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

[摘要] Let K be a quadratic imaginary number field, f is an element of N and let of be the order of conductor f in K. We consider the singular values of the Kleinian normalization phi of the Weierstrass sigma-function belonging to an arbitrary proper ideal of O-f. 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 Korpererweiterungen von imaginar-quadratischen Zahlkorpern, Dissertation, Universitat Augsburg, 19911 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. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2007-12-01

[关键词] Kleinian phi-function;principal ideal theorem;complex multiplication;factorization of the singular values of phi