• Invariant generalized ideal classes - structure theorems for $p$-class groups in $p$-extensions

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/127/01/0001-0034

• # Keywords

Number fields; class field theory; $p$-class groups; $p$-extensions; generalized classes; ambiguous classes; Chevalley’s formula.

• # Abstract

We give, in sections 2 and 3, an english translation of: Classes g\acute{e}n\acute{e}ralis\acute{e}es 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 $\mathbb{Z}_p[G]$-modules ($G\simeq\mathbb{Z}/p\mathbb{Z}$) obtained in: Sur les$\scr l$-classes d’id\acute{e}aux dans les extensions cycliques relatives de degr\acute{e} premier $\mathcal{l}$, 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 5, 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.

• # Author Affiliations

1. Villa la Gardette, chemin Chateau Gagniere, F-38520 Le Bourg d'Oisans, France

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 3
June 2019