Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields

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

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.

Citation

Burns, D., de Jeu, R., Gangl, H., Rahm, A. D., & Yasaki, D. (2021). Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields. Forum of Mathematics, Sigma, 9, Article e40. https://doi.org/10.1017/fms.2021.9

• Journal Article Type: Article
• Online Publication Date: May 24, 2021
• Publication Date: 2021
• Deposit Date: May 24, 2021
• Publicly Available Date: Nov 1, 2021
• Journal: Forum of Mathematics, Sigma
• Print ISSN: 2050-5094
• Electronic ISSN: 2050-5094
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 9
• Article Number: e40
• DOI: https://doi.org/10.1017/fms.2021.9
• Public URL: https://durham-repository.worktribe.com/output/1242265
• Related Public URLs: arxiv.org/abs/1909.09091