Resolve the general non-split Cartan case

Classify the rational points associated with the non-split Cartan case of exceptional -elladic Galois images of elliptic curves over \(\mathbb{Q}\) in general.

Background

The paper reviews the classification of possible images of -elladic Galois representations and the corresponding modular curves. It explains that the Borel and split Cartan cases have largely been resolved, while the non-split Cartan case is not settled in full generality. This remains a broader unresolved problem even though the paper completes the specific 7-adic classification addressed in its main theorem.