Multiple zeta-values, Galois groups, and geometry of modular varieties

TL;DR

This paper explores the properties of multiple zeta-values and polylogarithms at roots of unity, their Galois group actions, and their unexpected links to the geometry of modular varieties, advancing understanding in number theory and algebraic geometry.

Contribution

It introduces new insights into the relationship between multiple zeta-values, Galois actions, and modular variety geometry, connecting number theory with algebraic geometry.

Findings

Properties of multiple zeta-values and polylogarithms at roots of unity analyzed.

Galois group actions on fundamental groups studied.

Connections between number theory and modular variety geometry established.

Abstract

We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.