On Brauer–Manin obstruction to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global fields

Abstract

We give some new formulas via some exact sequences for computing an obstruction to the weak approximation on non-abelian cohomology sets and homogeneous spaces over global fields, with stabilizers belonging to some class of non-connected subgroups. As a consequence, we show that the Brauer–Manin obstruction to the weak approximation for such spaces is the only one. Along the way, we show that the Brauer–Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with stabilizers belonging to a certain class of non-necessarily connected groups are the only ones, extending some of Borovoi’s results obtained for number fields in this regard.