Generalized arithmetic duality and rational points

I will discuss extensions of the local and global arithmetic
duality theorems which hold for generalized Galois cohomology theories.
In particular, if X is a variety over a number field K, global duality
in a cohomology theory obtained from X carries information about the
Brauer-Manin obstruction, as well as some additional obstructions for
the existence of rational points and zero cycles on X. This is joint
work in progress with Tomer Schlank.