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

Takashi Miura; Kazuaki Murakami; Keiji Okano; Rei Otsuki

doi:10.5802/jtnb.1232

On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple

ℤ_{p}

-Extensions Over Imaginary Quadratic Fields

Takashi Miura ¹; Kazuaki Murakami ²; Keiji Okano ³; Rei Otsuki ⁴

¹ Department of Creative Engineering, National Institute of Technology, Tsuruoka College, 104 Sawada, Inooka, Tsuruoka, Yamagata 997-8511, Japan
² Department of Information Science, Toho University, 2-2-1 Miyama, Funabashi-shi, Chiba 274-8510 Japan
³ Department of Teacher Education, Tsuru University, 3-8-1 Tahara, Tsuru-shi, Yamanashi 402-0054, Japan
⁴ Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan

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

Abstract
Résumé

For an odd prime number $p$ , we study the number of generators of the unramified Iwasawa modules of the maximal multiple $ℤ_{p}$ -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.

Pour un nombre premier impair $p$ , on s’intéresse au nombre de générateurs des modules d’Iwasawa non ramifiés des $ℤ_{p}$ -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.

Received: 2021-07-04
Revised: 2021-12-24
Accepted: 2022-04-01
Published online: 2023-01-27

DOI: 10.5802/jtnb.1232

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

Author's affiliations:

Takashi Miura ¹; Kazuaki Murakami ²; Keiji Okano ³; Rei Otsuki ⁴

¹ Department of Creative Engineering, National Institute of Technology, Tsuruoka College, 104 Sawada, Inooka, Tsuruoka, Yamagata 997-8511, Japan
² Department of Information Science, Toho University, 2-2-1 Miyama, Funabashi-shi, Chiba 274-8510 Japan
³ Department of Teacher Education, Tsuru University, 3-8-1 Tahara, Tsuru-shi, Yamanashi 402-0054, Japan
⁴ Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522, Japan

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Takashi Miura and Kazuaki Murakami and Keiji Okano and Rei Otsuki},
     title = {On the {Cyclicity} of the {Unramified} {Iwasawa} {Modules} of the {Maximal} {Multiple} $\protect \mathbb{Z}_p${-Extensions} {Over} {Imaginary} {Quadratic} {Fields}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Takashi Miura; Kazuaki Murakami; Keiji Okano; Rei Otsuki. On the Cyclicity of the Unramified Iwasawa Modules of the Maximal Multiple $\protect \mathbb{Z}_p$-Extensions Over Imaginary Quadratic Fields. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 881-902. doi : 10.5802/jtnb.1232. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1232/

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