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

By Xia Gao

Abstract

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.MathematicsSCI(E)0ARTICLE6994-101913

Topics: Orders, Conductors, Binary cubic forms, Zeta functions, BINARY CUBIC FORMS, DISCRIMINANT, COEFFICIENTS, ORDERS
Publisher: journal of number theory
Year: 2011
DOI identifier: 10.1016/j.jnt.2010.11.010
OAI identifier: oai:localhost:20.500.11897/157594
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