On analogs of Cassels–Tate's exact sequence for connected reductive groups and Brauer-Manin obstruction for homogeneous spaces over global function fields

Abstract

We present some new analogs of Cassels–Tate dual exact sequence for connected reductive groups over global fields. We give also some extension of some important local-global exact sequences proved by Sansuc for connected linear algebraic groups over number fields, which are analogs of Cassels–Tate dual exact sequence, to the case of connected reductive groups over global function fields. As applications, 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 connected reductive stabilizers are the only ones, extending some of Borovoi's results over number fields in this regard.