Non-trivial elements in \(K_3\) of imaginary quadratic fields

We develop methods for constructing explicit generating elements, 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 first proven examples of explicit generators, modulo torsion, of any infinite \(K_3\)-group of a number field. As part of this approach, we also make several improvements to the theory of Bloch groups for \(K_3\) of any infinite 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. (link to paper)

\(\Delta=-303\) example

Let \(F\) be an imaginary quadratic field of discriminant \(\Delta < 0\) with ring of integers \(\mathcal{O}_F\), and define the elements \(d = \Delta/4\), \(\omega = \sqrt{d}\) if \(\Delta \equiv 0 \mod 4\) and \(d = \Delta\) and \(\omega = (1 + \sqrt{d})/2\) otherwise.

Belabas, Karim; Gangl, Herbert. Generators and relations for \(K_2\mathcal{O}_F\). K-Theory 31 (2004), no. 3, 195-231. (text)
Browkin, Jerzy; Gangl, Herbert. Tame and wild kernels of quadratic imaginary number fields. Math. Comp. 68 (1999), no. 225, 291-305. (text and table)

• \(\#K_2\mathcal{O}_F\): order of the Quillen \(K\)-group \(K_2\mathcal{O}_F\). Hover over the order to see the group structure.
• Alg data: Each algbelt file contains information about \(\beta_{\mathrm{alg}} = \sum_i n_i [z_i]\). The element is given as a list of pairs [..., [n_i, z_i] ,...], where x denotes \(\sqrt{d}\).
• Geom data: Data files for the Voronoi tessellation of \(\mathbb{H}_3\) for \(\mathrm{GL}_2(\mathcal{O}_F)\). In each geobelt and poly text file, w denotes \(\omega\) as defined above.
  • Each geobelt file contains information about \(\beta_{\mathrm{geo}} = \sum_i n_i [z_i]\). The element is given as a list of pairs [..., <n_i, z_i> ,...].
    geobelt_to_code.py: Script to convert geobelt data file to pari, magma, or sage format.
    python geobelt-to-code.py --pari bloch_neg3.txt > foo.gp
python geobelt-to-code.py --magma bloch_neg3.txt > foo.mag
python geobelt-to-code.py --sage bloch_neg3.txt > foo.sage
    
  • Each poly text file contains data about the 3-dimensional ideal polytopes that arise in the Voronoi tessellation and their stabilizers.
• Time Mem: Time (h:m:s) and memory usage to compute the Voronoi tessellation of \(\mathbb{H}_3\) for \(\mathrm{GL}_2(\mathcal{O}_F)\) and corresponding geometric element.
• Primes: For \(\beta_{\mathrm{geo}} = \sum_i n_i [z_i]\), the set of primes occuring in the factorizations of \(\mathrm{N}(z_i)\) or \(\mathrm{N}(1 - z_i)\) for some \(i\).

Sort by discriminant \(\Delta\). Sort by square-free \(d\). Sort by \(\#K_2\mathcal{O}\).