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

Mizusawa, Yasushi

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

Pour $k$ quadratique imaginaire, nous étudions le groupe de Galois $G\left(k_{\infty }\right)$ de la pro-$2$-extension non ramifiée maximale au-dessus de la ${\mathbb{Z}}_2$-extension cyclotomique $k_{\infty }$ de $k$. Nous déterminons des familles de tels corps imaginaires $k$ pour lesquels $G\left(k_{\infty }\right)$ est un pro-$2$-groupe métabélien et en donnons une présentation explicite ; nous précisons de même des familles pour lesquelles $G\left(k_{\infty }\right)$ 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 ${\mathbb{Z}}_2$-extension $k_{\infty }$ of an imaginary quadratic field $k$, we consider the Galois group $G\left(k_{\infty }\right)$ of the maximal unramified pro-$2$-extension over $k_{\infty }$. In this paper, we give some families of $k$ for which $G\left(k_{\infty }\right)$ is a metabelian pro-$2$-group with the explicit presentation, and determine the case that $G\left(k_{\infty }\right)$ 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.

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