Roots of unity and torsion points of abelian varieties

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 the property called (μ) in Hindry and Ratazzi (J Ramanujan Math Soc 25(1):81–111, 2010) holds for any abelian variety, while the same is not true for the stronger version of the property introduced in Hindry and Ratazzi (J Inst Math Jussieu 11(1):27–65, 2012)