Invitation to higher local fields, Part II, section 8: Higher local skew fields

Invitation to higher local fields, Part I, section 8: Explicit formulas for the Hilbert symbol

This is a review of the vast area of explicit formulas for the (wild) Hilbert symbol (not only in the one-dimensional case but in the higher dimensional case as well). An extensive bilbiography is included.

Invitation to higher local fields, Part I, section 10: Explicit higher local class field theory

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.

Invitation to higher local fields, Part I, section 16: Higher class field theory without using K-groups

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.

Invitation to higher local fields, Part II, section 4: Drinfeld modules and local fields of positive characteristic

This work is a survey of relations between Drinfeld modules and higher dimensional fields of positive characteristic. The main new result stated is the expression of vanishing orders of certain modular forms through partial zeta values.

Invitation to higher local fields, Part II, section 3: The Bruhat-Tits buildings over higher dimensional local fields

This work introduces author's theory of Bruhat-Tits buildings over higher dimensional local fields. The theory is illustrated with the buildings for PGL(2) and PGL(3) for one- and two-dimensional local fields.

Invitation to higher local fields, Part II, section 6: Phi-Gamma-modules and Galois cohomology

This is is a survey of applications of Fontaine's theory of p-adic representations of local fields (Phi-Gamma-modules) to Galois cohomology of local fields and explicit formulas for the Hilbert symbol in relation with two-dimensional local objects.

Invitation to higher local fields, Part I, section 14: Explicit abelian extensions of complete discrete valuation fields

This work presents author's explicit methods of constructing abelian extensions of complete discrete valuation fields. His approach to explicit equations of a cyclic extension of degree p^n which contains a given cyclic extension of degree p is explained.

Invitation to higher local fields, Part II, section 2: Adelic constructions for direct images of differentials and symbols

This work introduces adelic constructions of direct images of differentials and symbols in the two-dimensional case in the relative situation. In particular, reciprocity laws for relative residues of differentials and symbols are stated and applied to a construction of the Gysin map for Chow groups.

Invitation to higher local fields, Part I, section 6: Topological Milnor K-groups of higher local fields

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.

Invitation to higher local fields, Part I, section 13: Abelian extensions of absolutely unramified complete discrete valuation fields

This is an introduction to the author theory of cyclic p-extensions of an absolutely unramified complete discrete valuation field K with arbitrary residue field of characteristic p. In this theory a homomorphism is constructed from the p-part of the group of characters of K to Witt vectors over its residue field.

Invitation to higher local fields, Part I, section 4: Cohomological symbol for henselian discrete valuation fields of mixed characteristic

This is a sketch of main steps of the proof of Bloch--Kato's theorem which states that the norm residue homomorphism K_q(K)/m\to H^q(K,\Bbb Z/m(q)) is an isomorphism for a henselian discrete valuation field K of characteristic 0 with residue field of positive characteristic.

Invitation to higher local fields, Part II, section 5: Harmonic analysis on algebraic groups over two-dimensional local fields of equal characteristic

This work introduces author's approach to harmonic analysis on algebraic groups over functional two-dimensional local fields. For a two-dimensional local field a Hecke algebra which is formed by operators which integrate pro-locally-constant complex functions over a non-compact domain is defined and its properties are described.

Invitation to higher local fields, Part II, section 10: Galois modules and class field theory

This is a concise survey of links between Galois module theory and class field theory (CFT). It explores various uses of CFT in Galois module theory, it comments on the absence of CFT in contexts where it might be expected to play a role and indicates some lines of research that might bring CFT to play a more prominent role in Galois module theory.

Invitation to higher local fields, Part I, section 17: An approach to higher ramification theory

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.

Invitation to higher local fields, Part I, section 15: On the structure of the Milnor K-groups of complete discrete valuation fields

This work contains a list of all known results on the quotient filtration on the Milnor K-groups of a complete discrete valuation field in terms of differential modules over the residue field . Author's recent study of the case of a tamely ramified field of characteristic 0 with residue field of characteristic p by using an exponential map and a syntomic complex is explained.

Invitation to higher local fields, Part II, section 1: Higher dimensional local fields and L-functions

This work describes several first steps in extending Tate-Iwasawa's analytic method to define an L-function in higher dimensions. For generalizing this method the author advocates the usefulness of the classical Riemann-Hecke approach, his adelic complexes together with his generalization of Krichever's correspondence. He analyzes dimension 1 types of functions and discusses properties of the lattice of commensurable classes of subspaces in the adelic space associated to a divisor on an algebraic surface.

Invitation to higher local fields, Part I, section A: Appendix to Section 2

This appendix discusses some basic definitions and properties of differential forms and Kato's cohomology groups in characteristic p and a sketch of the proof of Bloch-Kato-Gabber's theorem which describes the differential symbol from the Milnor K-group K_n(F)/p of a field F of positive characteristic p to the differential module Ω_F^n.

Invitation to higher local fields, Part I, section 9: Exponential maps and explicit formulas

An exponential homomorphism for a complete discrete valuation field of characteristic zero which relates differential forms and the Milnor K-groups of the field is studied. An application to explicit formulas is included.

Invitation to higher local fields, Part I, section 2: p-primary part of the Milnor K-groups and Galois cohomology of fields of characteristic p

For an arbitrary field p-torsion and cotorsion of the Milnor groups K_n(F) and K_n^{t}(F)=K_n(F)/\cap_{l\ge1} lK_n(F) are discussed. The work contains further discussions of an analogue of Satz 90 for K_n(F) and K_n^{t}(F) and computation of H^{n+1}_m(F) where F is either the rational function field in one variable F=k(t) or the formal power series F=k((t)).

Invitation to higher local fields, Part I, section 12: Two types of complete discrete valuation fields

This work sketches the author classification of complete discrete valuation fields K of characteristic 0 with residue field of characteristic p into two classes depending on the behaviour of the torsion part of a differential module. For each of these classes, the quotient filtration of the Milnor K-groups of K is characterized for all sufficiently large members of the filtration, as a quotient of differential modules. For a higher local field the previous result and higher local class field theory imply certain restrictions on types of cyclic extensions of the field of sufficiently large degr…

Invitation to higher local fields, Part II, section 7: Recovering higher global and local fields from Galois groups - an algebraic approach

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 second application (which is a joint work with Fesenko) we analyze the arithmetic structure of fields with the same absolute …

Invitation to higher local fields (Introduction)

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

高秩径向分歧斜积的动力 Mordell--Lang 猜想

Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products

本文证明了复数域上径向分歧多项式斜积族的动力 Mordell--Lang 猜想，并建立了统一的 Skolem--Mahler--Lech 定理，适用于轨道与线性递推的联合多项式与有理关系。