Hyperbolic tessellations and generators of for imaginary quadratic fields

Abstract

We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

Describing all multivariable functional equations of dilogarithms

“…One shows as in the proof of [3,Theorem 3.23] that B(F ) = B(F )/ 3C F by replacing p(F )/ C F with p(F )/ 3C F and p(F ) with p(F ) in the diagram preceding the theorem of loc. cit., using that 3C F,c is in R 5,2 (F ).…”