Some remarks on pseudo-null submodules of tamely ramified Iwasawa modules

Fujii, Satoshi; Itoh, Tsuyoshi

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 2, pp. 533-555.

Nous donnons diverses observations sur la structure des modules d’Iwasawa modérément ramifiés pour une ${\mathbb{Z}}_p$-extension (ou une ${\mathbb{Z}}_p$-extension multiple) d’un corps de nombres. Dans cet article, nous considérons la question de savoir si un module d’Iwasawa modérément ramifié possède un sous-module fini (ou pseudo-nul) non-nul ou non. Pour la ${\mathbb{Z}}_p$-extension cyclotomique de $\mathbb{Q}$ (avec $p$ impair), nous pouvons obtenir une solution complète à cette question. Nous donnons également des conditions suffisantes pour avoir un sous-module pseudo-nul non-nul pour la ${\mathbb{Z}}_p^{\oplus 2}$-extension d’un corps quadratique imaginaire. Et nous donnons aussi une application de nos résultats à la « théorie d’Iwasawa non-abélienne » dans le sens d’Ozaki.

We will give several observations about the structure of tamely ramified Iwasawa modules for a ${\mathbb{Z}}_p$-extension (or a multiple ${\mathbb{Z}}_p$-extension) of an algebraic number field. In the present paper, we consider the question whether a given tamely ramified Iwasawa module has a non-trivial finite (or pseudo-null) submodule or not. For the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ (with odd $p$), we can obtain a complete answer to this question. We also give sufficient conditions for having a non-trivial pseudo-null submodule for the case of the ${\mathbb{Z}}_p^{\oplus 2}$-extension of an imaginary quadratic field. We also give an application of our results to the “non-abelian Iwasawa theory” in the sense of Ozaki.

Classification : 11R23
Mots-clés : Iwasawa modules, finite submodules, pseudo-null submodules

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