Geometry & Topology Monographs

Volume 3: Invitation to higher local fields

Ivan Fesenko and Masato Kurihara (editors)

Brief introduction

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.

Complete volume and complete parts I and II

Full introduction
Some conventions

Part I

1. Pages 5-18 : Higher dimensional local fields

2. Pages 19-29 : p-primary part of the Milnor K-groups and Galois cohomology of fields of characteristic p

3. Pages 31-41 : Appendix to Section 2

4. Pages 43-51 : Cohomological symbol for henselian discrete valuation fields of mixed characteristic

5. Pages 53-60 : Kato's higher local class field theory

6. Pages 61-74 : Topological Milnor K-groups of higher local fields

7. Pages 75-79 : Parshin's higher local class field theory in characteristic p

8. Pages 81-89 : Explicit formulas for the Hilbert symbol

9. Pages 91-94 : Exponential maps and explicit formulas

10. Pages 95-101 : Explicit higher local class field theory

11. Pages 103-108 : Generalized class formations and higher class field theory

12. Pages 109-112 : Two types of complete discrete valuation fields

13. Pages 113-116 : Abelian extensions of absolutely unramified complete discrete valuation fields

14. Pages 117-122 : Explicit abelian extensions of complete discrete valuation fields

15. Pages 123-135 : On the structure of the Milnor K-groups of complete discrete valuation fields

16. Pages 137-142 : Higher class field theory without using K-groups

17. Pages 143-150 : An approach to higher ramification theory

18. Pages 151-164 : On ramification theory of monogenic extensions


Pages 165-195 : Existence theorem for higher local class field theory

Part II

1. Pages 199-213 : Higher dimensional local fields and L-functions

2. Pages 215-221 : Adelic constructions for direct images of differentials and symbols

3. Pages 223-237 : The Bruhat-Tits buildings over higher dimensional local fields

4. Pages 239-253 : Drinfeld modules and local fields of positive characteristic

5. Pages 255-262 : Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic

6. Pages 263-272 : Phi-Gamma-modules and Galois cohomology

7. Pages 273-279 : Recovering higher global and local fields from Galois groups - an algebraic approach

8. Pages 281-292 : Higher local skew fields

9. Pages 293-298 : Local reciprocity cycles

10. Pages 299-304 : Galois modules and class field theory

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