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Residual Galois representations of elliptic curves with image contained in the normaliser of a nonsplit Cartan
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-05-20, DOI: 10.2140/ant.2021.15.747
Samuel Le Fourn, Pedro Lemos

It is known that if $p>37$ is a prime number and $E∕ℚ$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation

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

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 $p>1.4×1{0}^{7}$, the image of ${\stackrel{̄}{\rho }}_{E,p}$ 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 $ℚ$ and without complex multiplication admitting a degree $p$ isogeny defined over a number field of degree $\le d$. We show that, for $d\ge 1.4×1{0}^{7}$, we have

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