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

Seung-Hyeon Hyeon

TL;DR

This work advances local anabelian geometry by showing that for mixed-characteristic local fields, the isomorphism class of a field $K$ is determined by the maximal $2$-step solvable quotient $G_K^2$ of its absolute Galois group, when viewed as a filtered profinite group. It further proves that $K^m/K$ is functorially determined by $G_K^{m+2}$ (and by $G_K^{m+3}$ for $m=0,1$), extending Mochizuki’s filtration-based correspondences to metabelian and higher solvable quotients. Central to the approach is a robust, group-theoretic reconstruction of key invariants (e.g., $p_K$, $oldsymbol{ u}_K$, $a_K$, $d_K$, $e_K$, $f_K$) and the cyclotomic character from $G_K^2$, together with ramification-structure recovery via ramification subgroups and $K^ imes$-module realizations. The results yield strong local bi-anabelian statements, aligning with analogous global theorems for number fields and enriching the understanding of how ramification and cyclotomic data encode arithmetic geometry within restricted Galois data.

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. $m = 0, 1$).

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

TL;DR

This work advances local anabelian geometry by showing that for mixed-characteristic local fields, the isomorphism class of a field is determined by the maximal -step solvable quotient of its absolute Galois group, when viewed as a filtered profinite group. It further proves that is functorially determined by (and by for ), extending Mochizuki’s filtration-based correspondences to metabelian and higher solvable quotients. Central to the approach is a robust, group-theoretic reconstruction of key invariants (e.g., , , , , , ) and the cyclotomic character from , together with ramification-structure recovery via ramification subgroups and -module realizations. The results yield strong local bi-anabelian statements, aligning with analogous global theorems for number fields and enriching the understanding of how ramification and cyclotomic data encode arithmetic geometry within restricted Galois data.

Abstract

Let be a mixed-characteristic local field. For an integer , we denote by the maximal -step solvable extension of , and by the maximal -step solvable quotient of the absolute Galois group of . We regard 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 is determined by the isomorphism class of the filtered profinite group . In this paper, we prove that the isomorphism class of is determined by the isomorphism class of the maximal -step solvable quotient as a filtered profinite group, and furthermore, that is determined functorially by the filtered profinite group (resp. ) for (resp. ).
Paper Structure (7 sections, 22 theorems, 114 equations)

This paper contains 7 sections, 22 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminary notions
  3. Restoration of the cyclotomic character
  4. Ramification groups
  5. Abelian $p$-adic representations
  6. Proofs of the main theorems
  7. Center-freeness of $G_K^{m}$, $m \geq 2$

Key Result

Theorem 1.1

The image of $\eta$ coincides with $\mathrm{Out}_{\mathit{filt.}} (G_{K_\circ}, G_{K_\bullet})$. Equivalently, for an isomorphism of filtered profinite groups, there exists a unique isomorphism $\theta \colon {K_\circ^\mathrm{alg}} \to {K_\bullet^\mathrm{alg}}$ such that for every $\sigma \in G_{K_\circ}$. In particular, we have an isomorphism $\theta \vert_{K_\circ} \colon K_{\circ} \to K_{\bul

Theorems & Definitions (46)

  • Theorem 1.1: Mochizuki Mochizuki1997
  • Theorem 1.2: Saïdi-Tamagawa SaidiTamagawa2022
  • Theorem 1.3: Saïdi-Tamagawa SaidiTamagawa2022
  • Theorem 1.4: \ref{['theorem:2025.10.22.02.13.03']}
  • Theorem 1.5: \ref{['theorem:2025.10.22.02.13.10']}
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark
  • ...and 36 more