Brauer-Manin Obstruction on Generalized Kummer Varieties

Given an abelian variety $A$ over a number field, we consider the generalized Kummer varieties of $A$ coming from quotients of $A$ by an automorphism of prime order $p>2$. We prove that the Brauer-Manin obstruction on these generalized Kummer varieties only can come from the $p$-primary part of the Brauer group. This is applied to show that certain families of such varieties have no Brauer-Manin obstruction to the local-global principle.