Invitation to higher local fields, Part I, section 1: Higher dimensional local fields

These notes are an introduction to higher dimensional local fields and higher dimensional adeles. As well as the foundational theory, we summarise the theory of topologies on higher dimensional local fields and higher dimensional local class field theory.

The monograph "Invitation to higher local fields" is the result of the conference on higher local fields held in Muenster, August 29 to September 5, 1999. The aim is to provide an introduction to higher local fields (more generally complete discrete valuation fields with arbitrary residue field) and render the main ideas of this theory (Part I), as well as to discuss several applications and connections to other areas (Part II). The volume grew as an extended version of tal...

This is a presentation of main ingredients of Kato's higher local class field theory.

Certain topologies on Milnor K-groups of higher local fields K are studied. These are related to the topology on the multiplicative group and important for explicit higher local class field theory. The structure of the quotient of Milnor K-groups modulo the intersection of all neighbourhoods of zero is described.

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a secon...

This is a presentation of explicit methods to construct higher local class field theory by using topological K-groups, explicit symbols and a generalization of Neukirch-Hazewinkel's axiomatic approaches. The existence theorem is discussed as well.

This is a review of Parshin's higher local class field theory in characteristic p.

Notes for a course at the H.-C. R. I., Allahabad, 15 August 2008 -- 26 January 2009

This is an introduction to author's ramification theory of a complete discrete valuation field with residue field whose p-basis consists of at most one element. New lower and upper filtrations are defined; cyclic extensions of degree p may have non-integer ramification breaks. A refinement of the filtration for two-dimensional local fields which is compatible with the reciprocity map is discussed.

Author's generalization of one-dimensional class field theory to theory of abelian totally ramified p-extensions of a complete discrete valuation field with arbitrary non-separably p-closed residue field and its applications are described.