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

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.