On functional equations for Chow polylogarithms

TL;DR

This paper investigates the functional equations of Chow polylogarithms, demonstrating that complex relations can be derived from simpler properties and connecting these to classical polylogarithm equations.

Contribution

It shows that the key functional equations for Chow polylogarithms follow from basic properties and links these to classical polylogarithm functional equations.

Findings

Functional equations follow from skew-symmetry, functoriality, and multiplicativity.

Establishes an analogue of Beilinson-Soule vanishing conjecture.

Connects Chow polylogarithm equations to classical polylogarithm equations.

Abstract

Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity. To prove this, we study some analogue of Bloch's cycle complex and establish for this complex an analogue Beilinson-Soule vanishing conjecture. A. Goncharov defined a group of functional equations for classical polylogarithms. We show that any such functional equation formally follows from functional equations for Chow polylogarithms stated above.