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

Proceedings Mathematical Sciences

Georges Gras¹ 

[1] Villa la Gardette, chemin Chateau Gagniere, F-0 Le Bourg d'Oisans, France$$

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

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

We give, in sections 2 and 3, an english translation of:Classes gacute{e}nacute{e}ralisacute{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 ($Gsimeqmathbb{Z}/pmathbb{Z}$) obtained in: Sur les$scr l$-classes d’idacute{e}aux dans les extensions cycliques relatives de degracute{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.

【 授权许可】

Unknown