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 $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha )$ is the $\ell$-adic arboreal Galois representation ${\omega }_{E,\alpha ,{\ell }^{\infty }}:\mathrm{Gal}(\overline{F}/F)\to {\mathbb{Z}}_{\ell }^2⋊{\mathrm{GL}}_2({\mathbb{Z}}_{\ell })$ describing the action of $\mathrm{Gal}(\overline{F}/F)$ 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 $\ell$.

While the main result is similar to some results of Lombardo and Tronto (1909.05376), our results were obtained independently and using somewhat different methods. 8 pages