Galois groups of local fields and deformations of Lie algebras

Let $K=k((t))$ be a field of formal Laurent series in variable $t$ with coefficients in a finite field $k$ of characteristic $p>0$. Denote by $G_p$ the maximal quotient of the absolute Galois group of $K$ with the nilpotent class $<p$ and the period $p$. The nilpotent analogue of the Artin-Shreier theory allows us the Galois group $G_p$ with the group obtained from some profinite $F_p$-algebra Lie via the Campbell-Hausdorff composition law. This algebra Lie $L$ is provided with a (explicitly given) system of generators and this allows us to work efficiently with the elements of the Galois group $G_p$. In the talk it will be explained a new techniques allowing us to introduce an action of a formal group of order p on appropriate quotients of the algebra Lie $L$. The most surprising phenomenon: this action comes from “higher” differentiations of $K$. The main application: via the above results we can substantially simplify the appproach to the explicit description of the ramification filtration of the group $G_p$ obtained earlier by the author.