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Manin obstruction to strong approximation for homogeneous spaces. (English) Zbl 1271.14073
The main object of the paper under review is a homogeneous space $$X$$ of a connected algebraic $$k$$-group $$G$$, where $$k$$ is a number field, $$X$$ is not necessarily principal, $$G$$ is not necessarily linear, and the geometric stabilizer $$H$$ is assumed to be connected. The authors’ goal is to describe the obstruction to the strong approximation property for adelic points of $$X$$ away from a finite set $$S$$ of places of $$k$$ containing all archimedean places. Assuming that the Tate–Shafarevich group of the maximal abelian variety quotient of $$G$$ is finite, they prove that this can be done with the help of the Manin obstruction related to a certain subgroup of the Brauer group of $$X$$ (which may contain transcendental elements). In the case where $$S$$ contains at least one nonarchimedean place, it is shown that a smaller subgroup can be used, which does not contain transcendental elements. To show that the assumption on $$S$$ is necessary, an explicit example is provided.
These results extend some earlier work: namely, weak approximation for $$X$$ as above has been established by M. Borovoi, J.-L. Colliot-Thélène and A. N. Skorobogatov [Duke Math. J. 141, No. 2, 321–364 (2008; Zbl 1135.14013)]. The methods of the latter paper have been used in the paper under review to reduce to the case where $$X$$ is a $$k$$-torsor under a semiabelian variety (treated by D. Harari [Algebra Number Theory 2, No. 5, 595–611 (2008; Zbl 1194.14067)]) and to the case where $$X$$ is a homogeneous space of a simply connected semisimple group with connected geometric stabilizers (treated by J.-L. Colliot-Thélène and F. Xu [Compos. Math. 145, No. 2, 309–363 (2009; Zbl 1190.11036)]).

##### MSC:
 14M17 Homogeneous spaces and generalizations 11G35 Varieties over global fields 14F22 Brauer groups of schemes 20G30 Linear algebraic groups over global fields and their integers
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