Effective Kummer Theory for Elliptic Curves

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 $.

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

“…Remark. The paper [10], by Lombardo and Tronto which was finished at nearly the same time as this present paper, contains a similar result to the one above. In particular, Theorem 4.15 gives a bound on the index of the image of ω E,α,ℓ ∞ | Gal(K∞/T∞) in Z 2 ℓ in terms of the parameter n ℓ (the smallest positive integer so that the image of ρ E,ℓ ∞ contains all matrices ≡ I (mod ℓ n ℓ ).…”

“…The goal of the present paper is to prove uniformity results about the image of ω E,α,ℓ ∞ . In [10] (see Definition 4.1) the authors declare a point α ∈ E(F ) to be strongly ℓindivisible if α + T ∈ ℓE(F ) for any torsion point T ∈ E(F ) of ℓ-power order. This is a natural primitivity condition to impose.…”

Radical entanglement for elliptic curves

“…Our second goal is to apply Theorem 1.1 to some specific cases. In particular, we generalize the results of [13] to the case of arbitrary rank. Theorems 1.2 and 1.3 below follow from Theorems 6.14, 6.16 and 6.17 and Lemma 5.7.…”

“…In [13] Lombardo and the author were able to give an effective bound for the ratio (1) if G = E is an elliptic curve with End K (E) = Z and A = α has rank 1. Moreover, a uniform bound in the case K = Q, under some necessary assumptions on the divisibility of α in E(K)/E(K) tors , was given.…”

“…The goal of the present paper is twofold: firstly, we use the properties of r-extensions of abelian groups introduced by Palenstijn in [14] and [15] to generalize the methods of [13] to groups A of arbitrary finite rank and any commutative connected algebraic group G that satisfies the same properties mentioned above. The result we obtain is the following (see Theorem 5.9): Theorem 1.1.…”