The analogue of the Gauss class number problem in motivic cohomology

Abstract

The classical Gauss problem of determining all imaginary quadratic number fields of class number one has an analogue involving finite motivic cohomology groups attached to the ring of integers $$o_F$$ in a totally real number field $$F$$ . In the classical situation, the value of the zeta function of $$F$$ at $$0$$ can be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the class number. For a totally real field $$F$$ and an even integer $$n \ge 2$$ , the value of the zeta-function of $$F$$ at $$1-n$$ can also be written as a quotient of two—not necessarily coprime—integers, where the numerator is equal to the order $$h_n(F)$$ of the motivic cohomology group $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ . This order is always divisible by $$2^d$$ , where $$d$$ is the degree of $$F$$ . We determine all totally real fields $$F (\ne \mathbb{Q })$$ and all even integers $$n \ge 2$$ , for which the quotient $$\frac{h_n(F)}{2^d}$$ is equal to 1. There are no fields for $$n \ge 6$$ , there is only the field $$\mathbb{Q }(\sqrt{5})$$ for $$n = 4$$ , and there are 11 fields for $$n=2$$ . The motivic cohomology groups $$H_\mathcal M ^2(o_F,\mathbb Z (n))$$ contain the canonical subgroups $$WK^\mathcal M _{2n-2}(F)$$ , called motivic wild kernels, which are analogous to Tate–Shafarevic groups. For even integers $$n \ge 4$$ , there is again only the case $$n = 4$$ and the field $$\mathbb{Q }(\sqrt{5})$$ , for which the motivic wild kernel $$WK^\mathcal M _{2n-2}(F)$$ vanishes. However, for $$n=2$$ , among the totally real fields $$F$$ of degrees between $$2$$ and $$9$$ , we found 21 of them, for which $$WK^\mathcal M _{2}(F)$$ vanishes, the largest degree being $$5$$ . A basic assumption in our approach is the validity of the $$2$$ -adic Main Conjecture in Iwasawa theory for the trivial character, which so far has only been proven for abelian number fields (by A. Wiles).