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PARAMETRIZING ELLIPTIC CURVES BY MODULAR UNITS

Part of: Discontinuous groups and automorphic forms Curves Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 August 2015

FRANÇOIS BRUNAULT
FRANÇOIS BRUNAULT*
Affiliation:
ÉNS Lyon, UMPA, 46 allée d’Italie, 69007 Lyon, France email francois.brunault@ens-lyon.fr
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Abstract

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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.

Keywords

elliptic curvesmodular parametrizationmodular units

MSC classification

Primary: 11F03: Modular and automorphic functions
Secondary: 11G05: Elliptic curves over global fields 11G16: Elliptic and modular units 14H52: Elliptic curves
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 33 - 41
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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