Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields

TL;DR

This paper extends the understanding of homomorphisms between absolute Galois groups to mixed-characteristic fields, providing criteria for when such homomorphisms are geometric and establishing a weak isomorphism result.

Contribution

It introduces necessary and sufficient conditions for geometric homomorphisms between Galois groups of mixed-characteristic fields with perfect residue fields.

Findings

Criteria for geometric homomorphisms established

Weak-isomorphism results for Galois groups proved

Conditions applicable to fields with algebraic residue fields

Abstract

Although the analogue of the theorem of Neukirch-Uchida for $p$-adic local fields fails to hold as it is, Mochizuki proved a certain analogue of this theorem for the absolute Galois groups with ramification filtrations of $p$-adic local fields. Moreover, Mochizuki and Hoshi gave various (necessary and) sufficient conditions for homomorphisms between the absolute Galois groups of $p$-adic local fields to be "geometric" (i.e., to arise from homomorphisms of fields). In the present paper, we consider similar problems for general mixed-characteristic complete discrete valuation fields with perfect residue fields. One main result gives (necessary and) sufficient conditions for homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with residue fields algebraic over the prime fields to be geometric. We also give a "weak-Isom" anabelian result for homomorphisms between the absolute Galois groups of these fields.