Volume 3 (2000)

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ISSN (electronic): 1464-8997
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Invitation to higher local fields

Editors: Ivan Fesenko and Masato Kurihara

This monograph 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 talks given at the conference. The two parts are separated by a paper of K. Kato, an IHES preprint from 1980 which has never been published.

An n-dimensional local field is a complete discrete valuation field whose residue field is an (n-1)-dimensional local field; 0-dimensional local fields are just perfect (e.g. finite) fields of positive characteristic. Given an arithmetic scheme, there is a higher local field associated to a flag of subschemes on it. One of central results on higher local fields, class field theory, describes abelian extensions of an n-dimensional local field via (all in the case of finite 0-dimensional residue field; some in the case of infinite 0-dimensional residue field) closed subgroups of the n-th Milnor K-group of F.

We hope that the volume will be a useful introduction and guide to the subject. The contributions to this volume were received over the period November 1999 to August 2000 and the electronic publication date is 10 December 2000.

Ivan Fesenko and Masato Kurihara

Geometry & Topology Monographs 3 (2000)

DOI: 10.2140/gtm.2000.3


Masato Kurihara and Ivan Fesenko

Some conventions

Masato Kurihara and Ivan Fesenko

Part I
Higher dimensional local fields

Igor Zhukov

p–primary part of the Milnor K–groups and Galois cohomologies of fields of characteristic p

Oleg Izhboldin

Appendix to Section 2

Masato Kurihara and Ivan Fesenko

Cohomological symbol for henselian discrete valuation fields of mixed characteristic

Jinya Nakamura

Kato's higher local class field theory

Masato Kurihara

Topological Milnor K–groups of higher local fields

Ivan Fesenko

Parshin's higher local class field theory in characteristic p

Ivan Fesenko

Explicit formulas for the Hilbert symbol

Sergei V Vostokov

Exponential maps and explicit formulas

Masato Kurihara

Explicit higher local class field theory

Ivan Fesenko

Generalized class formations and higher class field theory

Michael Spieß

Two types of complete discrete valuation fields

Masato Kurihara

Abelian extensions of absolutely unramified complete discrete valuation fields

Masato Kurihara

Explicit abelian extensions of complete discrete valuation fields

Igor Zhukov

On the structure of the Milnor K–groups of complete discrete valuation fields

Jinya Nakamura

Higher class field theory without using K–groups

Ivan Fesenko

An approach to higher ramification theory

Igor Zhukov

On ramification theory of monogenic extensions

Luca Spriano

Existence theorem for higher local fields

Kazuya Kato

Part II
Higher dimensional local fields and L–functions

A N Parshin

Adelic constructions for direct images of differentials and symbols

Denis Osipov

The Bruhat–Tits building over higher dimensional local fields

A N Parshin

Drinfeld modules and local fields of positive characteristic

Ernst-Ulrich Gekeler

Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic

Mikhail Kapranov

Φ–Γ–modules and Galois cohomology

Laurent Herr

Recovering higher global and local fields from Galois groups – an algebraic approach

Ido Efrat

Higher local skew fields

Alexander Zheglov

Local reciprocity cycles

Ivan Fesenko

Galois modules and class field theory

Boas Erez