Galois Module Structure and Elliptic Functions

Abstract

Let K be a quadratic imaginary number field and R f the ring class field module f over K, f ∈ N . Let O f denote the order of conductor f in K and let g *, g be proper O f -ideals such that g * 2 ⊆ g ⊆ g *. Let τ denote the Weber function and let I denote an auxiliary O f -ideal. For certain extensions R f (τ(1 | I g ))/ R f (τ(1 | I g *)) it is shown that the ring of integers in R f (τ(1 | I g )) is a free rank one module over the associated order of R f (τ(1 | I g ))/ R f (τ(1 | I g *)).