The m-step Solvable Mono-anabelian Geometry of Number Fields
Yu Mao, Mohamed Saidi
TL;DR
The paper tackles the problem of determining a number field from quotients of its absolute Galois group by developing a mono-anabelian reconstruction framework. It builds a multi-layer pipeline comprising local invariant recovery from $G_K^{m+9}$, local-global synchronization of cyclotomes and Kummer containers, and a constructive reconstruction of the global fields $F_m(G^m)$ and $F(G^m)$ with explicit $G^m$-actions, culminating in a full reconstruction of $K$ from $G_K^{m+9}$ for $m\ge3$ (with a concrete result from $G_K^6$ when $K$ is $\mathbb{Q}$ or an imaginary quadratic field). The approach extends Saïdi–Tamagawa local theory and Hoshi's NF-monoid framework to explicit, computable reconstructions, and establishes functorial, isomorphism-type dependent correspondences that connect local data to global fields. This work advances mono-anabelian geometry by providing explicit procedures to recover number fields from solvable quotients of their Galois groups, with potential computational applications and broader implications for understanding the arithmetic of Galois groups.
Abstract
The goal of this paper is to develop a group-theoretic algorithm, to reconstruct a number field (together with its maximal m-step solvable ex- tension for some positive integer m \geq 3) from the maximal m+9-step solv- able quotient of its absolute Galois group. If K is an imaginary quadratic field or Q, we establish a group-theoretic reconstruction algorithm of K from the maximal 6-step solvable quotient of its absolute Galois group.