Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields

Abstract

Questions related to Brauer-Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces of tori over a number field are well-studied, generally using arithmetic duality theorems, starting with works of Sansuc and of Colliot-Th\'el\`ene. In this article, we prove the analogous statements (and include obstructions to strong approximation over finite places) in the general case of a commutative affine group scheme $G$ of finite type over a global field in any characteristic. We also study finiteness of different variants of the second Tate-Shafarevich kernel (such as $S$-kernels and $\omega$-kernels) of the Cartier dual of $G$. All this is made possible by some recent theoretical advancements in positive characteristic, namely the finiteness theorems of B. Conrad and the generalized Tate duality of Z. Rosengarten.