Motivic Multiple zeta values at roots of unity

Cyclotomic multiple zeta values (CMZV) are particularly interesting examples of periods (in the sens of Kontsevich-Zagier) and a fruitful recent approach is to look at their motivic version (MCMZV), which are motivic periods of the fundamental groupoid of $\mathbb{P}^{1}\diagup \left\lbrace 0, \mu_{N} , \infty \right\rbrace $. Notably, MCMZV have a Hopf comodule structure, dual of the action of the motivic Galois group on these specific motivic periods, which enables, via the period map (isomorphism under  Grothendieck's period conjecture), to deduce results on CMZV. After introducing this approach, we will apply some Galois descents ideas to the study of these motivic periods, and look how periods of the fundamental groupoid of $\mathbb{P}^{1} \diagup \left\lbrace 0, \mu_{N'} , \infty \right\rbrace $ are embedded into periods of $\pi_{1}\left( \mathbb{P}^{1} \diagup \left\lbrace 0, \mu_{N} , \infty \right\rbrace\right)  $, when $N' \vert N$. Finally, we would highlight how modular forms appear in relation with the length of multiple zeta values (MZV, i.e. $N=1$), and look at a simpler case: when we restrict to graded version of MZV with totally odd arguments.