Local abelian Kato-Parshin reciprocity law: A survey

Abstract

Let K denote an n-dimensional local field. The aim of this expository paper is to survey the basic arithmetic theory of the n-dimensional local field K together with its Milnor Ktheory and Parshin topological K-theory; to review Kato's ramification theory for finite abelian extensions of the n-dimensional local field K, and to state the local abelian higherdimensional K-theoretic generalization of local abelian class field theory of Hasse, which is developed by Kato and Parshin. The paper is geared toward non-abelian generalization of this theory.