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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 $$
\bar{\rho}_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow
\operatorname{GL}(E[p]) $$
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 $\bar{\rho}_{E,p}$ 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
$$
I(d)=\{p\text{ prime}:p\leq d-1\}. $$