Modularity over $\mathbb{C}$ implies modularity over $\mathbb{Q}$

TL;DR

This paper explains Mazur's proof that modularity over complex numbers for elliptic curves implies modularity over rationals, and discusses related open questions on elliptic curve uniformizations.

Contribution

It provides an exposition of Mazur's proof linking complex and rational modularity for elliptic curves and discusses open problems in the field.

Findings

Mazur's proof establishes the link between complex and rational modularity.

Discussion of open questions on noncongruence modular curves.

Clarification of the relationship between different modular parametrizations.

Abstract

We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generalised Fermat Equation" held online and organised by Bhaskaracharya Pratishthana in Pune, India.