A survey of the additive dilogarithm

Abstract

Borel’s construction of the regulator gives an injective map from the algebraic K–groups of a number field to its Deligne–Beilinson cohomology groups. This has many interesting arithmetic and geometric consequences. The formula for the regulator is expressed in terms of the classical polyogarithm functions. In this paper, we give a survey of the additive dilogarithm and the several different versions of the weight two regulator in the infinitesimal setting. We follow a historical approach which we hope will provide motivation for the definitions and the constructions.