Motivic fundamental group of ${\Bbb G}_m-\mu_N$ and modular manifolds

TL;DR

This paper explores the deep connections between the motivic Galois group actions on fundamental groups of punctured projective lines and the geometry of modular manifolds, using motivic correlators and cohomological methods.

Contribution

It introduces a canonical collection of motivic correlators at roots of unity and relates them to chains in cohomology of local systems on modular manifolds, establishing precise relationships for m up to 4.

Findings

Established relationships between motivic correlators and cohomology chains for m ≤ 4.

Connected motivic Galois actions with the geometry of modular manifolds.

Extended previous work on m=2, 3 cases to higher dimensions.

Abstract

We investigate geometric and combinatorial aspects of the mysterious relationship between the action of the motivic Galois group on the motivic fundamental group of the projective line punctured at zero, infinity, and N-th roots of unity, and the geometry of modular manifolds for the congruence subgroup Y(m;N) of GL(m,Z) fixing (0, ... , 0, 1) modulo N. To achieve this, we consider a canonical collection of elements in the image of the motivic Galois Lie algebra, spanning it over Q. These elements are the motivic correlators at N-th roots of unity. We assign to them chains in the complex computing cohomology of certain local systems on the space Y(m;N). Their geometric properties reflect, in a mysterious way, properties of motivic correlators. This construction allows to establish the relationship with high precision for m up to 4. The m=2, 3 cases were investigated by the author before in the Hodge and the Galois set-up.