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PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 100 / Issue 1 / February 2016
- Published online by Cambridge University Press:
- 19 August 2015, pp. 33-41
- Print publication:
- February 2016
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- Article
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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 W. 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.
