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

We study Galois descents for categories of mixed Tate motives over ${\mathcal{O}}_N[1\mathrm{/}N]$, for $N\in \{2,3,4,8\}$ 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 ${\mathcal{O}}_{k_N}[1\mathrm{/}N]$, for $N\in \left{2, 3, 4,8\right}$isspannedbythemotivicfundamentalgroupoidof$$\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 ${\mu }_N$.