On {$l$}-adic iterated integrals. I. Analog of Zagier conjecture

Abstract

We are studying some aspects of the action of Galois groups on the torsor of paths connecting two (possibly tangential) points on a projective line minus a finite number of points. We obtain objects which formally behave like classical iterated integrals and polylogarithms. We formulate an analog of Zagier conjecture for these $l$-adic analogs of iterated integrals and polylogarithms.