# Elliptic Curves over Totally Real Cubic Fields are Modular

## TL;DR

This paper proves that all elliptic curves over totally real cubic fields are modular, extending previous results from quadratic fields and recent breakthroughs in the field.

## Contribution

It establishes the modularity of elliptic curves over totally real cubic fields, advancing the understanding of elliptic curves in higher degree number fields.

## Key findings

- All elliptic curves over totally real cubic fields are modular.
- Builds on prior work for quadratic fields and recent breakthroughs.
- Extends modularity results to cubic fields.

## Abstract

We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.