Is the natural map from the modified Bloch group to indecomposable $K_3$ an isomorphism for every number field (Conjecture 3.33 of Burns–de Jeu–Gangl–Rahm–Yasaki)?

For a number field $F$ , Burns, de Jeu, Gangl, Rahm and Yasaki (Forum Math. Sigma 9 (2021) e40) constructed a 'modified Bloch group' $\overline{\mathcal{B}}(F)$ from symbols $[x]$ , $x\in F\setminus\{0,1\}$ , together with a natural homomorphism $\psi_F:\overline{\mathcal{B}}(F)\to K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ into the torsion-free part of the indecomposable third algebraic $K$ -group of $F$ . Injectivity of $\psi_{F,\mathrm{tf}}$ is proven for every number field, and the conjecture (their Conjecture 3.33) is that $\psi_{F,\mathrm{tf}}$ is an isomorphism for every number field $F$ ; it has been verified computationally for all imaginary quadratic fields of discriminant $>-1000$ . The problem is to prove surjectivity for all number fields, or to exhibit a number field for which the image has index strictly greater than $1$ .

Background

Algebraic $K$ -theory attaches to any field $F$ a sequence of abelian groups $K_n(F)$ introduced by Quillen. For $n=3$ there is a canonical map from Milnor $K$ -theory $K_3^{M}(F)$ (the degree- $3$ part of the graded ring built from $F^{\times}=F\setminus\{0\}$ modulo the Steinberg relation $x\otimes(1-x)$ ), and the cokernel $K_3(F)^{\mathrm{ind}}$ is the indecomposable $K_3$ . A theorem of Borel computes $K_3(F)^{\mathrm{ind}}$ for a number field $F$ (a finite extension of $\mathbb{Q}$ ) rationally: its torsion-free part $K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ is a free abelian group of rank $r_2(F)$ , the number of conjugate pairs of complex embeddings, and it is measured by the regulator, obtained by evaluating the Bloch–Wigner dilogarithm $D$ (the single-valued real-analytic function on $\mathbb{C}\setminus\{0,1\}$ defined by integrating $\log|w|\,d\arg(1-w)-\log|1-w|\,d\arg(w)$ , which satisfies the five-term functional equation) under each embedding $\sigma:F\to\mathbb{C}$ . Borel's result further identifies the size of the regulator lattice with the leading term $\zeta_F^{*}(-1)$ at $-1$ of the Dedekind zeta function of $F$ (the analogue for $F$ of the Riemann zeta function: the Dirichlet series $\sum_{\mathfrak{a}}N(\mathfrak{a})^{-s}$ over the non-zero ideals $\mathfrak{a}$ of $O_F$ , with $N(\mathfrak{a})$ the norm); the Lichtenbaum conjecture refines this to an integral formula $\zeta_F^{*}(-1)=\pm 2^{r_2+t_2(F)}|K_2(O_F)|/|K_3(O_F)_{\mathrm{tor}}|\cdot R_2(F)$ , where $O_F$ is the ring of integers of $F$ , $R_2(F)$ is the covolume of the regulator image, and the power of $2$ was pinned down conditionally in the same paper. The classical bridge between dilogarithms and $K_3$ is the Bloch group. The pre-Bloch group of $F$ is the free abelian group $\mathbb{Z}[F^{\flat}]$ on $F^{\flat}=F\setminus\{0,1\}$ modulo the five-term relations $[x]-[y]+[y/x]-[(1-y)/(1-x)]+[(1-y^{-1})/(1-x^{-1})]=0$ ; Suslin's Bloch group $B(F)$ is the kernel of the boundary map $[x]\mapsto (1-x)\wedge x$ into an exterior-square group of $F^{\times}$ . Suslin proved that for every infinite field $B(F)_{\mathrm{tf}}\cong K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ , which for number fields re-interprets Borel's theorem through the dilogarithm. Burns, de Jeu, Gangl, Rahm and Yasaki introduced a modified version designed to avoid the factor-of- $2$ losses in Goncharov's original $\mathrm{GL}_2$ -equivariant construction. They define

\widetilde{\wedge}^{2}F^{\times}=(F^{\times}\otimes_{\mathbb{Z}}F^{\times})/\langle(-x)\otimes x\rangle,

