Radical entanglement for elliptic curves

Abstract

Let G be a commutative connected algebraic group over a number field K, let A be a finitely generated and torsion-free subgroup of G(K) of rank r > 0 and, for n > 1, let K n −1 A be the smallest extension of K inside an algebraic closure K over which all the points P ∈ G(K) such that nP ∈ A are defined. We denote by s the unique non-negative integer such that G(K)[n] ∼ = (Z/nZ) s for all n 1. We prove that, under certain conditions, the ratio between n rs and the degree K n −1 A : K(G[n]) is bounded independently of n > 1 by a constant that depends only on the ℓ-adic Galois representations associated with G and on some arithmetic properties of A as a subgroup of G(K)/G(K)tors. In particular we extend the main theorems of [13] about elliptic curves to the case of arbitrary rank.