Hyperbolic tessellations and generators of K_3 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 the latter prediction is valid for all abelian number fields.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2019
- DOI:
- 10.48550/arXiv.1909.09091
- arXiv:
- arXiv:1909.09091
- Bibcode:
- 2019arXiv190909091B
- Keywords:
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- Mathematics - K-Theory and Homology;
- Mathematics - Number Theory;
- 11G55;
- 11R70;
- 19F27;
- 19D45 (primary);
- 11R42;
- 20H20;
- 51M20 (secondary)
- E-Print:
- 51 pages