The author defined multiple Dedekind zeta values in [J. Reine Angew. Math. 722, 65–104 (2017; Zbl 1391.11152)]. For a quadratic number field \(K\), multiple Dedekind zeta values are
\[ \zeta_{K;C}(s_1,s_2,\dots,s_m)=\sum_{\alpha_1,\dots,\alpha_m\in C}\frac{1}{N(\alpha_1)^{s_1}N(\alpha_1+\alpha_2)^{s_2}\dots N(\alpha_1+\dots+\alpha_m)^{s_m}}, \]
where \(s_1,\ldots, s_m\) are positive integers and \(s_m\ge 2\) and \(C\) is a cone generated by a totally positive unit \(\beta\) in \(K\) and \(1\), defined by
\[ C=\mathbb{N}\{1,\beta\}=\{\gamma\in K|\gamma=a+b\beta, \text{ for } a,b\geq 0\}. \]
In this paper, the author mainly proves that:
Theorem. Let \(K\) be a real quadratic field, and let \(C\) be a cone generated by a totally positive unit \(\beta\) in \(K\) and \(1\). Then the multiple Dedekind zeta value
\[ (\beta_2-\beta_1)^3\zeta_{K;C}(1,2) \]
is a period of mixed Tate motive over the ring of integers in \(K\).
The main idea is that by a special integral representation, \((\beta_2-\beta_1)^3\zeta_{K;C}(1,2)\) is interpreted as a period of
\[ H^6(\overline{\mathcal{M}}_{0,15}-A;B-A\cap B) \]
for some subvarities \(A\) and \(B\). Thus from A. B. Goncharov and Yu. I. Manin’s result [Compos. Math. 140, No. 1, 1–14 (2004; Zbl 1047.11063)], \((\beta_2-\beta_1)^3\zeta_{K;C}(1,2)\) is a mixed Tate motive over \(\mathcal{O}_K\).
For the entire collection see [Zbl 1388.00033].