On modular units
type: Item
instance of: scholarly article
title: On modular units
zbMATH Open document ID: 0663.10025
DOI: 10.1007/BF01455070
author: Anthony J. Scholl
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164614
review text: Let n be a positive integer, \(U_ n\) the group of modular units of level n, and \(U_{n/{\mathbb{Z}}}\) the subgroup of integral modular units, as defined by \textit{D. Kubert} and \textit{S. Lang} [Math. Ann. 218, 175-189 (1975; Zbl 0311.14005)]. Kubert and Lang determined the rank of \(U_{n/{\mathbb{Z}}}\) by finding generators for \(U_ n\) and calculating their q-expansions. In this note we use the geometry of the moduli schemes of elliptic curves (following an idea of Beilinson) to give a description of \(U_{n/{\mathbb{Z}}}\otimes {\mathbb{Q}}\) as a module over the Hecke algebra. In particular, this gives a simple way to calculate the rank of the group of integral modular units for any reasonable congruence subgroup of \(SL_ 2({\mathbb{Z}})\).
Mathematics Subject Classification ID: 11F99
Mathematics Subject Classification ID: 14K99
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 4083725
zbMATH Keywords: moduli schemes of elliptic curves
zbMATH Keywords: Hecke algebra
zbMATH Keywords: rank of the group of integral modular units
zbMATH Keywords: congruence subgroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4083725