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

TL;DR

This paper explores the structure of p-class groups in p-extensions, providing theoretical results, algorithms for computation, and applications to abelian extensions, with a focus on invariant classes and ramification theory.

Contribution

It offers new structure theorems for p-class groups, algorithms for their computation, and applications to abelian extensions, extending previous work with improvements and new insights.

Findings

Structure theorems for finite bZ_p[G]-modules

Algorithm for local normic computations of p-class groups

Application to the study of relative p-class groups in abelian extensions

Abstract

We give, in Sections 2 and 3, an english translation of: {\it Classes g\'en\'eralis\'ees invariantes}, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: {\it Class Field Theory: from theory to practice}, SMM, Springer-Verlag, $2^{\rm nd}$ 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: {\it Sur les $\ell$-classes d'id\'eaux dans les extensions cycliques relatives de degr\'e premier $\ell$}, 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: {\it 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.