Parametrizing elliptic curves by modular units

TL;DR

This paper proves that only finitely many elliptic curves over the rationals can be parametrized by modular units, providing a complete list for those with conductor up to 1000 and discussing related open questions.

Contribution

It establishes the finiteness of elliptic curves parametrized by modular units over  and lists all such curves with conductor up to 1000.

Findings

Finitely many elliptic curves over  are parametrized by modular units.

Complete list of such curves with conductor  1000 provided.

Raises open questions about modular parametrizations.

Abstract

It is well-known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to $1000$ parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.