Elliptic curves over totally real quartic fields not containing $\sqrt{5}$ are modular

TL;DR

This paper proves that all elliptic curves over certain degree 4 totally real fields are modular, focusing on fields not containing 75; it involves analyzing quartic points on modular curves.

Contribution

The paper establishes modularity of elliptic curves over specific quartic fields, extending previous results to new field classes.

Findings

All elliptic curves over these fields are modular.

Quartic points on modular curves are key to the proof.

The result excludes fields containing 75;

Abstract

We prove that every elliptic curve defined over a totally real number field of degree 4 not containing $\sqrt{5}$ is modular. To this end, we study the quartic points on four modular curves.