Some local-global principles in the arithmetic of algebraic groups over real function fields
type: Item
instance of: scholarly article
title: Some local-global principles in the arithmetic of algebraic groups over real function fields
zbMATH Open document ID: 0842.20038
DOI: 10.1007/BF02622095
published in: Mathematische Zeitschrift
publication date: 1996-07-21T00:00:00Z
review text: In his seminal papers [J. Reine Angew. Math. 171, 4-11 (1934; Zbl 0009.29103); ibid. 176, 31-44 (1936; Zbl 0015.05701)] \textit{E. Witt} was the first to explore the arithmetic of algebraic groups over a real function field in one variable, by establishing first the local-global principle (which subsequently is called also the Hasse principle) for Brauer groups and then for quadratic forms over such fields. These two results are of fundamental importance in the arithmetic of quadratic and hermitian forms and related algebraic groups over these fields. It is natural and important to study, continuing the study of Witt et al., the arithmetic of algebraic groups over function fields in general, and over real function fields in particular, due to their connections with problems in algebraic geometry, algebraic \(K\)-theory and theory of quadratic and hermitian forms over rings. Let \(G\) be a semisimple group defined over the field \(k = \mathbb{R}(t)\), \(V\) the set of all valuations of \(k\) trivial on \(\mathbb{R}\). Let \(C(A)\) denote the center of a group \(A\). We say that \(A\) is projectively simple if \(A/C(A)\) is simple as abstract group. In this paper we prove that if \(G\) is \(k\)-isotropic and simply connected then \(G\) satisfies weak approximation with respect to \(V\). We prove a Hasse principle for isotropicity of almost simple \(k\)-groups of type different from \(^6D_4\), i.e. an almost simple \(k\)-group \(G\) of type \(\neq{^6D_4}\) is isotropic if and only if it is over all completions \(k_v\), \(v \in V\), of \(k\). A vertex of a Dynkin diagram of an almost simple group defined over a field \(K\) is called (after Tits) isotropic (resp. quasi-isotropic) if it is circled (resp. if there is another vertex in its orbit under the \(*\)-action of \(\text{Gal}(\overline K/K)\)). We show that a vertex of Dynkin diagram of an almost simple \(k\)-group \(G\) of type different from \(^3 D_4\) (resp. \(^3 D_4\), \(^6 D_4\)) is isotropic (resp. quasi-isotr…
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 861930
zbMATH Keywords: arithmetic of algebraic groups
zbMATH Keywords: real function fields
zbMATH Keywords: Brauer groups
zbMATH Keywords: quadratic forms
zbMATH Keywords: semisimple groups
zbMATH Keywords: weak approximation
zbMATH Keywords: Hasse principle
zbMATH Keywords: almost simple \(k\)-groups
zbMATH Keywords: cohomological Hasse principle
author: Nguyêñ Quôć Thǎńg
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02622095
OpenAlex ID: W2159268227
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description: scientific article; zbMATH DE number 861930