Hyperbolic tessellations and generators of K3 for imaginary quadratic fields

David Burns, Rob de Jeu, Herbert Gangl, Alexander D. Rahm, Dan Yasaki

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Abstract

We develop methods for constructing explicit generators, modulo torsion, of the K3-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 K3-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for K3 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.

Original languageEnglish
Article numbere40
Pages (from-to)1-47
Number of pages47
JournalForum of Mathematics, Sigma
Volume9
Early online date24 May 2021
DOIs
Publication statusE-pub ahead of print - 24 May 2021

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