{
  "id": "1804.05859",
  "version": "v1",
  "published": "2018-04-16T18:00:13.000Z",
  "updated": "2018-04-16T18:00:13.000Z",
  "title": "The average number of rational points on genus two curves is bounded",
  "authors": [
    "Levent Alpoge"
  ],
  "comment": "37 pages. Comments and criticisms welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We prove that, when genus two curves $C/\\mathbb{Q}$ with a marked Weierstass point are ordered by height, the average number of rational points $\\#|C(\\mathbb{Q})|$ is bounded. The argument follows the same ideas as the sphere-packing proof of boundedness of the average number of integral points on (quasiminimal Weierstrass models of) elliptic curves. That is, we bound the number of small-height points by hand, the number of medium-height points by establishing an explicit Mumford gap principle and using the theorem of Kabatiansky-Levenshtein on spherical codes (this technique goes back to work of Silverman, Helfgott, and Helfgott-Venkatesh), and the number of large-height points by using Bombieri-Vojta's proof of Faltings' theorem. Explicitly, in dealing with non-small-height points we prove that the number of rational points $(x,y)$ on $C_f: y^2 = f(x)$ satisfying $h(x) > 8 h(f)$ is $\\ll 1.872^{\\mathrm{rank}(\\mathrm{Jac}(C)(\\mathbb{Q}))}$, which has finite average by the theorem of Bhargava-Gross on the average size of $2$-Selmer groups of Jacobians over this family. We note that our arguments in the small-height and large-height cases extend to general genera $g\\geq 2$, though for medium points we need to use Stoll's bounds on the non-Archimedean local height differences in genus $2$. For example, we prove that the number of rational points $P\\in C(\\mathbb{Q})$ with $h(P)\\gg_g h(C)$ on $C/\\mathbb{Q}$ smooth projective and of genus $g\\geq 2$ is $\\ll 1.872^{\\mathrm{rank}(\\mathrm{Jac}(C)(\\mathbb{Q}))}$, and that in fact the base of the exponent can be reduced to $1.311$ once $g\\gg 1$, though this is surely known to experts (the difference is the use of the Kabatiansky-Levenshtein bound in lieu of more elementary techniques).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-16T18:00:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational points",
      "average number",
      "non-archimedean local height differences",
      "explicit mumford gap principle",
      "quasiminimal weierstrass models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}