Motivic unipotent fundamental groupoid of Gm∖μNGm∖μN for N = 2,3,4,6,8 and Galois descents

We study Galois descents for categories of mixed Tate motives over ON[1/N]ON[1/N], for N∈{2,3,4,8}N∈{2,3,4,8} or ONON for N=6N=6, with ONON the ring of integers of the N  th cyclotomic field, and construct families of motivic iterated integrals with prescribed properties. In particular this gives a basis of multiple zeta values via multiple zeta values at roots of unity μNμN. It also gives a new proof, via Goncharov's coproduct, of Deligne's results [9]: the category of mixed Tate motives over OkN[1/N]OkN[1/N], for N∈{2,3,4,8}N∈{2,3,4,8} is spanned by the motivic fundamental groupoid of P1∖{0,μN,∞}P1∖{0,μN,∞} with an explicit basis. By applying the period map, we obtain a generating family for multiple zeta values relative to μNμN.