Abstract

Suppose K is an N-dimensional local field where N is a non-negative integer. By definition, if N=0 then K is just a finite field, otherwise, K is a complete discrete valuation field and its residue field is an (N-1)-dimensional local field. Let G be the absolute Galois group of K. If N=1 then the structure of the topological group G depends only on very weak invariants of K and is not sufficient to recover uniquely the field K. The situation becomes totally different if we take into account the filtration of G by its ramification subgroups. Then the corresponding functor from the category of 1-dimensional local fields to the category of profinite groups with decreasing filtration is fully faithful. In the talk it will be discussed an analog of this statement for higher local fields and its relation to the Grothendieck conjecture in the context of global fields.