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

TL;DR

This paper establishes explicit bounds on the size of the image of adelic Galois representations for non-CM elliptic curves over rationals, linking it to the Faltings height and improving previous bounds.

Contribution

It provides the first explicit sharp bounds on the adelic Galois image index in terms of the Faltings height, refining earlier results by Zywina and Lombardo.

Findings

Bound on the index: at most $10^{21} ( ext{height}+40)^{4.42}$.

As height tends to infinity, the index is bounded by the height to the power 3+o(1).

Classification of possible images when the local image is in the normaliser of a non-split Cartan.

Abstract

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 $\operatorname{h}_{\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\operatorname{GL}_2(\widehat{\mathbb{Z}}) : \operatorname{Im}\rho_E]$ 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 $\rho_{E,p^n}$ whenever $\operatorname{Im}\rho_{E,p}$ is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo.