Vol. 15, No. 3, 2021

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Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan

Samuel Le Fourn and Pedro Lemos

Vol. 15 (2021), No. 3, 747–771

It is known that if p > 37 is a prime number and E is an elliptic curve without complex multiplication, then the image of the mod p Galois representation

ρ̄E,p : Gal(¯) GL(E[p])

of E is either the whole of GL(E[p]), or is contained in the normaliser of a nonsplit Cartan subgroup of GL(E[p]). In this paper, we show that when p > 1.4 × 107, the image of ρ̄E,p is either GL(E[p]), or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For d 1, let I(d) denote the set of primes p for which there exists an elliptic curve defined over and without complex multiplication admitting a degree p isogeny defined over a number field of degree d. We show that, for d 1.4 × 107, we have

I(d) = {p  prime : p d 1}.
Galois representations of elliptic curves, Serre uniformity problem, nonsplit Cartan subgroup
Mathematical Subject Classification 2010
Primary: 11G05
Secondary: 11G18
Received: 5 February 2020
Revised: 19 August 2020
Accepted: 10 October 2020
Published: 20 May 2021
Samuel Le Fourn
Institut Fourier
Université Grenoble Alpes
Saint-Martin d’Hères
Pedro Lemos
Mathematics Department
University College London
United Kingdom