On the modularity of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension of some real quadratic fields

TL;DR

This paper extends the proof of elliptic curve modularity from the rational numbers to certain real quadratic fields within their cyclotomic $ ext{Z}_p$-extensions, under specific assumptions.

Contribution

It proves the modularity of elliptic curves over the $ ext{Z}_p$-extensions of some real quadratic fields, generalizing Thorne's results from $ ext{Q}$ to these fields.

Findings

Modularity of elliptic curves over specific real quadratic fields established.

Extension of Thorne's method to real quadratic fields demonstrated.

Conditional proof based on certain assumptions.

Abstract

The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime $ p $, every elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is modular. The method is to use some automorphy lifting theorems and study non-cusp points on some specific elliptic curves by Iwasawa theory for elliptic curves. Since the modularity of elliptic curves over real quadratic was proved, one may ask whether it is possible to replace $ \mathbb{Q} $ with a real quadratic field $ K $. Following Thorne's idea, we give some assumptions first and prove the modularity of elliptic curves over the $\mathbb{Z}_p$-extension of some real quadratic fields.