Effective Kummer Theory for Elliptic Curves

Let $E$ be an elliptic curve defined over a number field $K$, let $alpha in E(K)$ be a point of infinite order, and let $N^{-1}alpha$ be the set of $N$-division points of $alpha$ in $E(overline{K})$. We prove strong effective and uniform results for the degrees of the Kummer extensions $[K(E[N],N^{-1}alpha) : K(E[N])]$. When $K=mathbb Q$, and under a minimal (necessary) assumption on $alpha$, we show that the inequality $[mathbb Q(E[N],N^{-1}alpha) : mathbb Q(E[N])] geq cN^2$ holds for a positive constant $c$ independent of both $E$ and $alpha$.