{
  "id": "1307.3531",
  "version": "v2",
  "published": "2013-07-12T18:14:56.000Z",
  "updated": "2014-02-17T20:40:05.000Z",
  "title": "Average size of the 2-Selmer group of Jacobians of monic even hyperelliptic curves",
  "authors": [
    "Arul Shankar",
    "Xiaoheng Wang"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1208.1007 by other authors",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In [5], Manjul Bhargava and Benedict Gross considered the family of hyperelliptic curves over $\\Q$ having a fixed genus and a marked rational Weierstrass point. They showed that the average size of the 2-Selmer group of the Jacobians of these curves, when ordered by height, is 3. In this paper, we consider the family of hyperelliptic curves over $\\Q$ having a fixed genus and a marked rational non-Weierstrass point. We show that when these curves are ordered by height, the average size of the 2-Selmer group of their Jacobians is 6. This yields an upper bound of 5/2 on the average rank of the Mordell-Weil group of the Jacobians of these hyperelliptic curves. Finally using an equidistribution result, we modify the techniques of [16] to conclude that as $g$ tends to infinity, a proportion tending to 1 of these monic even-degree hyperelliptic curves having genus $g$ have exactly two rational points - the marked point at infinity and its hyperelliptic conjugate.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-17T20:40:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "average size",
      "monic even-degree hyperelliptic curves",
      "marked rational non-weierstrass point",
      "fixed genus",
      "marked rational weierstrass point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3531S"
    }
  }
}