Galois module structure and elliptic functions (Q1893470)

The subject of this paper is the problem of determining the Galois module structure of rings of integers \({\mathcal O}_L\) of abelian extensions \(L\) of a number field \(K\). By a classical result of Leopoldt \({\mathcal O}_L\) is free as a module over the associated order. More recently work of Cassou-Nogùes, Chan, Taylor, Schertz and Srivastav has dealt with cases where \(K\) and \(L\) are ray class fields over a quadratic imaginary field. The present paper establishes that the ring of integers is free over the associated order if \(K\) and \(L\) are generated over suitable ring class fields over quadratic imaginary number fields by values of Weber functions.