Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

2019/09/16 by Cerchia, Michael, Rouse, Jeremy
#11F80 (Primary) #11G05 #12G05 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1909.07468

Abstract

Let ℓ be a prime number and let F be a number field and E/F a non-CM elliptic curve with a point α∈ E(F) of infinite order. Attached to the pair (E,α) is the ℓ-adic arboreal Galois representation ω_E,α,ℓ^∞ : \rm Gal(F/F) → ℤ_ℓ² \rtimes \rm GL₂(ℤ_ℓ) describing the action of \rm Gal(F/F) on points β_n so that ℓⁿ β_n = α. We give an explicit bound on the index of the image of ω_E,α,ℓ^∞ depending on how ℓ-divisible the point α is, and the image of the ordinary ℓ-adic Galois representation. The image of ω_E,α,ℓ^∞ is connected with the density of primes \mathfrakp for which α∈ E(\mathbbF_\mathfrakp) has order coprime to ℓ.

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