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

Let ℓ\ell be a prime number, let FF be a number field, and let E/FE/F be a non-CM elliptic curve with a point α∈E(F)\alpha \in E(F) of infinite order. Attached to the pair (E,α)(E,\alpha ) is the ℓ\ell-adic arboreal Galois representation ωE,α,ℓ∞:Gal(F¯/F)→Zℓ2⋊GL2(Zℓ)\omega _{E,\alpha ,\ell ^{\infty }} : \mathrm {Gal}(\overline {F}/F) \to \mathbb {Z}_{\ell }^{2} \rtimes \mathrm {GL}_{2}(\mathbb {Z}_{\ell }) describing the action of Gal(F¯/F)\mathrm {Gal}(\overline {F}/F) on points βn\beta _{n} so that ℓnβn=α\ell ^{n} \beta _{n} = \alpha. We give an explicit bound on the index of the image of ωE,α,ℓ∞\omega _{E,\alpha ,\ell ^{\infty }} depending on how ℓ\ell-divisible the point α\alpha is, and the image of the ordinary ℓ\ell-adic Galois representation. The image of ωE,α,ℓ∞\omega _{E,\alpha ,\ell ^{\infty }} is connected with the density of primes p\mathfrak {p} for which α∈E(Fp)\alpha \in E(\mathbb {F}_{\mathfrak {p}}) has order coprime to ℓ\ell.