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 $ρ_E$. In particular, if $\operatorname{h}{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\operatorname{GL}2(\widehat{\mathbb{Z}}) : \operatorname{Im}ρ_E]$ is bounded above by $10^{21} (\operatorname{h}{\mathcal{F}}(E)+40)^{4.42}$, and, for $\operatorname{h}{\mathcal{F}}(E)$ tending to infinity, by $\operatorname{h}{\mathcal{F}}(E)^{3+o(1)}$. We also classify the possible (conjecturally non-existent) images of the representations $ρ{E,p^n}$ whenever $\operatorname{Im}ρ_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo.

💡 Research Summary

The paper studies the adelic Galois representation attached to a non‑CM elliptic curve E over ℚ. For such a curve the representation
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