scholarly journals HIGHER IDELES AND CLASS FIELD THEORY

2018 ◽  
Vol 236 ◽  
pp. 214-250 ◽  
Author(s):  
MORITZ KERZ ◽  
YIGENG ZHAO
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Duality Theorems ◽  
Class Field ◽  
Higher Dimensional ◽  
Universal Approach

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

scholarly journals REFINED SWAN CONDUCTORS OF ONE-DIMENSIONAL GALOIS REPRESENTATIONS

2019 ◽  
Vol 236 ◽  
pp. 134-182
Author(s):  
KAZUYA KATO ◽  
ISABEL LEAL ◽  
TAKESHI SAITO
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Galois Representations ◽  
Class Field Theory ◽  
Absolute Galois Group ◽  
Class Field ◽  
One Dimensional ◽  
Swan Conductor ◽  
Higher Dimensional ◽  
Valuation Field

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

scholarly journals Covering data and higher dimensional global class field theory

2009 ◽  
Vol 129 (10) ◽  
pp. 2569-2599 ◽  
Author(s):  
Moritz Kerz ◽  
Alexander Schmidt
Keyword(s):  
Field Theory ◽  
Class Field Theory ◽  
Class Field ◽  
Higher Dimensional

scholarly journals The Cohomology Groups of Tori in Finite Galois Extensions of Number Fields

1966 ◽  
Vol 27 (2) ◽  
pp. 709-719 ◽  
Author(s):  
J. Tate
Keyword(s):  
Field Theory ◽  
Multiplicative Group ◽  
Number Fields ◽  
Class Field Theory ◽  
Large Set ◽  
Cohomology Groups ◽  
Galois Extensions ◽  
Class Field ◽  
Higher Dimensional ◽  
The Given

Class field theory determines in a well-known way the higher dimensional cohomology groups of the idéies and idèle classes in finite Galois extensions of number fields. At the Amsterdam Congress in 1954 I announced [7] the corresponding result for the multiplicative group of the number field itself, but the proof has never been published. Meanwhile, Nakayama showed that results of this type have much broader implications than had been realized. In particular, his theorem allows us to generalize our result from the multiplicative group to the case of an arbitrary torus which is split by the given Galois extension. We also treat the case of “S-units” of the multiplicative group or torus, for a suitably large set of places S. It is a pleasure for me to publish this paper here, in recognition of Nakayama’s important contributions to our knowledge of the cohomological aspects of class field theory; his work both foreshadowed and generalized the theorem under discussion.

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