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

Let \ell be a prime number, let F be a number field, and let E/F be 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 }} : \textnormal {Gal}(\overline {F}/F) \to \mathbb{Z}_{\ell }^{2} \rtimes \textnormal {GL}_{2}(\mathbb{Z}_{\ell }) describing the action of \textnormal {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 .