On the modularity of elliptic curves over a composite field of some real quadratic fields

TL;DR

This paper establishes a sufficient condition for the modularity of all elliptic curves over a composite field formed from real quadratic fields, advancing understanding of elliptic curve modularity in composite fields.

Contribution

It provides a new criterion ensuring the modularity of elliptic curves over certain composite fields of real quadratic fields.

Findings

All elliptic curves over the specified composite fields are modular under the given condition.

The paper extends modularity results to a broader class of number fields.

It offers a framework for future research on elliptic curves over composite fields.

Abstract

Let $K$ be a composite field of some real quadratic fields. We give a sufficient condition on $K$ such that all elliptic curves over $K$ is modular.