Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Residual Galois representations of elliptic curves with image contained
  in the normaliser of a non-split Cartan   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
  $$
  I(d)=\{p\text{ prime}:p\leq d-1\}. $$

37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
16
The categories of this paper are: stat
15
The categories of this paper are: nucl-ex
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph