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

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 $\mu_{N}$.