Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one
type: Item
instance of: scholarly article
title: Hasse principles and approximation theorems for homogeneous spaces over fields of virtual cohomological dimension one
zbMATH Open document ID: 0857.20024
author: Claus Scheiderer
published in: Inventiones Mathematicae
publication date: 1996-11-10T00:00:00Z
review text: The main objects of the reviewed paper are connected linear groups defined over a perfect field \(k\) with cohomological dimension of \(k(\sqrt{-1})\) at most one. A typical example of such a field is the field of rational functions \(\mathbb{R}(C)\) of a smooth projective real curve \(C\). The goal is to establish the Hasse principle and approximation theorems for homogeneous spaces under such groups. The author's approach consists in taking as local objects the real closures of \(k\) rather than the completions at the closed points. Such a principle was established for quadrics by \textit{R. Elman}, \textit{T.-Y. Lam} and \textit{A. Prestel} [Math. Z. 134, 291-301 (1973; Zbl 0277.15013)] thus generalizing classical results of \textit{E. Witt} [J. Reine Angew. Math. 171, 4-11 (1934; Zbl 0009.29103), 176, 31-44 (1936; Zbl 0015.05701)], and for Hermitian forms by \textit{N. Q. Thang} [Manuscr. Math. 78, No. 1, 9-35 (1993; Zbl 0804.11034), 82, No. 3-4, 445-447 (1994; Zbl 0812.11026)]. In another direction, the Hasse principle is known to hold for norms of any finite extension of \(k=\mathbb{R}(C)\) (\textit{J. T. Knight} [Proc. Camb. Philos. Soc. 65, 635-650 (1969; Zbl 0176.50703)]) and more general for principal homogeneous spaces of \(k\)-tori (\textit{J.-L. Colliot-Thélène} [J. Reine Angew. Math. 474, 139-167 (1996; Zbl 0847.11017)]). In the latter paper it is conjectured that the Hasse principle and weak approximation should hold for homogeneous spaces under any connected linear group \(G\), and the problem is reduced to the case when \(G\) is semisimple simply connected. The author proves these conjectures. The Hasse principle is established for any homogeneous space, weak approximation is proved under the sole assumption that the (geometric) stabilizer is connected. He also proves component approximation (with real closures as local objects) without any restriction on stabilizers. These results have various applications (the author [Manuscr. Math. 89, No. 3, 373…
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 922468
zbMATH Keywords: connected linear groups
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Hasse principle
zbMATH Keywords: approximation theorems
zbMATH Keywords: homogeneous spaces
zbMATH Keywords: weak approximation
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
full work available at URL: s002220050077
OpenAlex ID: W2000225259
DOI: 10.1007/S002220050077
description: scientific article; zbMATH DE number 922468