Patching and local-global principles for homogeneous spaces over function fields of -adic curves

  • Jean-Louis Colliot-Thélène

    Université Paris-Sud, Orsay, France
  • Raman Parimala

    Emory University, Atlanta, USA
  • Venapally Suresh

    Emory University, Atlanta, USA

Abstract

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

(i) is a smooth projective quadric and is odd.

(ii) The group is the extension of a reductive group over the ring of integers of , and is a principal homogeneous space of .

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 -adic curves. Comment. Math. Helv. 87 (2012), no. 4, pp. 1011–1033

DOI 10.4171/CMH/276