On the local-global principle for twists of abelian varieties

TL;DR

This paper explores when local properties of certain twists of abelian varieties over number fields imply global properties, introducing a cohomological set that measures obstructions and establishing conditions for the local-global principle to hold.

Contribution

It defines a Tate-Shafarevich cohomology set for m-atic twists of abelian varieties and provides criteria for its finiteness and triviality, advancing understanding of the local-global principle.

Findings

The Tate-Shafarevich set is finite under mild assumptions.

Criteria are established for the triviality of the cohomology set.

Conditions are identified under which the local-global principle holds for m-atic twists.

Abstract

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.