The analogue of the Gauss class number problem in motivic cohomology
The classical Gauss problem is to determine all imaginary quadratic number fields of class number one. The paper under review studied the analogue of the Gauss class number problem in motivic cohomology. Let \(F\) be a number field and \(\zeta_F(s)\) be the Dedekind zeta function of \(F\). By Dirichlet's analytic class number formula and the functional equation, one has \[ \zeta_F(0)=-\frac{h(F)}{w(F)}, \] where \(h(F)\) is the class number of \(F\) and \(w(F)\) is the number of roots of unity in \(F\). In the paper under review, the authors study the analogous situation at negative integers. Let \(F\) be a totally real field with degree \(d\) and ring of integers \(O_F\). If we assume that the 2-primary Main Conjecture in Iwasawa theory is true for \(F\), the authors proved that, for \(n\geq 2\) even, \[ \zeta_F(1-n)=(-1)^{\frac{n}{2}}\frac{h_n(F)}{w_n(F)}, \] where \(h_n(F)\) is the order of the second motivic cohomology group \(H^2_\mathcal M(O_F,\mathbb Z(n))\), \(w_n(F)\) is the order of the Galois cohomology group \(H^0(F,\mathbb Q/\mathbb Z(n))\). Note that we have a surjective homomorphism [\textit{M. Kolster}, Math. Ann. 323, No. 4, 667--692 (2002; Zbl 1007.11068)] \[ H^2_\mathcal M(O_F,\mathbb Z(n))\rightarrow (\mathbb Z/2\mathbb Z)^d. \] The analogue of the classical Gauss class number problem is to determine all totally real number fields \(F\) with \(2^{-d}h_n(F)=1\). The authors showed that there are no such fields for \(n\geq 6\), there is only the field \(\mathbb Q(\sqrt{5})\) for \(n=4\), and there are 11 fields for \(n=2\). The \textit{motivic Gauss problem} is to determine the totally real fields \(F\) and even integers \(n\geq 2\) with \(h_n^\mathcal M(F)=1\), where \(h_n^\mathcal M(F)\) is the order of the so called \textit{motivic wild kernel} \(WK_{2n-2}^\mathcal M(F)\), which is the kernel of the homomorphism \[ H^2_\mathcal M(F,\mathbb Z(n))\rightarrow\bigoplus_{v\text{ non-complex}}H^2_\mathcal M(F_v,\mathbb Z(n)). \] The authors show that there is only the field \(\mathbb Q(\sqrt{5})\) for \(n\geq 4\). For \(n=2\), they showed that there are only finitely many such fields and they are all of degree \(d\leq 117\). Furthermore, they found 21 fields among the totally real fields of degrees between 2 and 9 for \(n=2\). The key part of their approach on the Gauss problem is to deduce the following two inequalities \[ D^{n-\frac{1}{2}}<\frac{h_n(F)}{w_n(F)}\left(\frac{(2\pi)^n}{2(n-1)!}\right)^d \] and \[ \delta_F<\left(\frac{2(2\pi)^n}{(n-1)^{1/2}(n-2)!}\left(\frac{h_n^\mathcal M(F)}{2w_n(F)}\right)^{1/d}\right)^{\frac{2}{2n-3}}, \] where \(D\) is the discriminant of \(F\) and \(\delta_F=D^{1/d}\) is the root discriminant of \(F\).
scientific article; zbMATH DE number 6350055
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