Class Formations and Higher Dimensional Local Class Field Theory

Local abelian Kato-Parshin reciprocity law: A survey

“…-The explicit approach of Fesenko [7][8][9] is based on extending the local abelian Hasse reciprocity law construction of Neukirch-Iwasawa [39,40] and on extending the local norm residue symbol construction of Hazewinkel [17] to the setting of n-dimensional local fields; -Kato's approach [22,25] is cohomological and extends Tate's construction of the local abelian Hasse reciprocity law [48]; -Koya on the other hand [30][31][32], using Lichtenbaum's complexes Z(i) [34], generalizes class formation approach of local abelian class field theory to construct the local abelian 2-dimensional class field theory, which is extended and streamlined by Spiess [47] to the n-dimensional setting; -The final approach, due to Parshin [42,44,45], which is the genesis of the whole program, generalizes Kawada-Satake construction of local abelian class field theory [29] to construct the local abelian n-dimensional class field theory in positive characteristic. In this work we shall review Fesenko's explicit approach, where as stated above, the idea is to generalize the classical Neukirch-Iwasawa and Hazewinkel methods to higherdimensional local fields, which will be recalled next with extra care following closely [10,11,13].…”

An introduction to higher dimensional local fields and adeles

“…Y. Koya [42,43] proved higher dimensional local class field theory using the language of class formations and complexes of Galois modules. His methods were further developed by M. Speiss [69,70], who gave the first proof that if L/F is a finite Galois extension (not necessarily abelian) of n-dimensional local fields, then…”

Topology on rational points over n-local fields

“…Regarding the topological approach, with which this work is concerned, the major achievement is local higher class field theory. This theory was established in the general case by Kato [20], [21], [22]; Fesenko [9], [10]; Spiess [29] and others. It uses Milnor K-groups endowed with a certain topology and the higher reciprocity map in order to describe abelian extensions of a higher local field.…”