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

Le Fourn, Samuel; Lemos, Pedro

doi:10.2140/ant.2021.15.747

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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

DOI: 10.2140/ant.2021.15.747

Abstract

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

{\bar{ρ}}_{E, p} : Gal (\bar{ℚ} ∕ ℚ) \to 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 $p > 1.4 \times 1 0^{7}$ , the image of ${\bar{ρ}}_{E, p}$ 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 $d \geq 1$ , let $I (d)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $ℚ$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$ . We show that, for $d \geq 1.4 \times 1 0^{7}$ , we have

I (d) = {p prime : p \leq d - 1} .

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Keywords

Galois representations of elliptic curves, Serre uniformity problem, nonsplit Cartan subgroup

Mathematical Subject Classification 2010

Primary: 11G05

Secondary: 11G18

Milestones

Received: 5 February 2020

Revised: 19 August 2020

Accepted: 10 October 2020

Published: 20 May 2021

Authors

Samuel Le Fourn
Institut Fourier Université Grenoble Alpes Saint-Martin d’Hères France

Pedro Lemos
Mathematics Department University College London United Kingdom

Vol. 15, No. 3, 2021