Parametrizing elliptic curves by modular units

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.

7 pages, comments welcome!

References in corpus (3)

• Regulator of modular units and Mahler measures
• On the ramification of modular parametrizations at the cusps
• Explicit lower bounds on the modular degree of an elliptic curve

Cited by in corpus (4)

• Regulators of Siegel units and applications
• Mahler measures of a family of non-tempered polynomials and Boyd's conjectures
• Mahler measure of a non-reciprocal family of elliptic curves
• Mahler's measure and elliptic curves with potential complex multiplication