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On modular units. (English) Zbl 0663.10025
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 D. Kubert and 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 \(SL_ 2({\mathbb{Z}})\).
Reviewer: A.J.Scholl

11F99 Discontinuous groups and automorphic forms
14K99 Abelian varieties and schemes
11F06 Structure of modular groups and generalizations; arithmetic groups
Full Text: DOI EuDML
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