Serre's uniformity question and proper subgroups of $C_{ns}^+(p)$

Abstract

Serre's uniformity question asks whether there exists a bound $N>0$ such that, for every non-CM elliptic curve $E$ over $\mathbb{Q}$ and every prime $p>N$, the residual Galois representation $\rho_{E,p}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{Aut}(E[p])$ is surjective. The work of many authors has shown that, for $p>37$, this representation is either surjective or has image contained in the normaliser of a non-split Cartan subgroup $C_{ns}^+(p)$. Zywina has further proved that, whenever $\rho_{E,p}$ is not surjective for $p>37$, its image is either $C_{ns}^+(p)$ or a certain subgroup $G(p)$ of $C_{ns}^+(p)$ of index $3$. Recently, Le Fourn and Lemos showed that the index-$3$ case cannot arise for $p>1.4 \cdot 10^7$. We strengthen this result by proving that the image of $\rho_{E, p}$ is not conjugate to $G(p)$ for any prime larger than $5$.