Higher class field theory and the connected component

Generalized class formations and higher class field theory

Kato established higher local reciprocity map (see section 5 and [K1, Th. 2 of §6] (two-dimensional case), [K2, Th. II], [K3, §4]) using in particular this theorem. In this section we deduce the reciprocity map for higher local fields from this theorem and Bloch–Kato’s theorem of section 4. Our approach which uses generalized class formations simplifies Kato’s original argument. We use the nota...

Kato ’ s higher local class field theory

We first recall the classical local class field theory. Let K be a finite extension of Qp or Fq((X)). The main theorem of local class field theory consists of the isomorphism theorem and existence theorem. In this section we consider the isomorphism theorem. An outline of one of the proofs is as follows. First, for the Brauer group Br(K), an isomorphism inv: Br(K) →̃Q/Z is established; it mainly...

Explicit higher local class field theory

Consider now an approach based on a generalization [F2] of Neukirch’s approach [N]. Below is a modified system of axioms of class formations (when applied to topological K -groups) which imposes weaker restrictions than the classical axioms (cf. section 11). (A1). There is a Ẑ-extension of F . In the case of higher local fields let F pur/F be the extension which corresponds to K sep 0 /K0: F pu...

the survey of the virtual higher education in iran and the ways of its development and improvement

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Class Field Theory