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Department of Mathematical Sciences

# Seminar Archives

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

Presented by Victor Abrashkin, Durham University

11 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.