Brauer-Manin obstructions for homogeneous spaces of commutative affine algebraic groups over global fields
Azur Đonlagić
TL;DR
The paper extends the Brauer-Manin obstruction framework from tori to homogeneous spaces of arbitrary commutative affine group schemes over global fields, leveraging Rosengarten's generalized Tate duality to connect the obstruction to the Cartier dual via a comparison map $\Phi_X:\Sh^2(\widehat{G})\rightarrow Be(X)$. It proves that the Brauer-Manin obstruction is the sole obstruction to the Hasse principle, as well as to weak/strong approximation, by intertwining Be(X) with the global Poitou-Tate pairing and establishing exact duality sequences involving closures of rational points and dual Tate-Shafarevich groups. The work also provides finiteness theorems for $\Sh_S^2(\widehat{G})$ and $\Sh_f^2(\widehat{G})$ under appropriate hypotheses (notably absence of non-finite wound unipotent quotients and connectedness), together with explicit positive-characteristic counterexamples that illustrate limits of these results. Overall, the paper unifies characteristic-free duality methods to describe Brauer-Manin obstructions for a broad class of homogeneous spaces and clarifies when finiteness and approximation statements hold or fail.
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-Thélène. 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 $ω$-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.