Vol. 15, No. 3, 2021

Download this article
Download this article For screen
For printing
Recent Issues

Volume 15
Issue 4, 821–1076
Issue 3, 569–820
Issue 2, 309–567
Issue 1, 1–308

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
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