Effective bounds for adelic Galois representations attached to elliptic curves over the rationals

Given an elliptic curve $E$ defined over $\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation ${\rho }_E$. In particular, if ${\mathrm{h}}_{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[{\mathrm{GL}}_2(\widehat{\mathbb{Z}}):\mathrm{Im}{\rho }_E]$ is bounded above by ${10}^{21}({\mathrm{h}}_{\mathcal{F}}(E)+40{)}^{4.42}$, and, for ${\mathrm{h}}_{\mathcal{F}}(E)$ tending to infinity, by ${\mathrm{h}}_{\mathcal{F}}(E{)}^{3+o(1)}$. We also classify the possible (conjecturally non-existent) images of the representations ${\rho }_{E,p^n}$ whenever $\mathrm{Im}{\rho }_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo.