Roots of unity and torsion points of abelian varieties

TL;DR

This paper investigates the intersection of cyclotomic extensions and torsion point extensions of abelian varieties, establishing a weak property $()$ for all abelian varieties and showing the stronger property does not hold.

Contribution

It proves a weak version of property $()$ for all abelian varieties and demonstrates the failure of a stronger property, addressing a question by Hindry and Ratazzi.

Findings

Weak property $()$ holds for any abelian variety.

Stronger property considered by Hindry and Ratazzi does not hold.

Provides insights into the intersection of cyclotomic and torsion point extensions.

Abstract

We answer a question raised by Hindry and Ratazzi concerning the intersection between cyclotomic extensions of a number field $K$ and extensions of $K$ generated by torsion points of an abelian variety over $K$. We prove that a weak version of the property $(\mu)$ they introduced holds for any abelian variety, while the same is not true for a slightly stronger version of the property they also considered.