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arXiv · 1504.00999

Parametrizing elliptic curves by modular units

François Brunault
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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.

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François Brunault. 2015-04-04. Parametrizing elliptic curves by modular units. https://doi.org/10.1017/s1446788715000233

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