Tamely ramified Iwasawa modules having no non-trivial pseudo-null submodules

Itoh, Tsuyoshi

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 859-872.

The present paper is a sequel to the previous paper [4] (by Satoshi Fujii and the author). Let $k$ be an algebraic number field, $p$ a prime number, and $k^c/k$ the cyclotomic ${\mathbb{Z}}_p$-extension. For a finite set $S$ of prime numbers which does not contain $p$, the Iwasawa module $X_S\left(k^c\right)$ (with respect to the maximal pro-$p$ abelian extension unramified outside $S$) has been studied in several papers. We will give some non-trivial examples such that $X_S\left(k^c\right)$ has no non-trivial finite submodules even when $k$ is totally real. We also give a similar example for the case of the ${\mathbb{Z}}_p^{\oplus 2}$-extension of an imaginary quadratic field. Moreover, weak analogs of Greenberg’s conjecture for $X_S\left(k^c\right)$ are also discussed in the appendix.

Ce travail fait suite à l’article [4] de Satoshi Fujii et l’auteur. Soient $k$ un corps de nombres, $p$ un nombre premier, et $k^c/k$ la ${\mathbb{Z}}_p$-extension cyclotomique. Pour un ensemble fini $S$ de nombres premiers qui ne contient pas $p$, le module d’Iwasawa (par rapport à la pro-$p$ extension abélienne maximale non ramifiée en dehors de $S$) a été étudié dans plusieurs articles. Nous donnons des exemples non-triviaux où $X_S\left(k^c\right)$ a un sous-module fini non-nul avec $k$ totalement réel. Nous donnons également un exemple similaire dans le cas de la ${\mathbb{Z}}_p^{\oplus 2}$-extension d’un corps quadratique imaginaire. De plus, nous discutons en appendice des analogues faibles de la conjecture de Greenberg pour $X_S\left(k^c\right).$

Classification: 11R23
Keywords: Iwasawa modules, non-existence of non-trivial pseudo-null submodules

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