "The Birch and Swinnerton-Dyer Formula for Elliptic Curves of Analytic Rank One",
  "authors": [
    "Dimitar Jetchev",
    "Christopher Skinner",
    "Xin Wan"
  ],
  "comment": "51 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be a semistable elliptic curve such that $\\mathrm{ord}_{s=1}L(E,s) = 1$. We prove the $p$-part of the Birch and Swinnerton-Dyer formula for $E/\\mathbb{Q}$ for each prime $p \\geq 5$ of good reduction such that $E[p]$ is irreducible: $$ \\mathrm{ord}_p \\left (\\frac{L'(E,1)}{\\Omega_E\\cdot\\mathrm{Reg}(E/\\mathbb{Q})} \\right ) = \\mathrm{ord}_p \\left (\\#\\mathrm{Sha}(E/\\mathbb{Q})\\prod_{\\ell\\leq \\infty} c_\\ell(E/\\mathbb{Q}) \\right ). $$ This formula also holds for $p=3$ provided $a_p(E)=0$ if $E$ has supersingular reduction at $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-21T22:26:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G40",
      "11G05",
      "11G07",
      "11R23",
      "11G15",
      "14H52",
      "14H25"
    ],
    "keywords": [
      "swinnerton-dyer formula",
      "analytic rank",
      "supersingular reduction",
      "semistable elliptic curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151206894J"
    }
  }
}