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

It is a survey paper on n-dimensional local fields and adeles on n-dimensional schemes.

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 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 work studies two dimensional local skew fields and their automorphisms.

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 di...

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)).

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in the boundary between number theory, algebra and topology. They use many definitions and theorems - more or less advanced - of general algebra and topology. Gradually, we will go from the general to the local, from the valued fields to the lo...

Viewing higher local fields as ring objects in the category of iterated pro-ind-objects, a definition of open subgroups in Milnor K-groups of the fields is given. The self-duality of the additive group of a higher local field is proved. By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the existence theorem is proved.

In this work, we prove the vanishing of the two cohomological group of the higher local field. This generalize the well-known propriety of finite field and one dimensional local field. We apply this result to study the arithmetic of curve defined over higher local field.

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.