Parametrizing elliptic curves by modular units

François Brunault 1
1 Algèbre. Théorie des nombres
UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
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 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. Parametrizing elliptic curves by modular units. Journal of the Australian Mathematical Society, 2016, 100, pp.33 - 41. ⟨10.1017/S1446788715000233⟩. ⟨ensl-01400935⟩

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