The motivic Galois group for a double zeta value

TL;DR

This paper computes the minimal motives and motivic Galois groups for double zeta values, providing insights into their algebraic relations and the structure of their periods within mixed Tate motives.

Contribution

It explicitly determines the minimal motives and motivic Galois groups for double zeta values, advancing understanding of their algebraic and motivic structures.

Findings

Computed minimal motives for double zeta values

Determined the motivic Galois groups and their dimensions

Provided period matrices and discussed algebraic relations

Abstract

We consider multiple zeta values, which are periods of mixed Tate motives over $\mathbb Z$. For a given multiple zeta value $\zeta$, there exists a unique minimal motive $M(\zeta)$ such that $\zeta$ is a period of $M(\zeta)$. In general, the motive $M(\zeta)$ is difficult to compute. In this article, we compute the minimal motive $M(a,b)$ associated to a given double zeta value $\zeta(a,b)$. We also compute the motivic Galois group $G(a,b)$ associated to $\zeta(a,b)$ and discuss its dimension. Moreover, we give a period matrix of $M(a,b)$. The period conjecture predicts that the dimension of $G(a,b)$ equals the transcendence degree of the algebra of periods of $M(a,b)$. Hence our results lead to conjectures about algebraic relations between single and double zeta values.