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

Source abstract

We introduce the weak Brauer equivalence relation on the set of rational points on homogeneous spaces under connected reductive groups and consider its connections with Brauer and R-equivalence relations on such spaces over local fields and give some formulas for computing the set of equivalence classes in some important cases. Especially, for $\textbf{R}$-homogeneous spaces $X$ under connected affine $\textbf{R}$-groups with connected $\textbf{R}$-stabilizers, any two elements $x, y \in X(\textbf{R})$ are directly R-equivalent. As an application, we give a partial affirmative answer to a question raised by Colliot-Thélène and Kunyavskiǐ.