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 $\overline{K}$ over which all the points $P\in G(\overline{K})$ such that $nP\in A$ are defined. We denote by $s$ the unique non-negative integer such that $G(\overline{K})[n]\cong (\mathbb{Z}/n\mathbb{Z}{)}^s$ for all $n\ge 1$. 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 $\ell$-adic Galois representations associated with $G$ and on some arithmetic properties of $A$ as a subgroup of $G(K)$ modulo torsion. In particular we extend the main theorems of [13] about elliptic curves to the case of arbitrary rank.

References in corpus (2)

• Reductions of points on algebraic groups
• Explicit Kummer Theory for Elliptic Curves