{
  "id": "1308.0395",
  "version": "v1",
  "published": "2013-08-02T02:37:17.000Z",
  "updated": "2013-08-02T02:37:17.000Z",
  "title": "Most hyperelliptic curves over Q have no rational points",
  "authors": [
    "Manjul Bhargava"
  ],
  "comment": "33 pages. arXiv admin note: text overlap with arXiv:1208.1007",
  "categories": [
    "math.NT"
  ],
  "abstract": "By a hyperelliptic curve over Q, we mean a smooth, geometrically irreducible, complete curve C over Q equipped with a fixed map of degree 2 to P^1 defined over Q. Thus any hyperelliptic curve C over Q of genus g can be embedded in weighted projective space P(1,1,g+1) via an equation of the form C : z^2 = f(x,y) = f_0 x^n + f_1 x^{n-1} y + ... + f_n y^n where n=2g+2, the coefficients f_i lie in Z, and f factors into distinct linear factors over Q-bar. Define the height H(C) of C by H(C):=max{|f_i|}, and order all hyperelliptic curves over Q of genus g by height. Then we prove that, as g tends to infinity: 1) a density approaching 100% of hyperelliptic curves of genus g have no rational points; 2) a density approaching 100% of those hyperelliptic curves of genus g that have points everywhere locally fail the Hasse principle; and 3) a density approaching 100% of hyperelliptic curves of genus g have empty Brauer set, i.e., have a Brauer-Manin obstruction to having a rational point. We also prove positive proportion results of this type for individual genera, including g = 1.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-08-02T02:37:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "14H25"
    ],
    "keywords": [
      "hyperelliptic curve",
      "rational point",
      "density approaching",
      "empty brauer set",
      "distinct linear factors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.0395B"
    }
  }
}