{
  "id": "1312.7333",
  "version": "v1",
  "published": "2013-12-27T20:40:05.000Z",
  "updated": "2013-12-27T20:40:05.000Z",
  "title": "The average number of elements in the 4-Selmer groups of elliptic curves is 7",
  "authors": [
    "Manjul Bhargava",
    "Arul Shankar"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove that when all elliptic curves over $\\mathbb{Q}$ are ordered by height, the average size of their 4-Selmer groups is equal to 7. As a consequence, we show that a positive proportion (in fact, at least one fifth) of all 2-Selmer elements of elliptic curves, when ordered by height, do not lift to 4-Selmer elements, and thus correspond to nontrivial 2-torsion elements in the associated Tate--Shafarevich groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-27T20:40:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11R45"
    ],
    "keywords": [
      "elliptic curves",
      "average number",
      "associated tate-shafarevich groups",
      "correspond",
      "consequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7333B"
    }
  }
}