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

DOI10.1112/S0010437X0800376XzbMath1190.11036arXiv0712.1957OpenAlexW1998551054MaRDI QIDQ3628243

Fei Xu, Jean-Louis Colliot-Thélène

Publication date: 19 May 2009

Published in: Compositio Mathematica (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0712.1957

zbMATH Keywords

Galois cohomologyHasse principleintegral pointsBrauer-Manin obstructionintegral quadratic formshomogeneous spaces of linear algebraic groupsspinor exceptions

Mathematics Subject Classification ID

General ternary and quaternary quadratic forms; forms of more than two variables (11E20) Quadratic forms over global rings and fields (11E12) Linear algebraic groups over global fields and their integers (20G30) Varieties over global fields (11G35) Brauer groups of schemes (14F22) Global ground fields in algebraic geometry (14G25) Classical groups (11E57) Galois cohomology of linear algebraic groups (11E72) Representation problems (11D85)

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Cites Work

• Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen
• Darstellungsmaße indefiniter quadratischer Formen
• Bounded generation of \(S\)-arithmetic subgroups of isotropic orthogonal groups over number fields.
• Stable trace formula: Elliptic singular terms
• La descente sur les variétés rationnelles. II. (The descent on rational varieties. II)
• Darstellung durch Spinorgeschlechter ternärer quadratischer Formen
• Fibration method and Manin obstruction
• Representations of indefinite ternary quadratic forms over number fields
• The defect of weak approximation for homogeneous spaces
• On representations of spinor genera. II
• Hardy-Littlewood varieties and semisimple groups
• Exceptional integers for genera of integral ternary positive definite quadratic forms
• Halbeinfache Gruppenschemata über Dedekindringen
• Products in sheaf-cohomology
• Ext-products and edge-morphisms
• Die Ähnlichkeitsklassen indefiniter Gitter
• Sums of Three Integer Squares in Complex Quadratic Fields
• On Indefinite Ternary Quadratic Forms
• Indefinite quadratic Diophantine equations
• Groupe de Picard et groupe de Brauer des compactifications lisses d’espaces homogènes
• Sums of three squares over imaginary quadratic fields
• On representations of spinor genera
• Diophantine Equations Reducible to Quadratics
• On representations of integers by indefinite ternary quadratic forms.

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