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Hyperbolic tessellations and generators of K3 for imaginary quadratic fields

Research output: Contribution to journalArticlepeer-review

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

Original languageEnglish
Article numbere40
Pages (from-to)1-47
Number of pages47
JournalForum of Mathematics, Sigma
Volume9
Early online date24 May 2021
DOIs
Accepted/In press6 Jan 2021
E-pub ahead of print24 May 2021

Bibliographical note

Publisher Copyright: © The Author(s), 2021. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

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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.

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