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

Abstract

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

${\bar{\rho }}_{E,p}:Gal\left(\overline{\mathbb{Q}}∕\mathbb{Q}\right)\to GL\left(E\left[p\right]\right)$

of $E$ is either the whole of $GL\left(E\left[p\right]\right)$, or is contained in the normaliser of a nonsplit Cartan subgroup of $GL\left(E\left[p\right]\right)$. In this paper, we show that when , the image of ${\bar{\rho }}_{E,p}$ is either $GL\left(E\left[p\right]\right)$, or the full normaliser of a nonsplit Cartan subgroup. We use this to show the following result, partially settling a question of Najman. For $d\ge 1$, let $I\left(d\right)$ denote the set of primes $p$ for which there exists an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\le d$. We show that, for $d\ge 1.4\times 10^7$, we have

$I\left(d\right)=\left\{p\text{ prime} : p\le d-1\right\}.$