Modified proof of a local analogue of the Grothendieck conjecture

TL;DR

This paper presents a modified proof of a local analogue of the Grothendieck Conjecture, extending the case to characteristic 2 fields, simplifying techniques, and focusing on maximal pro-p quotients of Galois groups.

Contribution

It introduces a new approach to the local Grothendieck Conjecture, covering characteristic 2 fields and simplifying the proof by using maximal pro-p quotients.

Findings

Extended the conjecture to characteristic 2 fields

Simplified the proof technique for the conjecture

Clarified the recovery of field isomorphisms from Galois group isomorphisms

Abstract

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was considered by Mochizuki several years ago. Then the author proved it by different method if $p>2$ (but $\operatorname{char}K=0$ or $p$). This paper represents a modified approach: it covers the case $p=2$, contains considerable technical simplifications and replaces the Galois group of $K$ by its maximal pro-$p$-quotient. Special attention is paid to the procedure of recovering field isomorphisms coming from isomorphisms of Galois groups, which are compatible with the corresponding ramification filtrations.