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

TL;DR

This paper demonstrates that the isomorphism class of mixed-characteristic local fields can be fully characterized by their maximal 2-step solvable Galois quotients, extending Mochizuki's results to a finer level of the Galois group structure.

Contribution

It proves that the isomorphism class of a local field is determined by its maximal 2-step solvable Galois quotient as a filtered profinite group, and that higher solvable extensions are functorially determined by higher quotients.

Findings

The isomorphism class of $K$ is determined by $G_K^2$.

Extensions $K^m / K$ are functorially determined by $G_K^{m+2}$ or $G_K^{m+3}$.

Extends Mochizuki's result to a finer Galois-theoretic classification.

Abstract

Let $K$ be a mixed-characteristic local field. For an integer $m \geq 0$, we denote by $K^m / K$ the maximal $m$-step solvable extension of $K$, and by $G_K^m$ the maximal $m$-step solvable quotient of the absolute Galois group $G_K$ of $K$. We regard $G_K$ and its quotients as filtered profinite groups via the respective upper-numbering ramification filtrations. 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 that the isomorphism class of $K$ is determined by the isomorphism class of the maximal $2$-step solvable quotient $G_K^2$ as a filtered profinite group, and furthermore, 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. $m = 0, 1$).