Patching and local-global principles for homogeneous spaces over function fields of \(p\)-adic curves
type: Item
instance of: scholarly article
title: Patching and local-global principles for homogeneous spaces over function fields of \(p\)-adic curves
zbMATH Open document ID: 1332.11065
DOI: 10.4171/CMH/276
author: Jean-Louis Colliot-Thélène
author: Venapally Suresh
author: R. Parimala
published in: Commentarii Mathematici Helvetici
publication date: 2012-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0812.3099
review text: Summary: 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 [\textit{D. Harbater} et al., Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)]. There is a connection to injectivity properties of the Rost invariant and a result of \textit{K. Kato} [J. Reine Angew. Math. 366, 142--183 (1986; Zbl 0576.12012)].
Mathematics Subject Classification ID: 11G99
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 6104833
zbMATH Keywords: local-global principle
zbMATH Keywords: curves over local fields
zbMATH Keywords: homogeneous spaces
Wikidata QID: Q105471113
MaRDI profile type: Publication
arXiv ID: 0812.3099
description: scientific article