Manin obstruction to strong approximation for homogeneous spaces

Abstract

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k , we prove a theorem of strong approximation for the adelic points of X in the Brauer–Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group \operatorname{Br}(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k . Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Thélène and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.