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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)]).

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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