Modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7

TL;DR

This paper proves that elliptic curves over certain abelian totally real fields unramified at 3, 5, and 7 are modular, extending the understanding of modularity in these specific number fields.

Contribution

It establishes the modularity of elliptic curves over abelian totally real fields unramified at 3, 5, and 7, broadening the class of fields where modularity is confirmed.

Findings

Elliptic curves over specified fields are proven to be modular.

The result extends previous modularity theorems to new field classes.

The paper improves upon earlier work on modularity of elliptic curves.

Abstract

This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation". Let $E$ be an elliptic curve over an abelian totally real field $K$ unramified at 3,5, and 7. We prove that $E$ is modular.