G\'en\'eralisation d'un Th\'eor\`eme d'Iwasawa

TL;DR

This paper generalizes Iwasawa's classical theorem to finite quotients of noetherian Lambda-modules, computes associated characters in cyclotomic towers, and extends Greenberg's genus theory results for wild ramification.

Contribution

It extends Iwasawa's asymptotic results to new algebraic contexts and relates these to classical invariants and ramification decomposition via the Spiegelungssatz.

Findings

Extended Iwasawa's theorem to finite quotients of Lambda-modules.

Computed characters related to T-infinitesimal S-classes in cyclotomic towers.

Generalized Greenberg's genus theory to wild ramification cases.

Abstract

We extend to convenient finite quotients of a noetherian Lambda-module the classical result of K. Iwasawa giving the asymptotic expression of the l-part of the number of ideal class groups in Zl-extensions of number fields. Then, in the arithmetic context, we compute the three characters associated by this way to the l-groups of T-infinitesimal S-classes in the cyclotomic tower and relate them to the classical invariants and the decomposition characters associated to the finite sets of places S and T. A main tool in this study is the so-called Spiegelungssatz of Georges Gras, which exchanges (wild or tame) ramification and decomposition. The main results of this arithmetical part extend those we obtained with Christian Maire in a previous article. The most intricate study of the wild contribution of the sets S and T involves a generalization of a classical result of R. Greenberg on the genus theory of cyclotomic towers.