Periods and mixed motives

TL;DR

This paper introduces motivic multiple polylogarithms, establishes their double shuffle relations, and explores their connections to the motivic fundamental group, Galois actions, and modular varieties, advancing understanding of mixed Tate motives.

Contribution

It defines motivic multiple polylogarithms, proves their double shuffle relations, and links these to the motivic fundamental group and Galois actions, providing new insights into mixed Tate motives.

Findings

Proved double shuffle relations for motivic multiple polylogarithms

Analyzed the motivic fundamental group of the multiplicative group minus roots of unity

Gained new understanding of Galois actions on fundamental groups

Abstract

We define motivic multiple polylogarithms and prove the double shuffle relations for them. We use this to study the motivic fundamental group of the multiplicative group - {N-th roots of unity} and relate it to geometry of modular varieties. In particular we get new information about the actionof the Galois group on the pro-l completion of the above fundamental group. This paper is the second part of "Multiple polylogarithms and mixed Tate motives" math.AG/0103059