On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple ${\mathbb{Z}}_p$-Extensions Over Imaginary Quadratic Fields

Miura, Takashi; Murakami, Kazuaki; Okano, Keiji; Otsuki, Rei

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 3, pp. 881-902.

Pour un nombre premier impair $p$, on s’intéresse au nombre de générateurs des modules d’Iwasawa non ramifiés des ${\mathbb{Z}}_p$-extensions multiples maximales sur l’algèbre d’Iwasawa. Dans notre article précédent, sous plusieurs hypothèses sur un corps quadratique imaginaire, nous avons obtenu une condition nécessaire et suffisante de cyclicité du module d’Iwasawa sur l’algèbre d’Iwasawa. Le présent travail fournit des méthodes de calcul et des exemples numériques des modules d’Iwasawa qui sont cycliques en tant que modules sur l’algèbre d’Iwasawa. Nous remarquons que nos méthodes ne supposent pas la véracité de la conjecture de Greenberg généralisée.

For an odd prime number $p$, we study the number of generators of the unramified Iwasawa modules of the maximal multiple ${\mathbb{Z}}_p$-extensions over the Iwasawa algebra. In our previous paper, under several assumptions for an imaginary quadratic field, we obtained a necessary and sufficient condition for the cyclicity of the Iwasawa module over the Iwasawa algebra. The present work provides computational methods and numerical examples of Iwasawa modules that are cyclic as modules over the Iwasawa algebra. We remark that our methods do not require the assumption that Greenberg’s generalized conjecture holds.

Classification : 11R23
Mots clés : Iwasawa modules, Imaginary quadratic fields, Multiple $\mathbb{Z}_p$-extensions

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