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: {\it Classes généralisées 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, $2^{\mathrm{nd}}$ 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'idéaux dans les extensions cycliques relatives de degré 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.

References in corpus (7)

• Computing Arakelov class groups
• The Ambiguous Class Number Formula Revisited
• The second p-class group of a number field
• Principalization of ideals in abelian extensions of number fields
• l-Class groups of cyclic extensions of prime degree l
• On the Bertrandias-Payan module in a p-extension -- capitulation kernel
• No Riemann-hurwitz formula for the p-ranks of relative class groups

Cited by in corpus (12)

• p-adic approach of Greenberg's conjecture (p-split totally real case)
• $\ell$-Class groups of fields in Kummer towers
• Practice of incomplete p-ramification over a number field -- History of abelian p-ramification
• Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg
• Genus theory and $\epsilon$-conjectures on p-class groups
• The Chevalley-Herbrand formula and the real abelian Main Conjecture
• No Riemann-hurwitz formula for the p-ranks of relative class groups
• The Chevalley-Gras formula over global fields
• On the Real Abelian Main Conjecture in the non semi-simple case
• Initial layer of the anti-cyclotomic ${\mathbb{Z}}_3$-extension of $\mathbb{Q}(\sqrt{-m})$ and capitulation phenomenon
• Note on some p-invariants of Q(N^{1/p}) using reflection theorem
• Principalization of logarithmic class groups