For a homogeneous space (not necessarily principal) of a connected algebraic group (not necessarily linear) over a number field , we prove a theorem of strong approximation for the adelic points of in the Brauer–Manin set. Namely, for an adelic point of orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group of with respect to the Manin pairing, we prove a strong approximation property for away from a finite set of places of . 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.
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Mikhail Borovoi, Cyril Demarche, Manin obstruction to strong approximation for homogeneous spaces. Comment. Math. Helv. 88 (2013), no. 1, pp. 1–54DOI 10.4171/CMH/277