Summary: AN ANALOGUE OF THE FIELDOFNORMS FUNCTOR
AND OF THE GROTHENDIECK CONJECTURE
Abstract. The paper contains a construction of an analogue of the FontaineWintenberger
fieldofnorms functor for higher dimensional local fields. This construction is done
completely in terms of the ramification theory of such fields. It is applied to deduce
the mixed characteristic case of a local analogue of the Grothendieck Conjecture for
these fields from its characteristic p case, which was proved earlier by the author.
Throughout all this paper p is a fixed prime number.
The fieldofnorms functor [FW1,2] allows to identify the Galois groups of some
infinite extensions of Q p with those of complete discrete valuation fields of char
acteristic p. This functor is an essential component of Fontaine's theory of ##
modules --- one of most powerful tools in the modern study of padic representations.
Other areas of very impressive applications are the Galois cohomology of local fields
[He], arithmetic aspects of dynamical systems [LMS], explicit reciprocity formulae
[Ab2,3], [Ben], a description of the structure of ramification filtration [Ab7], the
proof of an analogue of the Grothendieck Conjecture for 1dimensional local fields
A local analogue of the Grothendieck Conjecture establishes an opportunity to