{
  "id": "1701.02458",
  "version": "v1",
  "published": "2017-01-10T07:47:40.000Z",
  "updated": "2017-01-10T07:47:40.000Z",
  "title": "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves",
  "authors": [
    "Manjul Bhargava",
    "Arul Shankar",
    "Takashi Taniguchi",
    "Frank Thorne",
    "Jacob Tsimerman",
    "Yongqiang Zhao"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_{\\epsilon}(|{\\rm Disc}(K)|^{1/2+\\epsilon})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quartic fields of bounded discriminant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-10T07:47:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29",
      "11G05"
    ],
    "keywords": [
      "integral points",
      "class groups",
      "higher degree number fields",
      "yields corresponding improvements",
      "quartic fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}