Abrashkin: Modified proof of a local analogue of the Grothendieck conjecture

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Victor Abrashkin
Modified proof of a local analogue of the Grothendieck conjecture
Journal de théorie des nombres de Bordeaux, 22 no. 1 (2010), p. 1-50, doi: 10.5802/jtnb.703
Article PDF | Reviews MR 2675872 | Zbl pre05831861

Résumé - Abstract

A local analogue of the Grothendieck Conjecture is an equivalence between 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 studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p>2$ (but with no restrictions on the characteristic of $K$). In this paper we suggest 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.

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