Vol. 15, No. 3, 2021

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

Volume 16
Issue 9, 2005–2264
Issue 8, 1777–2003
Issue 7, 1547–1776
Issue 6, 1327–1546
Issue 5, 1025–1326
Issue 4, 777–1024
Issue 3, 521–775
Issue 2, 231–519
Issue 1, 1–230

Volume 15, 10 issues

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
This article is available for purchase or by subscription. See below.
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}.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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