On the zeta function associated with module classes of a number field

JOURNAL OF NUMBER THEORY

卷:131

Article

Gao, Xia

关键词: Orders;    Conductors;    Binary cubic forms;    Zeta functions;   

DOI  :  10.1016/j.jnt.2010.11.010

来源: Elsevier

【 摘 要 】

Text. The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d - 1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d - 1 of K and the integral ideals of width < d - 1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields. Video. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE. (C) 2011 Published by Elsevier Inc.

【 授权许可】

Free