Multiple Dedekind Zeta Values are Periods of Mixed Tate Motives

TL;DR

This paper proves that multiple Dedekind zeta values are periods of mixed Tate motives, extending the understanding of their algebraic and geometric nature over number fields, especially for totally real fields.

Contribution

It establishes that all multiple Dedekind zeta values over any number field are periods of mixed Tate motives, generalizing previous results for quadratic fields.

Findings

Multiple Dedekind zeta values are periods of mixed Tate motives.

For totally real fields, these values can be unramified over the ring of algebraic integers.

The motives are constructed using moduli spaces of genus zero curves with marked points.

Abstract

Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value over any number field K is a period of a mixed Tate motive. Moreover, if K is a totally real number field, then we can choose a cone C so that every multiple Dedekind zeta associated to the pair (K;C) is unramified over the ring of algebraic integers in K. In [7], the author proves similar statements in the special case of a real quadratic fields for a particular type of a multiple Dedekind zeta values. The mixed motives are defined over K in terms of a the Deligne-Mumford compactification of the moduli space of curves of genus zero with n marked points.