Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions
We give, in sections 2 and 3 , an english translation of: Classes généralisées invariantes , J. Math. Soc. Japan , 46 , 3 ( 1994 ), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: From Theory to Practice , SMM, Springer-Verlag, 2nd corrected...
Gespeichert in:
Veröffentlicht in: | Proceedings of the Indian Academy of Sciences. Mathematical sciences 2017-02, Vol.127 (1), p.1-34 |
---|---|
1. Verfasser: | |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | We give, in sections
2
and
3
, an english translation of:
Classes généralisées invariantes
,
J. Math. Soc. Japan
,
46
, 3 (
1994
), with some improvements and with notations and definitions in accordance with our book:
Class Field Theory: From Theory to Practice
, SMM, Springer-Verlag, 2nd corrected printing 2005. We recall, in section
4
, some structure theorems for finite
ℤ
p
[
G
]
-modules (
G
≃
ℤ
/
p
ℤ
) obtained in: Sur les
ℓ
-classes d’idéaux dans les extensions cycliques relatives de degré premier
ℓ
,
Annales de l’Institut Fourier
,
23
, 3 (
1973
). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a
p
-class group in a cyclic extension of degree
p
. In section
1973
, we apply this to the study of the structure of relative
p
-class groups of Abelian extensions of prime to
p
degree, using the Thaine–Ribet–Mazur–Wiles–Kolyvagin ‘principal theorem’, and the notion of ‘admissible sets of prime numbers’ in a cyclic extension of degree
p
, from: Sur la structure des groupes de classes relatives,
Annales de l’Institut Fourier
,
43
, 1 (
1993
). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the
p
-ramification theory (as dual form of non-ramification theory) and which have become standard in a
p
-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject. |
---|---|
ISSN: | 0253-4142 0973-7685 |
DOI: | 10.1007/s12044-016-0324-1 |