Some cases of Serre's uniformity problem

Abstract

We show that if $E/\mathbb{Q}$ is an elliptic curve without complex multiplication and for which there is a prime $q$ such that the image of $\bar{\rho}_{E,q}$ is contained in the normaliser of a split Cartan subgroup of $\rm{GL}_2(\mathbb{F}_q)$, then $\bar{\rho}_{E,p}$ surjects onto $\rm{GL}_2(\mathbb{F}_p)$ for every prime $p>37$. This result complements a previous result by the author. We also prove analogue results for certain families of $\mathbb{Q}$-curves, building on results of Ellenberg (2004) and Le Fourn (2016).