Residual Galois representations of elliptic curves with image contained in the normaliser of a non-split Cartan

Abstract

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$ \barρ_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}(E[p]) $$ of $E$ is either the whole of $\operatorname{GL}(E[p])$, or is \emph{contained} in the normaliser of a non-split Cartan subgroup of $\operatorname{GL}(E[p])$. In this paper, we show that when $p>1.4\times 10^7$, the image of $\barρ_{E,p}$ is either $\operatorname{GL}(E[p])$, or the \emph{full} normaliser of a non-split 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 $\mathbb{Q}$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\leq d$. We show that, for $d\geq 1.4\times 10^7$, we have $$ I(d)=\{p\text{ prime}:p\leq d-1\}. $$