{
  "id": "1312.7859",
  "version": "v1",
  "published": "2013-12-30T20:43:23.000Z",
  "updated": "2013-12-30T20:43:23.000Z",
  "title": "The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1",
  "authors": [
    "Manjul Bhargava",
    "Arul Shankar"
  ],
  "comment": "32 pages. arXiv admin note: text overlap with arXiv:1312.7333",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we prove that the average rank of elliptic curves over $\\mathbb{Q}$, when ordered by height, is less than $1$ (in fact, less than $.885$). As a consequence of our methods, we also prove that at least four fifths of all elliptic curves over $\\mathbb{Q}$ have rank either 0 or 1; furthermore, at least one fifth of all elliptic curves in fact have rank 0. The primary ingredient in the proofs of these theorems is a determination of the average size of the $5$-Selmer group of elliptic curves over $\\mathbb{Q}$; we prove that this average size is $6$. Another key ingredient is a new lower bound on the equidistribution of root numbers of elliptic curves; we prove that there is a family of elliptic curves over $\\mathbb{Q}$ having density at least $55\\%$ for which the root number is equidistributed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-30T20:43:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11R45"
    ],
    "keywords": [
      "elliptic curves",
      "average rank",
      "average size",
      "root number",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7859B"
    }
  }
}