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

TL;DR

This work delivers explicit, sharp bounds for the index of the adelic Galois image of non-CM elliptic curves over $\mathbb{Q}$ in terms of the stable Faltings height and, separately, the conductor. It combines a refined group-theoretic analysis of $p$-adic images (Cartan and non-split Cartan cases), effective surjectivity theorems, and control of entanglement among division fields to produce global adelic bounds. The main results improve prior bounds by Zywina and Lombardo and provide effective, quantitative descriptions of the possible $p$-adic images when the residual image lies in the normaliser of a non-split Cartan. The paper also extends height-conductor comparisons and uses modular-curve integral-point results to handle exceptional and small-prime cases, yielding both asymptotic and explicit finite bounds with potential computational applications in determining open-image indices from $j$-invariants.

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 $ρ_E$. 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.

Theorems & Definitions (147)

• Conjecture 1.2
• Theorem 1.3: Mazur
• Theorem 1.4
• Theorem 1.5
• proof
• Theorem 1.6: Zywina
• Theorem 1.7: Lombardo
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1