The $m$-step solvable anabelian geometry of mixed-characteristic local fields

Abstract: Let $K$ be a mixed-characteristic local field, and let $m$ be an integer $\geq 0$. We shall denote by $K^m/K$ the maximal $m$-step solvable extension of $K$, and write $G_K^{m}$ for the maximal $m$-step solvable quotient of the absolute Galois group $G_K$ of $K$; we will regard $G_K$ and its quotients as filtered profinite groups by equipping them with the respective ramification filtrations (in upper numbering). It is known from the previous result due to Mochizuki that the isomorphism class of $K$ is determined by the isomorphism class of the filtered profinite group $G_K$. In this paper, we prove (with some mono-anabelian results) that the isomorphism class of $K$ is determined by the isomorphism class of the maximal metabelian quotient $G_K^2$ as a filtered profinite group, and further that $K^{m}/K$ is determined functorially by the filtered profinite group $G_{K}^{m+2}$ (resp. $G_{K}^{m+3}$) for $m\geq 2$ (resp. $m = 0, 1$).