Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms
By J.-L. Colliot-Th'elene, Fei Xu. Compositio Mathematica, 2007
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Abstract 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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3fdcc8d6035145310952d30841d1bb71a6df21c8
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In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Manin obstruction to strong approximation for homogeneous spaces (2009, methodology): “This is very close to a result of Colliot-Thélène and Xu, see [11], Theorem 3.”