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

摘要

An integer may be represented by a quadratic form over each ring of p-adicintegers and over the reals without being represented by this quadratic formover the integers. More generally, such failure of a local-global principle mayoccur for the representation of one integral quadratic form by another integralquadratic form. We show that many such examples may be accounted for by aBrauer-Manin obstruction for the existence of integral points on schemesdefined over the integers. For several types of homogeneous spaces of linearalgebraic groups, this obstruction is shown to be the only obstruction to theexistence of integral points. ----- Une forme quadratique enti\`ere peut \^etre repr\'esent\'ee par une autreforme quadratique enti\`ere sur tous les anneaux d'entiers p-adiques et sur lesr\'eels, sans l'\^etre sur les entiers. On en trouve de nombreux exemples dansla litt\'erature. Nous montrons qu'une partie de ces exemples s'explique aumoyen d'une obstruction de type Brauer-Manin pour les points entiers. Pourplusieurs types d'espaces homog\`enes de groupes alg\'ebriques lin\'eaires,cette obstruction est la seule obstruction \`a l'existence d'un point entier.