{
  "id": "1203.1075",
  "version": "v4",
  "published": "2012-03-05T23:57:31.000Z",
  "updated": "2014-01-27T16:53:53.000Z",
  "title": "Hasse Principle for Simply Connected Groups over Function Fields of Surfaces",
  "authors": [
    "Yong Hu"
  ],
  "comment": "39 pages, final version. Accepted for publication in Journal of the Ramanujan Mathematical Society",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\\'el\\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of $K$, $X$ has a point over the completion $K_v$, then $X$ has a $K$-rational point. The main result of this paper is the proof of this conjecture for groups of some classical types. In particular, we prove the conjecture when $G$ is of one of the following types: (1) ${}^2A_n^*$, i.e. $G=\\mathbf{SU}(h)$ is the special unitary group of some hermitian form $h$ over a pair $(D, \\tau)$, where $D$ is a central division algebra of square-free index over a quadratic extension $L$ of $K$ and $\\tau$ is an involution of the second kind on $D$ such that $L^{\\tau}=K$; (2) $B_n$, i.e., $G=\\mathbf{Spin}(q)$ is the spinor group of quadratic form of odd dimension over $K$; (3) $D_n^*$, i.e., $G=\\mathbf{Spin}(h)$ is the spinor group of a hermitian form $h$ over a quaternion $K$-algebra $D$ with an orthogonal involution. Our method actually yields a parallel local-global result over the fraction field of a 2-dimensional, henselian, excellent local domain with finite residue field, under suitable assumption on the residue characteristic.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-01-27T16:53:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E72",
      "11E57"
    ],
    "keywords": [
      "function field",
      "hasse principle",
      "hermitian form",
      "spinor group",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.1075H"
    }
  }
}