Motivic unipotent fundamental groupoid of $\mathbb{G}_{m} \setminus \mu_{N}$ for $N=2,3,4,6,8$ and Galois descents

TL;DR

This paper explores Galois descents for mixed Tate motives over specific rings, constructing bases of motivic iterated integrals and providing new proofs and generating families for multiple zeta values relative to roots of unity.

Contribution

It introduces explicit bases for motivic fundamental groupoids over rings related to cyclotomic fields, offering new proofs and constructions in the theory of multiple zeta values.

Findings

Constructed families of motivic iterated integrals with prescribed properties.

Provided a new proof of Deligne's results using Goncharov's coproduct.

Generated bases for multiple zeta values relative to roots of unity.

Abstract

We study Galois descents for categories of mixed Tate motives over $\mathcal{O}_{N}[1/N]$, for $N\in \left\{2, 3, 4, 8\right\}$ or $\mathcal{O}_{N}$ for $N=6$, with $\mathcal{O}_{N}$ the ring of integers of the $N^{\text{th}}$ cyclotomic field, and construct families of motivic iterated integrals with prescribed properties. In particular this gives a basis of honorary multiple zeta values (linear combinations of iterated integrals at roots of unity $\mu_{N}$ which are multiple zeta values). It also gives a new proof, via Goncharov's coproduct, of Deligne's results: the category of mixed Tate motives over $\mathcal{O}_{k_{N}}[1/N]$, for $N\in \left\{2, 3, 4,8\right\}$ is spanned by the motivic fundamental groupoid of $\mathbb{P}^{1}\setminus\left\{0,\mu_{N},\infty \right\}$ with an explicit basis. By applying the period map, we obtain a generating family for multiple zeta values relative to $\mu_{N}$.