Local-Global principles for certain images of Galois representations

Abstract: Let $K$ be a number field and let $E/K$ be an elliptic curve whose mod $\ell$ Galois representation locally has image contained in a group $G$, up to conjugacy. We classify the possible images for the global Galois representation in the case where $G$ is a Cartan subgroup or the normalizer of a Cartan subgroup. When $K = \mathbf{Q}$, we deduce a counterexample to the local-global principle in the case where $G$ is the normalizer of a split Cartan and $\ell = 13$. In particular, there are at least three elliptic curves (up to twist) over $\mathbf{Q}$ whose mod $13$ image of Galois is locally contained in the normalizer of a split Cartan, but whose global image is not.