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

Abstract

Let F = K(X) be the function field of a smooth projective curve over a p-adic field K. To each rank one discrete valuation of F one may associate the completion Fv. Given an F-variety Y which is a homogeneous space of a connected reductive group G over F, one may wonder whether the existence of Fv-points on Y for each v is enough to ensure that Y has an F-point. In this paper we prove such a result in two cases : (i) Y is a smooth projective quadric and p is odd. (ii) The group G is the extension of a reductive group over the ring of integers of K, and Y is a principal homogeneous space of G. 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.