Elliptic curves over $\mathbb{Q}_\infty$ are modular

TL;DR

This paper proves that all elliptic curves over the cyclotomic Z_p extension of Q are modular, extending the understanding of elliptic curves in infinite extensions.

Contribution

It establishes the modularity of elliptic curves over the cyclotomic Z_p extension of Q, a significant generalization of previous modularity results.

Findings

All elliptic curves over the cyclotomic Z_p extension of Q are modular.

The result applies to any prime p, broadening the scope of modularity.

Supports the conjecture that elliptic curves over certain infinite extensions are modular.

Abstract

We show that if p is a prime, then all elliptic curves defined over the cyclotomic Z_p extension of Q are modular.