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2021 Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan
Samuel Le Fourn, Pedro Lemos
Algebra Number Theory 15(3): 747-771 (2021). DOI: 10.2140/ant.2021.15.747

Abstract

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 d1, 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 d1.4×107, we have

I(d)={p  prime : pd1}.

Citation

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Samuel Le Fourn. Pedro Lemos. "Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan." Algebra Number Theory 15 (3) 747 - 771, 2021. https://doi.org/10.2140/ant.2021.15.747

Information

Received: 5 February 2020; Revised: 19 August 2020; Accepted: 10 October 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/ant.2021.15.747

Subjects:
Primary: 11G05
Secondary: 11G18

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

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.15 • No. 3 • 2021
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