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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 assumption on $\alpha$, we show that the inequality $[{\mathbb{Q}}(E[N],N^{-1}\alpha) : {\mathbb{Q}}(E[N])] {\geqslant}cN^2$ holds with a constant $c$ independent of both $E$ and $\alpha$.'
address:
- 'Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy'
- 'Mathematics Research Unit, University of Luxembourg, 6 av. de la Fonte, 4364 Esch-sur-Alzette, Luxembourg'
author:
- Davide Lombardo
- Sebastiano Tronto
bibliography:
- 'biblio.bib'
title: Explicit Kummer Theory for Elliptic Curves
---

Introduction
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Setting
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Let $E$ be an elliptic curve defined over a number field $K$ (for which we fix an algebraic closure ${\overline{K}}$) and let $\alpha\in E(K)$ be a point of infinite order. The purpose of this paper is to study the extensions of $K$ generated by the division points of $\alpha$; in order to formally introduce these extensions we need to set some notation.

Given a positive 