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

2024/12/13 by Furio, Lorenzo · 4 citations
#11F80 #11G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2412.10340

Abstract

Given an elliptic curve E defined over 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 h_F(E) is the stable Faltings height of E, we show that [GL₂(\widehatℤ) : Imρ_E] is bounded above by 10²¹ (h_F(E)+40)^4.42, and, for h_F(E) tending to infinity, by h_F(E)^3+o(1). We also classify the possible (conjecturally non-existent) images of the representations ρ_E,pⁿ whenever Imρ_E,p is contained in the normaliser of a non-split Cartan. This result improves previous work of Zywina and Lombardo.

Cited by

• Isolated points on modular curves of prime-power level
• On 7-adic Galois representations for elliptic curves over ℚ
• Effective open image theorem and a Linnik type problem for elliptic curves
• Rational isolated j-invariants from X₁(ℓⁿ) and X₀(ℓⁿ)

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