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

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