{
  "id": "math/0401140",
  "version": "v2",
  "published": "2004-01-14T07:15:36.000Z",
  "updated": "2004-01-16T07:03:31.000Z",
  "title": "Descente de torseurs, gerbes et points rationnels - Descent of torsors, gerbes and rational points",
  "authors": [
    "Stephane Zahnd"
  ],
  "comment": "144 pages, uses xypic, in french, thesis",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $k$ be a field of characteristic 0 and $G$ a linear algebraic $k$-group. When $G$ is abelian, it is well known that torsors under $G_{X}$ over a $k$-scheme $\\pi:X\\to \\textup{Spec} k$ provide an obstruction to the existence of $k$-rational points on $X$, since Leray spectral sequence gives rise (when $X$ is 'nice', e.g. $X$ smooth and proper) to an exact sequence of groups (5-term exact sequence associated). This sequence gives an obstruction for a $\\bar{G}_{X}$-torsor $\\bar{P}\\to\\bar{X}$ with field of moduli $k$ to be defined over $k$, i.e. to be obtained by extension of scalars to the algebraic closure $\\bar{k}$ of $k$ from a $G_{X}$-torsor $P\\to X$. This obstruction is measured by a gerbe, which is neutral if $X$ possesses a $k$-rational point. We try to extend this result to the non-commutative case, and in some cases, we deduce non-abelian cohomological obstruction to the existence of $k$-rational points on $X$, and results about descent of torsors.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-01-16T07:03:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14A20",
      "14F20",
      "18G50"
    ],
    "keywords": [
      "rational point",
      "points rationnels",
      "exact sequence",
      "leray spectral sequence",
      "deduce non-abelian cohomological obstruction"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 144,
      "language": "fr",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1140Z"
    }
  }
}