The modularity of elliptic curves over all but finitely many totally real fields of degree 5

TL;DR

This paper proves that elliptic curves over almost all totally real fields of degree 5 are modular, using finiteness results for rational points on certain modular curves and their quotients.

Contribution

It establishes the modularity of elliptic curves over all but finitely many degree 5 totally real fields, extending previous modularity results.

Findings

Finiteness of low degree points on modular curves proven

Criterion for finiteness of degree 5 rational points established

Modularity of elliptic curves over most degree 5 totally real fields confirmed

Abstract

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the way, we prove a criterion for the finiteness of rational points of degree $5$ on a curve of large genus over a number field using the results of Abramovich--Harris and Faltings on subvarieties of Jacobians.