Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

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...

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Bibliographic Details
Published in:arXiv.org 2020-11
Main Authors: Burns, David, de Jeu, Rob, Gangl, Herbert, Rham, Alexander D, Yasaki, Dan
Format: Article
Language:English
Subjects:
Fields (mathematics)
Generators
Number theory
Torsion
Online Access:Get full text
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Summary: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 the latter prediction is valid for all abelian number fields.
ISSN:2331-8422
DOI:10.48550/arxiv.1909.09091