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

@article{Cerchia2019UniformBO, title={Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves}, author={Michael Cerchia and Jeremy A. Rouse}, journal={arXiv: Number Theory}, year={2019} }

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…

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