{
  "id": "1605.08245",
  "version": "v1",
  "published": "2016-05-26T12:03:38.000Z",
  "updated": "2016-05-26T12:03:38.000Z",
  "title": "On the $p$-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\\mathbb{Q}(\\sqrt{-3})$",
  "authors": [
    "Yukako Kezuka"
  ],
  "comment": "32 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study infinite families of quadratic and cubic twists of the elliptic curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the value of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-26T12:03:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curve",
      "complex multiplication",
      "birch-swinnerton-dyer conjecture",
      "lower bound",
      "adic valuation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}