JournalscmhVol. 87, No. 4pp. 1011–1033

Patching and local-global principles for homogeneous spaces over function fields of <em>p</em>-adic curves

  • Jean-Louis Colliot-Thélène

    Université Paris-Sud, Orsay, France
  • Raman Parimala

    Emory University, Atlanta, USA
  • Venapally Suresh

    Emory University, Atlanta, USA
Patching and local-global principles for homogeneous spaces  over  function fields of <em>p</em>-adic curves cover
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Abstract

Let F=K(X)F=K(X) be the function field of a smooth projective curve over a pp-adic field KK. To each rank one discrete valuation of FF one may associate the completion FvF_v. Given an FF-variety YY which is a homogeneous space of a connected reductive group GG over FF, one may wonder whether the existence of FvF_v-points on YY for each vv is enough to ensure that YY has an FF-point. In this paper we prove such a result in two cases:

OL.withroman { list-style-type: lower-roman }

  1. YY is a smooth projective quadric and pp is odd.
  2. The group GG is the extension of a reductive group over the ring of integers of KK, and YY is a principal homogeneous space of GG.

An essential use is made of recent patching results of Harbater, Hartmann and Krashen. There is a connection to injectivity properties of the Rost invariant and a result of Kato.

Cite this article

Jean-Louis Colliot-Thélène, Raman Parimala, Venapally Suresh, Patching and local-global principles for homogeneous spaces over function fields of <em>p</em>-adic curves. Comment. Math. Helv. 87 (2012), no. 4, pp. 1011–1033

DOI 10.4171/CMH/276