The $m$-step solvable anabelian geometry of number fields

TL;DR

This paper demonstrates that the isomorphism type of a number field can be uniquely determined by the structure of its maximal 3-step solvable Galois group, refining previous theorems in anabelian geometry.

Contribution

It proves that the isomorphism class of a number field is determined by its maximal 3-step solvable Galois group, and establishes functorial determination of intermediate extensions from higher Galois groups.

Findings

Number field is determined by G_K^3

Extensions K_m/K are determined by G_K^{m+3} or G_K^{m+4}

Local theory characterizes decomposition groups from Galois groups

Abstract

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the isomorphy type of $K$ is determined by the isomorphy type of $G_K^3$. Further, we prove that $K_m/K$ is determined functorially by $G_K^{m+3}$ (resp. $G_K^{m+4}$) for $m\geq 2$ (resp. $m \leq 1$). This is a substantial sharpening of a famous theorem of Neukirch and Uchida. A key step in our proof is the establishment of the so-called local theory, which in our context characterises group-theoretically the set of decomposition groups (at nonarchimedean primes) in $G_K^m$, starting from $G_K^{m+2}$.