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 $\mathrm{Br}(X)$ 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.