{
  "id": "1310.7692",
  "version": "v2",
  "published": "2013-10-29T06:29:20.000Z",
  "updated": "2017-02-24T21:35:54.000Z",
  "title": "A positive proportion of locally soluble hyperelliptic curves over $\\mathbb Q$ have no point over any odd degree extension",
  "authors": [
    "Manjul Bhargava",
    "Benedict H. Gross",
    "Xiaoheng Wang",
    "Tim Dokchitser",
    "Vladimir Dokchitser"
  ],
  "comment": "42 pages, to appear in JAMS, with an appendix by Tim and Vladimir Dokchitser)\\",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "A hyperelliptic curve over $\\mathbb Q$ is called \"locally soluble\" if it has a point over every completion of $\\mathbb Q$. In this paper, we prove that a positive proportion of hyperelliptic curves over $\\mathbb Q$ of genus $g\\geq 1$ are locally soluble but have no points over any odd degree extension of $\\mathbb Q$. We also obtain a number of related results. For example, we prove that for any fixed odd integer $k > 0$, the proportion of locally soluble hyperelliptic curves over $\\mathbb Q$ of genus $g$ having no points over any odd degree extension of $\\mathbb Q$ of degree at most $k$ tends to 1 as $g$ tends to infinity. We also show that the failures of the Hasse principle in these cases are explained by the Brauer-Manin obstruction. Our methods involve a detailed study of the geometry of pencils of quadrics over a general field of characteristic not equal to 2, together with suitable arguments from the geometry of numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-29T06:29:20.000Z",
      "title": "Pencils of quadrics and the arithmetic of hyperelliptic curves",
      "comment": "40 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "Manjul Bhargava",
        "Benedict H. Gross",
        "Xiaoheng Wang"
      ]
    },
    {
      "version": "v2",
      "updated": "2017-02-24T21:35:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G30",
      "14G05"
    ],
    "keywords": [
      "odd degree extension",
      "arithmetic",
      "proportion",
      "odd integer",
      "general field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.7692B"
    }
  }
}