The conjecture that the modified Bloch group computes indecomposable K3

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Let FF be a number field. Let B‾(F)\overline{B}(F) be the modified Bloch group and let K3(F)tfindK_3(F)^{\mathrm{ind}}_{\mathrm{tf}} be the torsion-free quotient of the indecomposable part of K3(F)K_3(F). The canonical homomorphism is

ψF,tf:B‾(F)tf⟶K3(F)tfind.\psi_{F,\mathrm{tf}}:\overline{B}(F)_{\mathrm{tf}}\longrightarrow K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}.

Modified Bloch group conjecture. The homomorphism ψF,tf\psi_{F,\mathrm{tf}} is an isomorphism. Extensive computations for imaginary quadratic fields motivate the claim that the modified Bloch group accounts for all of the indecomposable K3K_3-group modulo torsion. The source gives no resolution, so the conjecture remains open.

References

Primary source

David Burns, Rob de Jeu, Herbert Gangl, Alexander D. Rham and Dan Yasaki, “Hyperbolic tessellations and generators of K_3 for imaginary quadratic fields”, arXiv:1909.09091 (2020).

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