Galois module structure of Kummer orders and elliptic units (Q1325188)

Let \(K\) be a quadratic imaginary number field. Let \(f\) denote a rational integer which is coprime to 6. Let \(K(f^ 2)\) (resp. \(K(f)\)) denote the ray class field of \(K\) of conductor \(f^ 2\) (resp. \(f\)). Let \(\Gamma\) denote the Galois group of \(K(f^ 2)/ K(f)\) and let \({\mathfrak M}\) denote the unique maximal order of \(K(f)\Gamma\). It is shown that the Kummer order \(\widetilde {\mathcal O}_{K(f^ 2)}= \{\alpha\in K(f^ 2)\mid \alpha{\mathfrak M}\subseteq {\mathcal O}_{K(f^ 2)}\}\) is a free, rank one \({\mathfrak M}\)-module. An explicit generator is given by singular values of certain modular functions.