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The 6-step Solvable Mono-anabelian Reconstruction of Abelian Number Fields

Yu Mao

TL;DR

The paper addresses reconstructing abelian number fields from their maximal $6$-step solvable Galois groups $G_K^{6}$ and presents a self-contained, mono-anabelian reconstruction method that does not rely on prior Saïdi–Tamagawa results. It develops a local-to-global framework using NF-type and $ ext{GSC}^{6}$-type groups, proving that local decomposition data and cyclotomic information can be recovered from high-level solvable quotients, then assembles these into a global field $F(G)$ isomorphic to $K$ via a detailed chain of objects: $k(D)$, $F_n(D)$, $F_{ abla}(G)$, and the cyclotomic character to form $H_{ ext{cycl}}$ and $F(G)$. The main contribution is a concrete method to reconstruct abelian number fields from $G_K^{6}$, offering a new mono-anabelian pathway that is independent of previous bi-anabelian results. This advances the understanding of how solvable Galois quotients encode full field data and provides a practical blueprint for recovering $K$ from solvable Galois information.

Abstract

In this paper, we develop a new method to reconstruct an abelian number field $K$ from the maximal $6$-step solvable quotient of $G_K$ group- theoretically. The new aspect of this paper is that the results in this paper are independent from the bi-anabelian results proved proven by Saidi and Tamagawa in [ST22].

The 6-step Solvable Mono-anabelian Reconstruction of Abelian Number Fields

TL;DR

The paper addresses reconstructing abelian number fields from their maximal -step solvable Galois groups and presents a self-contained, mono-anabelian reconstruction method that does not rely on prior Saïdi–Tamagawa results. It develops a local-to-global framework using NF-type and -type groups, proving that local decomposition data and cyclotomic information can be recovered from high-level solvable quotients, then assembles these into a global field isomorphic to via a detailed chain of objects: , , , and the cyclotomic character to form and . The main contribution is a concrete method to reconstruct abelian number fields from , offering a new mono-anabelian pathway that is independent of previous bi-anabelian results. This advances the understanding of how solvable Galois quotients encode full field data and provides a practical blueprint for recovering from solvable Galois information.

Abstract

In this paper, we develop a new method to reconstruct an abelian number field from the maximal -step solvable quotient of group- theoretically. The new aspect of this paper is that the results in this paper are independent from the bi-anabelian results proved proven by Saidi and Tamagawa in [ST22].

Paper Structure

This paper contains 3 sections, 10 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Some Local Reconstructions
  3. Reconstruction of Abelian Number Fields

Key Result

Theorem 1.1

Let $K,L$ be number fields. Let $m \geq 0$ be an integer. Let $\tau_{m+3}: G_K^{m+3} \xrightarrow{\sim} G_L^{m+3}$ be an isomorphism. Then the followings hold: (i) There exists a field isomorphism $\sigma_{m} : K_m \xrightarrow{\sim} L_m$ such that $\tau_m(g) = \sigma_mg\sigma_m^{-1}$ for every $g \

Theorems & Definitions (23)

  • Theorem 1.1: Saïdi-Tamagawa, Theorem 2 in ST1
  • Theorem 1.2
  • Definition 2.1: c.f. Definition 3.2 in Ho1
  • Definition 2.2: c.f. Definition 3.2 in Ho1, Definition 2.1 in MS
  • Theorem 2.3: c.f. Theorem 1.25 in ST1
  • Definition 2.4: c.f. Definition 2.3 in MS
  • Proposition 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 13 more