{
  "id": "1302.2377",
  "version": "v3",
  "published": "2013-02-10T23:07:21.000Z",
  "updated": "2015-09-20T20:35:49.000Z",
  "title": "Lois de réciprocité supérieures et points rationnels",
  "authors": [
    "Jean-Louis Colliot-Thélène",
    "Raman Parimala",
    "Venapally Suresh"
  ],
  "comment": "Final version, in French, to appear in the Transactions of the American Mathematical Society",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let C be the complex field and K=C((x,y)) or K=C((x))(y). Let G be a connected linear algebraic group over K. Under the assumption that the K-variety G is K-rational, i.e. that the function field is purely transcendant, it was proved that a principal homogeneous space of G has a rational point over K as soon as it has one over each completion of K with respect to a discrete valuation. In this paper we show that one cannot in general do without the K-rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-22T19:36:18.000Z",
      "comment": "Slightly revised, 22nd november 2013",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-09-20T20:35:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G99",
      "14G99",
      "11E72",
      "14F22"
    ],
    "keywords": [
      "points rationnels",
      "principal homogeneous space",
      "rational point",
      "supérieures",
      "connected linear algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "fr",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2377C"
    }
  }
}