# class field theory for K-groups of number fields

Let $F$ be a number field. The classical class field theory gives an isomorphism between the ideal class group $Cl(F)$ and $\mathrm{Gal}(H/F)$, where $H$ is the Hilbert class field (maximal unramified abelian extension) of $F$. Recall that $Cl(F)$ is the torsion part of $K_0(\mathcal{O}_F)$. Quillen and Borel determined the abelian group structure of higher K-groups of $\mathcal{O}_F$. We also knot that they are finitely generated and its size of torsion part is connected with zeta-values like ideal class groups.

My question: do we have a ''class field theory'' describing the (torsion part) of higher K-groups of $\mathcal{O}_F$ by some ''higher Hilbert class field''?

• So in their paper, on page 422, they speak of the "most important two-dimensional local field", which they write as $(\varprojlim_n\mathbb{Z}/p^n\mathbb{Z}[[q]][q^{-1}])\otimes_\mathbb{Z}\mathbb{Q}$, doesn't even seem to be a field. Ie, what they describe seems to be $\mathbb{Z}_p((q))[1/p]$. For example, the element $p+q$ doesn't seem to have an inverse - the coefficients of the inverse would have "unbounded denominators". What's going on here? Commented Apr 4, 2017 at 3:02
• @oxeimon My impression is that the inverse limit $\varprojlim_n \mathbb Z / p^n \mathbb Z [[q]] [q^{-1}]$ is not $\mathbb Z_p ((q))$, because the inverse limit contains formal power series in $q$ with infinitely many nonzero negative terms, as long as they tend to zero. For example, $\sum_{n \le 0} p^n q^{-n}$ is in the inverse limit, but not in $\mathbb Z_p ((q))$. Commented Apr 4, 2017 at 4:26
• @user94041 You mean $\sum_{n\geqslant0}$? Commented Apr 4, 2017 at 4:32