Mono-anabelian Reconstruction of Number Fields with Restricted Ramification

TL;DR

The paper develops a mono-anabelian reconstruction program for number fields with maximal unramified outside $S$ extensions in the density $1$ setting. Starting from an abstract profinite group $G$ isomorphic to $G_{K,S}$, it constructs a field $F_S(G)$ with a $G$-action and a fixed field $F(G)$ so that $G \cong \mathrm{Gal}(F_S(G)/F(G))$, and it embeds this into a commuting diagram linked to the original $K_S/K$. The work advances a step-by-step reconstruction: (i) local invariants from decomposition groups, (ii) the $\Sigma$-cyclotome and Kummer-type containers, (iii) the maximal pro-solvable outer extension $\mathbb{Q}_{\Sigma}^{\text{sol}}$ and its decomposition data, and (iv) a global reconstruction of $K_S$ via $S(G)$-standard subfields and Neukirch-Uchida-type arguments. The approach leverages and extends prior frameworks (Ho1/Ho2, Shi2) to the density-one, restricted-ramification setting, yielding a group-theoretic algorithm to recover both the field and its maximal unramified outside $S$ extension, with functoriality and explicit obstructions explained for the non-density-one case.

Abstract

In this paper, we apply Hoshi's mono-anabelian reconstruction of number fields to establish a group-theoretic reconstruction of a number field K together with its maximal unramified outside S extension K_S for a density 1 subset of primes of K starting from the profinite group G_{K,S}.

Theorems & Definitions (60)

• Theorem 1.1: The Neukirch-Uchida theorem
• Theorem 1.2: Hoshi, Ho1 and Ho2
• Theorem 1.3: Shimizu, Shi2 Theorem 2.4
• Theorem 1.4: c.f. Theorem 5.6
• Definition 2.1
• Proposition 2.2
• proof
• Remark 2.3
• Theorem 2.4
• proof