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Iwasawa invariants on non-cyclotomic \(\mathbb Z_p\)-extensions of CM fields. (English) Zbl 1163.11073

Let \(K\) be an abelian CM field of degree \(2m\) and let \(p\) be a prime that splits completely in \(K\) as \(\mathfrak p_1\overline{\mathfrak p}_1\cdots \mathfrak p_m\overline{\mathfrak p}_m\). Let \(K_{\infty}\) be a \(\mathbb Z_p\)-extension of \(K\) such that the \(n\)th layer \(K_n\) is contained in the ray class field of \(K\) modulo \(\mathfrak p_1^{n+1}\cdots \mathfrak p_m^{n+1}\), and assume that \(K_{\infty}/K\) is totally ramified at \(\mathfrak p_1,\dots,\mathfrak p_m\). Let \(A_n\) be the \(p\)-part of the class group of \(K_n\), let \(B_n\) be the classes in \(A_n\) that are fixed by \(\text{Gal}(K_{\infty}/K)\), and let \(D_n\) be the classes in \(A_n\) that contain ideals all of whose prime factors lie above \(p\). The author uses methods of R. Greenberg [Am. J. Math. 98, 263–284 (1976; Zbl 0334.12013)] to show that \(| A_n| \) is bounded as \(n\to \infty\) if and only if \(B_n=D_n\) for all sufficiently large \(n\). This generalizes work of T. Fukuda and K. Komatsu [Exp. Math. 11, No. 4, 469–475 (2002; Zbl 1162.11390)]. Also, results of A. Inatomi [Kodai Math. J. 12, No.3, 420–422 (1989; Zbl 0697.12005)] are used to obtain a formula for \(| B_n| \).

MSC:

11R23 Iwasawa theory
11R29 Class numbers, class groups, discriminants

Keywords:

Iwasawa theory
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References:

[1] T. Fukuda and K. Komatsu, Noncyclotomic \({\mathbf Z}_ p\)-extensions of imaginary quadratic fields, Experiment. Math. 11 (2002), no. 4, 469-475 (2003). · Zbl 1162.11390 · doi:10.1080/10586458.2002.10504699
[2] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, 263-284. · Zbl 0334.12013 · doi:10.2307/2373625
[3] A. Inatomi, On \({\mathbf Z}_ p\)-extensions of real abelian fields, Kodai Math. J. 12 (1989), no. 3, 420-422. \beginthebibliography9 · Zbl 0697.12005 · doi:10.2996/kmj/1138039105
[4] T. Fukuda and K. Komatsu, Noncyclotomic \(\mathbf{Z}_{p}\)-Extensions of Imaginary Quadratic Fields, Experiment. Math. Vol.11 (2002), 469-475 · Zbl 1162.11390 · doi:10.1080/10586458.2002.10504699
[5] R. Greenberg, On the Iwasawa invariants of totally Real Number Fields, Amer. J. Math. 98 (1976), 263-284 · Zbl 0334.12013 · doi:10.2307/2373625
[6] A. Inatomi, On \(\mathbf{Z}_{p}\)-Extensions of Real Abelian Fields, Kodai Math. J. 12 (1989), 420-422 · Zbl 0697.12005 · doi:10.2996/kmj/1138039105
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