Masato Kurihara, Ivan Fesenko
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.
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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.
This work studies two dimensional local skew fields and their automorphisms.
Basic concepts of higher local fields and topologies on their additive and multiplicative groups are introduced.
This is a presentation of main ingredients of Kato's higher local class field theory.
Ramification theory of monogenic extensions of complete discrete valuation fields is presented. Relations to Kato's conductor are discussed.
This is a review of Parshin's higher local class field theory in characteristic p.
Koya's and author's approach to the higher local reciprocity map as a generalization of the classical class formations approach to the level of complexes of Galois modules.
This is an introduction to noncommutative local reciprocity maps for totally ramified Galois extensions with arithmetically profinite group. These maps in general are not homomorphisms but Galois cycles; a description of their image and kernel is included.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)).
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…
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 …
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 …
Given a point $ξ$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $ξ$. When $(A, ξ)$ varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often $ξ$ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when $ξ$ is a torsi…
In this appendix we present an expanded version of Section 4 of our paper arXiv:1404.4596, including the proofs of all of the technical lemmas.
Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundament…
The special uniformity of zeta functions claims that pure non-abelian zeta functions coincide with group zeta functions associated to the special linear groups. Naturally associated are three aspects, namely, the analytic, arithmetic, and geometric aspects. In the first paper of this series, we expose intrinsic geometric structures of our zetas by counting semi-stable bundles on curves defined over finite fields in terms of their automorphism groups and global sections. We show that such a counting maybe read from Artin zetas which are abelian in nature. This paper also contains an appendix wr…
This is an update of the first version. We clarify that the main results apply to more general smooth projective varieties X than products of elliptic curves (briefly: X is of "abelian type", e.g. an abelian variety or a product of curves, and the Tate conjecture holds). We also deduce the Gersten conjecture for dvrs whose residue field is the function field of such an X. In an appendix, we construct some functoriality for etale motivic cohomology. Finally, some errors which were in Section 5 of the first version are corrected.
Equivariant Vector Bundles on the Drinfeld Upper Half Space over a Local Field of Positive Characteristic
本文将Orlik关于p-adic域的齐次向量丛整体截面工作推广至正特征局部域，通过子表示过滤和改编的函子刻画子商，并确定了K^×的局部解析与连续特征。
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