Dedekind Zeta motives for totally real fields

TL;DR

This paper constructs Dedekind zeta motives for totally real fields, computes their periods, and relates these to special values of Dedekind zeta functions, advancing the understanding of motivic Ext groups and hyperbolic geometry.

Contribution

It introduces explicit Dedekind zeta motives in the category of mixed Tate motives over totally real fields and links their periods to zeta function values, confirming conjectured motivic calculations.

Findings

Period of the motive is a rational multiple of ^{n[k:A0Q]} ^*_k(1-n)

Ext^1 group is generated by cohomology of a quadric relative to hyperplanes

Special zeta value relates to volumes of hyperbolic simplices over k

Abstract

Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $\pi^{n[k:\Q]}\zeta^*_k(1-n)$, where $\zeta^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculate the groups $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$. In particular, the special value of the Dedekind zeta function is a determinant of volumes of geodesic hyperbolic simplices defined over $k$.