Torsion of Abelian varieties over solvable extensions of number fields

Abstract: Let $K$ be a number field, and let $A$ be an Abelian variety over $K$ which has no CM over $\overline{K}$. We prove that $A$ has only finitely many torsion points over the maximal $n$-step-solvable extension of $K$ for any $n$ and only finitely many torsion points of prime order over the maximal prosolvable extension of $K$.