Modularity of elliptic curves over cyclotomic $\mathbb{Z}_p$-extensions of real quadratic fields

TL;DR

This paper proves the modularity of elliptic curves over cyclotomic $Z_p$-extensions of real quadratic fields, assuming certain $p$-adic unit conditions on twisted $L$-functions, generalizing prior results by Zhang and Thorne.

Contribution

It extends modularity results to a broader class of elliptic curves over real quadratic fields under specific $p$-adic conditions.

Findings

All elliptic curves over the specified extensions are modular under the assumptions.

Generalizes Zhang's real quadratic analogue of Thorne's result.

Provides new insights into the $p$-adic properties of $L$-functions for elliptic curves.

Abstract

We prove that all elliptic curves defined over the cyclotomic $\mathbb{Z}_p$-extension of a real quadratic field are modular under the assumption that the algebraic part of the central value of a twisted $L$-function is a $p$-adic unit. Our result is a generalization of a result of X. Zhang, which is a real quadratic analogue of a result of Thorne.