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

摘要

WearestudyingrepresentationsobtainedfromactionsofGaloisgroupsontorsorsofpathsonaprojectivelineminusafinitenumberofpoints.Usingtheseactionsontorsorsofpaths,weconstructgeometricallyrepresentationsofGaloisgroupswhichrealize$ell$-adicallytheassociatedgradedLiealgebraofthefundamentalgroupofthetannakiancategoryofmixedTatemotivesover${ mSpec},bb$,${ mSpec},bb[i]$,${ mSpec},bb[frac{1}{q}]$,${ mSpec},Oc_{Qbb(sqrt{-q})}$foranyprimenumber$q$($q eq2$inthelastcase)andover${ mSpec},Oc_{Qbb(sqrt{-q})}[frac{1}{q}]$foranyprimenumber$q$congruentto$3$modulo$4$andalsofor$q=2.$