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...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Proceedings of the Indian Academy of Sciences. Mathematical sciences 2017-02, Vol.127 (1), p.1-34
1. Verfasser: GRAS, GEORGES
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
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