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

TL;DR

This paper extends the study of Brauer-Manin obstructions from tori to general commutative affine algebraic groups over global fields, proving analogous results and exploring related finiteness properties.

Contribution

It generalizes known results on Brauer-Manin obstructions from tori to all commutative affine algebraic groups over global fields, including obstructions to strong approximation.

Findings

Proved analogous Brauer-Manin obstruction results for commutative affine groups.

Established finiteness of certain Tate-Shafarevich kernels.

Extended duality theorems to positive characteristic settings.

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.