{
  "id": "1610.05759",
  "version": "v1",
  "published": "2016-10-18T19:48:33.000Z",
  "updated": "2016-10-18T19:48:33.000Z",
  "title": "The average size of the 3-isogeny Selmer groups of elliptic curves $y^2 = x^3 + k$",
  "authors": [
    "Manjul Bhargava",
    "Noam Elkies",
    "Ari Shnidman"
  ],
  "comment": "25 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The elliptic curve $E_k \\colon y^2 = x^3 + k$ admits a natural 3-isogeny $\\phi_k \\colon E_k \\to E_{-27k}$. We compute the average size of the $\\phi_k$-Selmer group as $k$ varies over the integers. Unlike previous results of Bhargava and Shankar on $n$-Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on $k$; this sensitivity can be precisely controlled by the Tamagawa numbers of $E_k$ and $E_{-27k}$. As consequences, we prove that the average rank of the curves $E_k$, $k\\in\\mathbb Z$, is less than 1.21 and over $23\\%$ (resp. $41\\%$) of the curves in this family have rank 0 (resp. 3-Selmer rank 1).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-18T19:48:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11R45",
      "11E76"
    ],
    "keywords": [
      "elliptic curve",
      "selmer group",
      "average size",
      "average rank",
      "tamagawa numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}