An explicit factorization formula for the singular values of φ and the generalized principal ideal theorem

Let K be a quadratic imaginary number field, f∈N and let Of be the order of conductor f in K. We consider the singular values of the Kleinian normalization φ of the Weierstrass σ-function belonging to an arbitrary proper ideal of Of. The factorization of these singular values goes back to K. Ramachandra, R. Schertz and W. Bley. But the factorization formula in [W. Bley, Konstruktion von Ganzheitsbasen in abelschen Körpererweiterungen von imaginär-quadratischen Zahlkörpern, Dissertation, Universität Augsburg, 1991] is very implicit and not easy to handle in view of many practical applications. In this paper we provide an explicit factorization formula and give different tools to control this factorization. As an immediate application we prove the generalized principal ideal theorem in the ring class field situation.