The Brauer-Manin obstruction to the local-global principle for the embedding problem

We study an analogue of the Brauer-Manin obstruction to the
local-global principle for embedding problems over global fields. We will
prove the analogues of several fundamental structural results. In
particular we show that the Brauer-Manin obstruction is the only one to
strong approximation when the embedding problem has abelian kernel and show
that the analogue of the algebraic Brauer-Manin obstruction is equivalent
to the analogue of the abelian descent obstruction. In the course of our
investigations we give a new, elegant description of the Tate duality
pairing and prove a new theorem on the cup product in group cohomology.