Periods of Mixed Tate Motives over Real Quadratic Number Rings

TL;DR

This paper constructs explicit examples of mixed Tate motives over the ring of integers in real quadratic fields, linking their periods to multiple Dedekind zeta values at specific points, thus advancing understanding of motives over number rings.

Contribution

It explicitly constructs non-trivial mixed Tate motives over real quadratic number rings and relates their periods to multiple Dedekind zeta values, providing concrete examples.

Findings

Explicit examples of mixed Tate motives over real quadratic fields.

Periods expressed as multiple Dedekind zeta values at (1,2).

Connection established between motives and special zeta values.

Abstract

Recently, the author defined multiple Dedekind zeta values \cite{MDZF} associated to a number $K$ field and a cone $C$. In this paper we construct explicitly non-trivial examples of mixed Tate motives over the ring of integers in $K$, for a real quadratic number field $K$ and a particular cone C. The period of such a motive is a multiple Dedekind zeta values at $(s_1,s_2)=(1,2)$, associated to the pair $(K;C)$, times a nonzero element of $K$.