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 ${\mathrm{GL}}_2({\mathbb{F}}_{\mathrm{q}})$, then ${\bar{\rho }}_{E,p}$ surjects onto ${\mathrm{GL}}_2({\mathbb{F}}_{\mathrm{p}})$ 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).

Changes made: some typos were corrected