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 $F_v$. 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 $F_v$-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.