Some cases of Serre's uniformity problem

TL;DR

This paper investigates conditions under which elliptic curves over rationals have Galois representations with large images, proving surjectivity for all primes greater than 37 under certain subgroup containment conditions, and extends results to families of Q-curves.

Contribution

It establishes new uniformity results for Galois representations of elliptic curves without complex multiplication, including for Q-curves, complementing previous work.

Findings

Surjectivity of Galois representations for primes > 37 under specific subgroup conditions.

Extension of uniformity results to certain families of Q-curves.

Complementary to previous results by the author.

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).