On modular units
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 \(SL_ 2({\mathbb{Z}})\).
rank of the group of integral modular units
scientific article; zbMATH DE number 4083725