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
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Abstract:
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

Mathematics Subject Classification (2000): 11G55, 11G99, 14G32
Mathematical Reviews Number: MR2351366

Received: 2004-12-15