Effective Kummer Theory for Elliptic Curves

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