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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

##### MSC:
 11F99 Discontinuous groups and automorphic forms 14K99 Abelian varieties and schemes 11F06 Structure of modular groups and generalizations; arithmetic groups
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##### References:
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