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Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

Part of: Diophantine equations Arithmetic problems. Diophantine geometry (Co)homology theory Arithmetic algebraic geometry Forms and linear algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  01 March 2009

Jean-Louis Colliot-Thélène  and
Fei Xu
Jean-Louis Colliot-Thélène
Affiliation:
CNRS, UMR 8628, Mathématiques, Bâtiment 425, Université Paris-Sud, F-91405 Orsay, France (email: jlct@math.u-psud.fr)
Fei Xu
Affiliation:
Academy of Mathematics and System Science, CAS, Beijing 100080, P.R. China (email: xufei@math.ac.cn)
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Abstract

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An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

Résumé

Une forme quadratique entière peut être représentée par une autre forme quadratique entière sur tous les anneaux d’entiers p-adiques et sur les réels, sans l’être sur les entiers. On en trouve de nombreux exemples dans la littérature. Nous montrons qu’une partie de ces exemples s’explique au moyen d’une obstruction de type Brauer–Manin pour les points entiers. Pour plusieurs types d’espaces homogènes de groupes algébriques linéaires, cette obstruction est la seule obstruction à l’existence d’un point entier.

Keywords

integral quadratic formsintegral pointsHasse principleBrauer–Manin obstructionspinor exceptionshomogeneous spaces of linear algebraic groupsGalois cohomology

MSC classification

Secondary: 11D85: Representation problems 11E12: Quadratic forms over global rings and fields 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11E57: Classical groups 11E72: Galois cohomology of linear algebraic groups 11G35: Varieties over global fields 14G25: Global ground fields 14F22: Brauer groups of schemes 20G30: Linear algebraic groups over global fields and their integers
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 309 - 363
Copyright
Copyright © Foundation Compositio Mathematica 2009

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