Effective Kummer Theory for Elliptic Curves

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