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Galois coinvariants of the unramified Iwasawa modules of multiple \(\mathbb {Z}_p\)-extensions

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Abstract

For a CM-field K and an odd prime number p, let \(\widetilde{K}'\) be a certain multiple \(\mathbb {Z}_p\)-extension of K. In this paper, we study several basic properties of the unramified Iwasawa module \(X_{\widetilde{K}'}\) of \(\widetilde{K}'\) as a \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}'/K)\rrbracket \)-module. Our first main result is a description of the order of a Galois coinvariant of \(X_{\widetilde{K}'}\) in terms of the characteristic power series of the unramified Iwasawa module of the cyclotomic \(\mathbb {Z}_p\)-extension of K under a certain assumption on the splitting of primes above p. The second result is that if K is an imaginary quadratic field and if p does not split in K, then, under several assumptions on the Iwasawa \(\lambda \)-invariant and the ideal class group of K, we determine a necessary and sufficient condition such that \(X_{\widetilde{K}}\) is \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}/K)\rrbracket \)-cyclic. Here, \(\widetilde{K}\) is the \(\mathbb {Z}_p^2\)-extension of K.

Résumé

Pour un corps CM K et un nombre premier impair p, soit \(\widetilde{K}'\) une certaine \(\mathbb {Z}_p\)-extension multiple de K. Dans cet article, nous étudions plusieurs propriétés de base du module \(X_{\widetilde{K}'}\) d’Iwasawa non ramifié de \(\widetilde{K}'\) en tant que \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}'/K)\rrbracket \)-module. Notre premier résultat principal est une description de l’ordre d’un coinvariant de Galois de \(X_{\widetilde{K}'}\) en termes de la série des puissances caractéristique du module d’Iwasawa non ramifié de la \(\mathbb {Z}_p\)-extension cyclotomique de K sous une certaine hypothèse sur la décomposition des idéaux premiers au-dessus de p. Un deuxième résultat est que, si K est un corps quadratique imaginaire et si p ne se factorise pas en K, alors, sous plusieurs hypothèses sur l’invariant \(\lambda \) d’Iwasawa et le groupe des classes d’idéaux de K, nous déduisons une condition nécessaire et suffisante pour que \(X_{\widetilde{K}}\) soit \(\mathbb {Z}_p\llbracket \mathrm{Gal}(\widetilde{K}/K)\rrbracket \)-cyclique. Ici \(\widetilde{K}\) est la \(\mathbb {Z}_p^2\)-extension de K.

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Acknowledgements

The authors would like to thank Masato Kurihara, Satoshi Fujii, and Takamichi Sano for their encouragement and useful comments. The authors also would like to express their thanks to the referee for reading this article carefully and for giving many useful comments.

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Authors and Affiliations

  1. Department of Creative Engineering, National Institute of Technology, Tsuruoka College, 104 Sawada, Inooka, Tsuruoka, Yamagata, 997-8511, Japan

    Takashi Miura

  2. Department of Mathematical Sciences, Graduate School of Science and Engineering, Keio University, Hiyoshi, Kohoku-ku, Yokohama, Kanagawa, 223-8522, Japan

    Kazuaki Murakami

  3. Department of Teacher Education, Tsuru University, 3-8-1 Tahara, Tsuru-shi, Yamanashi, 402-0054, Japan

    Keiji Okano

  4. Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8522, Japan

    Rei Otsuki

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  1. Takashi Miura
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  2. Kazuaki Murakami
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  4. Rei Otsuki
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Correspondence to Keiji Okano.

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Miura, T., Murakami, K., Okano, K. et al. Galois coinvariants of the unramified Iwasawa modules of multiple \(\mathbb {Z}_p\)-extensions. Ann. Math. Québec 45, 407–431 (2021). https://doi.org/10.1007/s40316-020-00150-6

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