## On the Galois actions on torsors of paths I, Descent of Galois representations

J. Math. Sci. Univ. Tokyo
Vol. 14 (2007), No. 2, Page 177--259.

Wojtkowiak, Zdzis{\l}aw
On the Galois actions on torsors of paths I, Descent of Galois representations
We are studying representations obtained from actions of Galois groups on torsors of paths on a projective line minus a finite number of points. Using these actions on torsors of paths, we construct geometrically representations of Galois groups which realize $\ell$-adically the associated graded Lie algebra of the fundamental group of the tannakian category of mixed Tate motives over ${\rm Spec}\, \Zbb$, ${\rm Spec}\, \Zbb [i]$, ${\rm Spec}\, \Zbb [\frac{1}{q}]$, ${\rm Spec}\,\Oc _{\Qbb ( \sqrt {-q})}$ for any prime number $q$ ($q\neq 2$ in the last case) and over ${\rm Spec}\,\Oc _{\Qbb ( \sqrt {-q})}[\frac{1}{q}]$ for any prime number $q$ congruent to $3$ modulo $4$ and also for $q=2.$
Keywords: Galois group, fundamental group, torsor of paths, $\ell$-adic polylogarithms, Lie algebra, Lie algebra of derivations