{
  "id": "1302.0061",
  "version": "v3",
  "published": "2013-02-01T03:19:45.000Z",
  "updated": "2014-06-15T17:44:24.000Z",
  "title": "Most odd degree hyperelliptic curves have only one rational point",
  "authors": [
    "Bjorn Poonen",
    "Michael Stoll"
  ],
  "comment": "24 pages; to appear in Annals of Math",
  "categories": [
    "math.NT"
  ],
  "abstract": "Consider the smooth projective models C of curves y^2=f(x) with f(x) in Z[x] monic and separable of degree 2g+1. We prove that for g >= 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g tends to infinity. Finally, we show that C(Q) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty's method that shows that certain computable conditions imply #C(Q)=1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava-Gross equidistribution theorem for nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p=2.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-06-15T17:44:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "14G25",
      "14G40",
      "14K15",
      "14K20"
    ],
    "keywords": [
      "odd degree hyperelliptic curves",
      "rational point",
      "p-adic analysis",
      "bhargava-gross equidistribution theorem",
      "chabautys method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.0061P"
    }
  }
}