{
  "id": "math/0411413",
  "version": "v1",
  "published": "2004-11-18T18:10:03.000Z",
  "updated": "2004-11-18T18:10:03.000Z",
  "title": "There are genus one curves of every index over every number field",
  "authors": [
    "Pete L. Clark"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We show that there exist genus one curves of every index over the rational numbers, answering affirmatively a question of Lang and Tate. The proof is \"elementary\" in the sense that it does not assume the finiteness of any Shafarevich-Tate group. On the other hand, using Kolyvagin's construction of a rational elliptic curve whose Mordell-Weil and Shafarevich-Tate groups are both trivial, we show that there are infinitely many curves of every index over every number field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-11-18T18:10:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "number field",
      "shafarevich-tate group",
      "rational elliptic curve",
      "rational numbers",
      "kolyvagins construction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11413C"
    }
  }
}