Functional equations of the dilogarithm in motivic cohomology

TL;DR

This paper establishes functional equations for the dilogarithm within motivic cohomology, enabling the explicit construction and detection of torsion cycles in higher Chow groups of number fields.

Contribution

It introduces new relations between fractional linear cycles that help generate and analyze torsion elements in motivic cohomology of number fields.

Findings

Derived explicit relations for dilogarithm functional equations.

Constructed explicit higher Chow cycles for certain number fields.

Verified non-triviality of torsion cycles via regulator maps.

Abstract

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH^2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.