Localabelian 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 $K$-theory 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 higher-dimensional $K$-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.