Mixed Tate motives and cyclotomic multiple zeta values of level $2^n$ or $3^n$

TL;DR

This paper demonstrates that the ring of motivic periods for certain mixed Tate motives is generated by cyclotomic multiple zeta values at levels that are powers of 2 or 3, extending known results to new levels.

Contribution

It generalizes previous results by showing the spanning of the motivic period ring for levels $2^n$ and $3^n$, and proves the faithfulness of the Galois action on the fundamental group.

Findings

The motivic period ring is spanned by cyclotomic multiple zeta values of level $N$ for $N$ a power of 2 or 3.

The Galois action on the fundamental group is faithful for these levels.

Extension of known results to new levels beyond the previously studied cases.

Abstract

Let $N$ be a power of $2$ or $3$, and $\mu_{N}$ the set of $N$-th roots of unity. We show that the ring of motivic periods of Mixed Tate motives over $\mathbb{Z}[\mu_{N},\frac{1}{N}]$ is spanned by the motivic cyclotomic multiple zeta values of level $N$. This implies that the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[\mu_{N},\frac{1}{N}]$ on the motivic fundamental group of $\mathbb{G}_{m}-\mu_{N}$ is faithful. This is a generalization of the known results for $N\in\{1,2,3,4,8\}$ by Deligne and Brown. We also discuss cyclotomic multiple zeta values of weight $2$ of other levels.