On the local-global principle for twists of abelian varieties

This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for $m$ a positive integer, locally $m$-atic twists of an abelian variety $A$ over a number field $K$ are globally $m$-atic. We define and study a “Tate-Shafarevich cohomology set” that governs the obstruction to the local-global principle for $m$-atic twists. We prove that, under some mild assumptions, this set is finite, and give criteria for it to be trivial, i.e. for the local-global principle to be satisfied.