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

TL;DR

This paper investigates the failure of local-global principles in representing integral quadratic forms and shows that many such failures can be explained by a Brauer-Manin obstruction, which is often the only obstruction for certain homogeneous spaces.

Contribution

It demonstrates that a Brauer-Manin obstruction accounts for many failures of representing quadratic forms integrally and proves it is the only obstruction for specific homogeneous spaces of linear algebraic groups.

Findings

Many failures of local-global principles explained by Brauer-Manin obstruction.

For certain homogeneous spaces, the Brauer-Manin obstruction is the only obstacle.

Provides a unified framework for understanding integral quadratic form representations.

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. ----- Une forme quadratique enti\`ere peut \^etre repr\'esent\'ee par une autre forme quadratique enti\`ere sur tous les anneaux d'entiers p-adiques et sur les r\'eels, sans l'\^etre sur les entiers. On en trouve de nombreux exemples dans la litt\'erature. 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 homog\`enes de groupes alg\'ebriques lin\'eaires, cette obstruction est la seule obstruction \`a l'existence d'un point entier.