On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field

Mizusawa, Yasushi

doi:10.5802/jtnb.707

On the maximal unramified pro-2-extension over the cyclotomic

ℤ_{2}

-extension of an imaginary quadratic field

Mizusawa, Yasushi ¹

¹ Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya, Aichi 466-8555, JAPAN

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 115-138.

Résumé
Abstract

Pour $k$ quadratique imaginaire, nous étudions le groupe de Galois $G (k_{\infty})$ de la pro- $2$ -extension non ramifiée maximale au-dessus de la $ℤ_{2}$ -extension cyclotomique $k_{\infty}$ de $k$ . Nous déterminons des familles de tels corps imaginaires $k$ pour lesquels $G (k_{\infty})$ est un pro- $2$ -groupe métabélien et en donnons une présentation explicite ; nous précisons de même des familles pour lesquelles $G (k_{\infty})$ est un pro- $2$ -groupe métacyclique non abélien. Nous calculons enfin en termes de Théorie d’Iwasawa les groupes de Galois de $2$ -tours de corps de classes de certaines $2$ -extensions cyclotomiques.

For the cyclotomic $ℤ_{2}$ -extension $k_{\infty}$ of an imaginary quadratic field $k$ , we consider the Galois group $G (k_{\infty})$ of the maximal unramified pro- $2$ -extension over $k_{\infty}$ . In this paper, we give some families of $k$ for which $G (k_{\infty})$ is a metabelian pro- $2$ -group with the explicit presentation, and determine the case that $G (k_{\infty})$ becomes a nonabelian metacyclic pro- $2$ -group. We also calculate Iwasawa theoretically the Galois groups of $2$ -class field towers of certain cyclotomic $2$ -extensions.

MR Zbl

DOI : 10.5802/jtnb.707

Affiliations des auteurs :

Mizusawa, Yasushi ¹

¹ Department of Mathematics Nagoya Institute of Technology Gokiso, Showa, Nagoya, Aichi 466-8555, JAPAN

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     title = {On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {115--138},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.707},
     zbl = {1221.11215},
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     url = {http://www.numdam.org/articles/10.5802/jtnb.707/}
}

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Mizusawa, Yasushi. On the maximal unramified pro-2-extension over the cyclotomic $\mathbb{Z}_2$-extension of an imaginary quadratic field. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 115-138. doi : 10.5802/jtnb.707. http://www.numdam.org/articles/10.5802/jtnb.707/

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