On rational points on homogeneous spaces over local and global fields and their Brauer and R-equivalence relations. II

Source abstract

We study the weak Brauer and R-equivalence relation on the set of rational points on homogeneous spaces under connected reductive groups defined over global fields and construct several exact sequences connecting various arithmetic-geometric invariants. We show that the Brauer loci on such spaces enjoy the weak approximation property, extending an earlier result proved for connected reductive algebraic groups, that if $X$ is defined over a real number field $k$ and $X(k) \ne \emptyset$, any element from $X(\textbf{R})$ can be arbitrarily well-approximated by a $k$-point of $X$, which is Brauer and R-equivalent to the pointed unit class of $X(k)$. As an application, we give a partial affirmative answer from a global point of view, to a question raised by Colliot-Thélène and Kunyavskiǐ.