Radical entanglement for elliptic curves

TL;DR

This paper extends previous results on elliptic curves to general algebraic groups, analyzing the degree of field extensions generated by points related to a subgroup and establishing bounds based on Galois representations.

Contribution

It generalizes main theorems from elliptic curves to arbitrary commutative algebraic groups, providing bounds on degrees of certain field extensions.

Findings

The ratio between n^{rs} and the extension degree is bounded independently of n.

The bounds depend only on Galois representations and arithmetic properties of the subgroup.

Extends results from elliptic curves to algebraic groups of arbitrary rank.

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 $n\geq 1$. 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.