Commentarii Mathematici Helvetici


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Volume 87, Issue 4, 2012, pp. 1011–1033
DOI: 10.4171/CMH/276

Published online: 2012-10-10

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

Jean-Louis Colliot-Thélène[1], Raman Parimala[2] and Venapally Suresh[3]

(1) Université Paris-Sud, Orsay, France
(2) Emory University, Atlanta, USA
(3) Emory University, Atlanta, USA

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:

  1. $Y$ is a smooth projective quadric and $p$ is odd.
  2. 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.

Keywords: Local-global principle, curves over local fields, homogeneous spaces

Colliot-Thélène Jean-Louis, Parimala Raman, Suresh Venapally: Patching and local-global principles for homogeneous spaces over function fields of p-adic curves. Comment. Math. Helv. 87 (2012), 1011-1033. doi: 10.4171/CMH/276