Effective Kummer Theory for Elliptic Curves

Abstract

Abstract 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 $.