Radical entanglement for elliptic curves

Let $G$ be a commutative connected algebraic group over a number field $K$, let $A$ be a finitely generated and torsion-free subgroup of $G(K)$ of rank $r>0$ and, for $n>1$, let $K(n^{-1}A)$ be the smallest extension of $K$ inside an algebraic closure $\overline K$ over which all the points $P\in G(\overline K)$ such that $nP\in A$ are defined. We denote by $s$ the unique non-negative integer such that $G(\overline K)[n]\cong (\mathbb Z/n\mathbb Z)^s$ for all $n\geq 1$. We prove that, under certain conditions, the ratio between $n^{rs}$ and the degree $[K(n^{-1}A):K(G[n])]$ is bounded independently of $n>1$ by a constant that depends only on the $\ell$-adic Galois representations associated with $G$ and on some arithmetic properties of $A$ as a subgroup of $G(K)$ modulo torsion. In particular we extend the main theorems of [13] about elliptic curves to the case of arbitrary rank.