# Seminar Archives

## Arithmetic Study Group: Galois groups of local fields, Lie algebras and ramification

4 November 2014 14:00 in *CM103*

Let K be a complete discrete valuation field with a finite

residue field of characteristic p>0.

If G_K(p) is the Galois group of the maximal p-extension of K then its

structure is completely known :

it is either free or Demushkin's group. However, this result is not

completely satisfactory because

the appropriate functor from the fields K to the pro-p-groups G_K(p) is

not fully faithful. In other words, in this setting

the Galois group does not reflect essential invariants of the original

field K. The situation becomes completely different

if we take into account an additional structure on G_K(p) given by its

decreasing filtration by ramification

subgroups G_K(p)^{(v)}, v\ge 0. The importance of explicit description

of this filtration

was pointed out in 1960-1970's by A.Weil, I.Shafarevich, P.Deligne etc.

In particular,

if K has characteristic p then we should invent a way to specify a

special choice of free generators

of G_K(p) which reflects arithmetic properties of K. Suppose C_s, where

s\ge 1, is the closure of the subgroup of

commutators of order at least s in G_K(p). Then the above problem of

"arithmetic description" of

G_K(p) can be considered modulo subgroups C_s. If s=1 it is trivial and

if s=2 the answer is given

via class field theory. In the case s>2 we obtain a long-standing

problem of constructing

a "nilpotent class field theory". In the talk we discuss the case s=p,

in particular, the author new results

related to the mixed characteristic case (i.e. when K is a finite

extension of Q_p). The quotient G_K(p)/C_p

(together with the induced ramification filtration) is complicated

enough to reflect the invariants of

K. At the same time this quotient comes from a profinite Lie algebra via

Campbell-Hausdorff composition law.

The description of the appropriate ramification filtration essentially

uses this structure of Lie algebra.

Contact athanasios.bouganis@durham.ac.uk for more information