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

TL;DR

This paper establishes explicit bounds on the size of the image of arboreal Galois representations attached to non-CM elliptic curves, linking the bounds to the divisibility properties of a point and the Galois image.

Contribution

It provides the first explicit bounds on the index of the arboreal Galois representation image for non-CM elliptic curves, depending on point divisibility and Galois image.

Findings

Bound depends on the divisibility of the point α by powers of ℓ

Bound relates to the image of the ordinary ℓ-adic Galois representation

Results connect the image size to prime density for certain point orders

Abstract

Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $\beta_{n}$ so that $\ell^{n} \beta_{n} = \alpha$. We give an explicit bound on the index of the image of $\omega_{E,\alpha,\ell^{\infty}}$ depending on how $\ell$-divisible the point $\alpha$ is, and the image of the ordinary $\ell$-adic Galois representation. The image of $\omega_{E,\alpha,\ell^{\infty}}$ is connected with the density of primes $\mathfrak{p}$ for which $\alpha \in E(\mathbb{F}_{\mathfrak{p}})$ has order coprime to $\ell$.