with class written $a\,\tilde{\wedge}\,b$ , and the boundary $\delta_{2,F}:\mathbb{Z}[F^{\flat}]\to\widetilde{\wedge}^{2}F^{\times}$ , $[x]\mapsto(1-x)\,\tilde{\wedge}\,x$ . The modified pre-Bloch group is $\mathfrak{p}(F)=\mathbb{Z}[F^{\flat}]$ modulo the five-term relations above and the additional inversion relations $[x]+[x^{-1}]=0$ and $[x]+[1-x]=0$ , and the modified Bloch group is $\overline{\mathcal{B}}(F)=\ker(\delta_{2,F}:\mathfrak{p}(F)\to\widetilde{\wedge}^{2}F^{\times})$ . A theorem of Burns–de Jeu–Gangl (2012) gives a homomorphism $\varphi_F:\ker(\delta_{2,F})\to K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ , natural up to a universal sign, characterized by the fact that, after fixing the sign, composing with the regulator induced by an embedding $\sigma$ sends each $[x]$ to $iD(\sigma(x))$ . Theorem 3.25 of the paper shows $\varphi_F$ descends to a homomorphism $\psi_F:\overline{\mathcal{B}}(F)\to K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ . For a number field $F$ , the kernel of $\psi_F$ is exactly the torsion subgroup of $\overline{\mathcal{B}}(F)$ , so the induced $\psi_{F,\mathrm{tf}}:\overline{\mathcal{B}}(F)_{\mathrm{tf}}\to K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ is injective and its cokernel is finite; combining with Suslin's isomorphism and the paper's Theorem 3.23 ( $\overline{\mathcal{B}}(F)_{\mathrm{tf}}\cong B(F)_{\mathrm{tf}}$ ) yields an injective endomorphism of the free abelian group $K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ of rank $r_2(F)$ , whose index is finite and computable in examples. The conjecture is supported by explicit geometry and computation. For an imaginary quadratic field $k$ (with ring of integers $O_k$ ), the group $\mathrm{GL}_2(O_k)$ acts on hyperbolic $3$ -space $\mathbb{H}^{3}$ , and the Voronoi tessellation of the cone of positive-definite binary Hermitian forms over $O_k$ gives an explicit finite cell decomposition of a quotient of $\mathbb{H}^{3}$ . From the cells, their finite stabiliser subgroups, and the cross-ratios of their ideal vertices, the authors construct a 'geometric' element $\beta_{\mathrm{geo}}\in\overline{\mathcal{B}}(k)$ with $\mathrm{reg}_2(\psi_k(\beta_{\mathrm{geo}}))/(2\pi i)=-12\zeta'_k(-1)$ , generating a subgroup of index $|K_2(O_k)|$ in $K_3(k)^{\mathrm{ind}}_{\mathrm{tf}}$ ; dividing by $|K_2(O_k)|$ using 'exceptional $S$ -units' (an $S$ -unit, for a finite set $S$ of primes of $O_k$ , is an element of $k$ that is a unit at every prime outside $S$ ; it is exceptional when $1-x$ is also an $S$ -unit) produces generators whenever the division is possible in $\overline{\mathcal{B}}(k)$ . In this way the conjecture was verified for all imaginary quadratic fields of discriminant $>-1000$ , the first proven explicit generators, modulo torsion, of an infinite $K$ -group of a number field.

Problem Statement

Prove or refute Conjecture 3.33 of Burns–de Jeu–Gangl–Rahm–Yasaki (2021). Let $F$ be a number field, $F^{\flat}=F\setminus\{0,1\}$ , $\widetilde{\wedge}^{2}F^{\times}=(F^{\times}\otimes_{\mathbb{Z}}F^{\times})/\langle(-x)\otimes x:x\in F^{\times}\rangle$ , and $\delta_{2,F}:\mathbb{Z}[F^{\flat}]\to\widetilde{\wedge}^{2}F^{\times}$ the homomorphism $[x]\mapsto(1-x)\,\tilde{\wedge}\,x$ . Let $\mathfrak{p}(F)$ be $\mathbb{Z}[F^{\flat}]$ modulo the five-term relations $[x]-[y]+[y/x]-[(1-y)/(1-x)]+[(1-y^{-1})/(1-x^{-1})]=0$ ( $x\neq y$ in $F^{\flat}$ ) and the inversion relations $[x]+[x^{-1}]=0$ , $[x]+[1-x]=0$ ; let $\overline{\mathcal{B}}(F)=\ker(\delta_{2,F}:\mathfrak{p}(F)\to\widetilde{\wedge}^{2}F^{\times})$ be the modified Bloch group, and let $\psi_F:\overline{\mathcal{B}}(F)\to K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ be the homomorphism of Theorem 3.25 of that paper (induced by the sign-normalized map $\varphi_F$ of Burns–de Jeu–Gangl, under which the regulator composed with an embedding $\sigma$ sends $[x]$ to $iD(\sigma(x))$ ).

Since $\psi_{F,\mathrm{tf}}$ is known to be injective for every number field and both sides are free abelian of rank $r_2(F)$ , the conjecture is exactly the assertion that

\operatorname{im}(\psi_{F,\mathrm{tf}})=K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}\quad\text{for every number field }F,

equivalently $[K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}:\operatorname{im}(\psi_{F,\mathrm{tf}})]=1$ for every number field $F$ , equivalently the composite endomorphism of $K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}$ obtained from Suslin's isomorphism $K_3(F)^{\mathrm{ind}}_{\mathrm{tf}}\cong B(F)_{\mathrm{tf}}$ , Theorem 3.23, and $\psi_{F,\mathrm{tf}}$ is an isomorphism (conjecturally multiplication by $\pm1$ ).

References

David Burns, Rob de Jeu, Herbert Gangl, Alexander D. Rahm, and Dan Yasaki, Hyperbolic tessellations and generators of $K_3$ for imaginary quadratic fields, Forum of Mathematics Sigma 9 (2021), e40. DOI 10.1017/fms.2021.9; arXiv:1909.09091; https://arxiv.org/abs/1909.09091
A. A. Suslin, $K_3$ of a field and the Bloch group, Proc. Steklov Inst. Math. 183 (1991), 217–239; Russian original: Tr. Mat. Inst. Steklova 183 (1990), 180–199. Zbl 0741.19005; https://zbmath.org/0741.19005
David Burns, Rob de Jeu, and Herbert Gangl, On special elements in higher algebraic $K$ -theory and the Lichtenbaum–Gross Conjecture, Advances in Mathematics 230 (2012), no. 3, 1502–1529. DOI 10.1016/j.aim.2012.03.014; https://doi.org/10.1016/j.aim.2012.03.014
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