No CrossRef data available.
Article contents
On metabelian 2-class field towers over
$\mathbb Z_2$-extensions of real quadratic fields
Published online by Cambridge University Press: 06 October 2021
Abstract
We give a family of real quadratic fields such that the 2-class field towers over their cyclotomic
$\mathbb Z_2$
-extensions have metabelian Galois groups of abelian invariants
$[2,2,2]$
. We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.
MSC classification
Primary:
11R23: Iwasawa theory
- Type
- Article
- Information
- Copyright
- © Canadian Mathematical Society 2021
References
Azizi, A. and Mouhib, A.,
Sur le rang du 2-groupe de classes de
$Q(\sqrt{m},\sqrt{d})$
où
$\ m= 2\ {}$
ou un premier
$p \equiv 1\ (mod\ 4)$
. Trans. Amer. Math. Soc. 353(2001), no. 7, 2741–2752.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline650.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline651.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline652.png?pub-status=live)
Azizi, A., Rezzougui, M., and Zekhnini, A.,
On the maximal unramified pro-
$2$
-extension of certain cyclotomic
$~{\mathbb{Z}}_2$
-extensions
. Period. Math. Hungar. 83(2021), no. 1, 54–66.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline653.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline654.png?pub-status=live)
Benjamin, E., Lemmermeyer, F., and Snyder, C.,
Real quadratic fields with abelian
$2$
-class field tower
. J. Number Theory 73(1998), no. 2, 182–194.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline655.png?pub-status=live)
Benjamin, E. and Snyder, C.,
Real quadratic number fields with
$2$
-class group of type
$\left(2,2\right)$
. Math. Scand. 76(1995), no. 2, 161–178.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline656.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline657.png?pub-status=live)
Fukuda, T.,
Remarks on
$\textbf{Z}_p$
-extensions of number fields
. Proc. Japan Acad. Ser. A 70(1994), 264–266.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline658.png?pub-status=live)
Fukuda, T. and Komatsu, K.,
On the Iwasawa
$\lambda$
-invariant of the cyclotomic
$\textbf{Z}_2$
-extension of a real quadratic field
. Tokyo J. Math. 28(2005), no. 1, 259–264.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline659.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline660.png?pub-status=live)
Gamble, G., Nickel, W., O’Brien, E. and Horn, M., ANUPQ, ANU p-Quotient – a GAP package, Version 3.2.1. 2019. https://gap-packages.github.io/anupq/
Google Scholar
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.1. 2021. https://www.gap-system.org
Google Scholar
Greenberg, R.,
On the Iwasawa invariants of totally real number fields
. Amer. J. Math. 98(1976), no. 1, 263–284.CrossRefGoogle Scholar
Iwasawa, K.,
A note on class numbers of algebraic number fields
. Abh. Math. Sem. Univ. Hamburg 20(1956), 257–258.CrossRefGoogle Scholar
Kisilevsky, H.,
Number fields with class number congruent to 4 mod 8 and Hilbert’s theorem 94
. J. Number Theory 8(1976), no. 3, 271–279.CrossRefGoogle Scholar
Kumakawa, N.,
On the Iwasawa
$\lambda$
-invariant of the cyclotomic
$~{\mathbb{Z}}_2$
-extension of
$\,\mathbb{Q}(\sqrt{pq})$
and the
$2$
-part of the class number of
$\,\mathbb{Q}(\sqrt{pq},\sqrt{2+\sqrt{2}})$
. Int. J. Number Theory 17(2021), no. 4, 931–958.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline661.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline662.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline663.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline664.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline665.png?pub-status=live)
Lemmermeyer, F.,
The ambiguous class number formula revisited
. J. Ramanujan Math. Soc. 28(2013), no. 4, 415–421.Google Scholar
Mizusawa, Y.,
On the maximal unramified pro-
$2$
-extension of
$~{\mathbb{Z}}_2$
-extensions of certain real quadratic fields II
. Acta Arith. 119(2005), no. 1, 93–107.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline666.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline667.png?pub-status=live)
Mizusawa, Y.,
On unramified Galois
$2$
-groups over
${\mathbb{Z}}_2$
-extensions of real quadratic fields
. Proc. Amer. Math. Soc. 138(2010), no. 9, 3095–3103.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline668.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline669.png?pub-status=live)
Mizusawa, Y.,
A note on semidihedral
$2$
-class field towers and
${\mathbb{Z}}_2$
-extensions
. Ann. Math. Qué. 38(2014), no. 1, 73–79.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline670.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline671.png?pub-status=live)
Mizusawa, Y.,
Tame pro-
$2$
Galois groups and the basic
${\mathbb{Z}}_2$
-extension
. Trans. Amer. Math. Soc. 370(2018), no. 4, 2423–2461.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline672.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline673.png?pub-status=live)
Mouhib, A.,
Sur la
$2$
-extension maximale non ramifié de la
$\textbf{Z}_2$
-extension cyclotomique de certains corps quadratiques
. An. Şt. Univ. Ovidius Constanţa 22(2014), no. 1, 207–214.Google Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline674.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline675.png?pub-status=live)
Mouhib, A. and Movahhedi, A.,
On the
$p$
-class tower of a
$\textbf{Z}_p$
-extension
. Tokyo J. Math. 31(2008), no. 2, 321–332.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline676.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline677.png?pub-status=live)
Nishino, Y.,
On the Iwasawa invariants of the cyclotomic
$\textbf{Z}_2$
-extensions of certain real quadratic fields
. Tokyo J. Math. 29(2006), no. 1, 239–245.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline678.png?pub-status=live)
Ozaki, M.,
Non-abelian Iwasawa theory of
${\mathbb{Z}}_p$
-extensions
. J. Reine Angew. Math. 602(2007), 59–94.Google Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline679.png?pub-status=live)
Ozaki, M. and Taya, H.,
On the Iwasawa
${\lambda}_2$
-invariants of certain families of real quadratic fields
. Manuscripta Math. 94(1997), no. 4, 437–444.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline680.png?pub-status=live)
The PARI Group, PARI/GP version 2.13.1, Univ. Bordeaux. 2021. http://pari.math.u-bordeaux.fr/
Google Scholar
Salle, L.,
Sur les pro-
$p$
-extensions à ramification restreinte au-dessus de la
${\mathbb{Z}}_p$
-extension cyclotomique d’un corps de nombres
. J. Théor. Nombres Bordeaux 20(2008), no. 2, 485–523.CrossRefGoogle Scholar
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline681.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20220823021702597-0070:S0008439521000862:S0008439521000862_inline682.png?pub-status=live)