Local-Global principles for certain images of Galois representations

TL;DR

This paper classifies the possible global images of Galois representations of elliptic curves over number fields when local images are contained in Cartan subgroups or their normalizers, revealing counterexamples to the local-global principle over .

Contribution

It provides a classification of global Galois images for elliptic curves with local images in Cartan subgroups, and constructs explicit counterexamples over .

Findings

Classified possible global Galois images for certain local conditions.

Constructed explicit counterexamples to the local-global principle over .

Identified at least three elliptic curves with specific Galois image properties.

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.