Osaka Journal of Mathematics

The commutativity of Galois groups of the maximal unramified pro-$p$-extensions over the cyclotomic $\mathbb{Z}_{p}$-extensions II

Keiji Okano

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Abstract

Let $p$ be an odd prime number and $K_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of a Galois $p$-extension $K$ over an imaginary quadratic field. We consider the Galois group $\tilde{X}(K_{\infty})$ of the maximal unramified pro-$p$-extension of $K_{\infty}$. In this paper, under certain assumptions, we give certain $K$ such that $\tilde{X}(K_{\infty})$ is abelian. Also, we give an example such that a special value of the characteristic polynomial of the Iwasawa module of $K_{\infty}$ determines whether $\tilde{X}(K_{\infty})$ is abelian or not.

Article information

Source
Osaka J. Math., Volume 49, Number 2 (2012), 271-295.

Dates
First available in Project Euclid: 20 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1340197926

Mathematical Reviews number (MathSciNet)
MR2887619

Zentralblatt MATH identifier
1333.11101

Subjects
Primary: 11R23: Iwasawa theory
Secondary: 11R37: Class field theory

Citation

Okano, Keiji. The commutativity of Galois groups of the maximal unramified pro-$p$-extensions over the cyclotomic $\mathbb{Z}_{p}$-extensions II. Osaka J. Math. 49 (2012), no. 2, 271--295. https://projecteuclid.org/euclid.ojm/1340197926


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