Hasse principle for principal homogeneous spaces under classical groups over a field of virtual cohomological dimension at most one
type: Item
instance of: scholarly article
title: Hasse principle for principal homogeneous spaces under classical groups over a field of virtual cohomological dimension at most one
zbMATH Open document ID: 0846.20047
DOI: 10.1007/BF02567522
author: Antoine Ducros
published in: Manuscripta Mathematica
publication date: 1996-06-10T00:00:00Z
full work available at URL: https://eudml.org/doc/156167
review text: The main objects of the 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 for principal homogeneous spaces under such groups. The author follows Scheiderer's approach 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, 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, 639-650 (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)]. In the latter paper it is conjectured that the Hasse principle should hold for principal 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 this conjecture for all classical groups by case-by-case consideration following the approach by \textit{E. Bayer-Fluckiger} and \textit{R. Parimala} [Invent. Math. 122, No. 2, 195-229 (1995)] who treated the case of a ground field of cohomological dimension at most 2. One should note that recently the results of the paper reviewed were generalized in two directions: the above conjecture was proved by C. Scheiderer for any \(G\) an…
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 20G10
zbMATH DE Number: 871039
zbMATH Keywords: connected linear groups
zbMATH Keywords: perfect fields
zbMATH Keywords: cohomological dimension
zbMATH Keywords: field of rational functions
zbMATH Keywords: smooth projective real curves
zbMATH Keywords: Hasse principle for principal homogeneous spaces
reviewed by: Boris Kunyavskiĭ
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 871039