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

Abstract

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 $\mathbb {Z}_{p}[G]$-modules ($G \simeq \mathbb {Z}/p\,\mathbb {Z}$) 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.