{
  "id": "1311.5306",
  "version": "v1",
  "published": "2013-11-21T03:50:17.000Z",
  "updated": "2013-11-21T03:50:17.000Z",
  "title": "Quadratic Twists of Elliptic Curves with 3-Selmer Rank 1",
  "authors": [
    "Zane Kun Li"
  ],
  "comment": "21 pages, to appear in IJNT",
  "doi": "10.1142/S1793042114500213",
  "categories": [
    "math.NT"
  ],
  "abstract": "A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which force a positive proportion of the quadratic twists of E to have 3-Selmer rank 1 and global root number -1. We then give four nonisomorphic infinite families of elliptic curves $E_{m, n}$ which satisfy these criteria. Conditional on the rank part of the Birch and Swinnerton-Dyer conjecture, this verifies the aforementioned conjecture for infinitely many elliptic curves. Our elliptic curves are easy to give explicitly and we state precisely which quadratic twists d to use. Furthermore, our methods have the potential of being generalized to elliptic curves over other number fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-21T03:50:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H52",
      "11G05"
    ],
    "keywords": [
      "elliptic curve",
      "quadratic twists",
      "positive proportion",
      "nonisomorphic infinite families",
      "global root number"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5306L"
    }
  }
}