Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields

David Burns , Rob M. H. de Jeu (1) , Herbert Gangl (2) , Alexander D. Rahm (3) , Dan Yasaki

1 Faculteit Exacte Wetenschappen Afdeling Wiskunde
2 Department of Mathematical Sciences, University of Durham
3 GAATI - Laboratoire de Géométrie Algébrique et Applications à la Théorie de l'Information

We develop methods for constructing explicit generating elements, 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 infinite 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.

Domaines: K-théorie et homologie [math.KT]; Théorie des nombres [math.NT]

David Burns, Rob M. H. de Jeu, Herbert Gangl, Alexander D. Rahm, Dan Yasaki. Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields. Forum of Mathematics, Sigma, 2021, 9 (E40), ⟨10.1017/fms.2021.9⟩. ⟨hal-02550079⟩