Tokyo Journal of Mathematics

Ideal Class Groups of CM-fields with Non-cyclic Galois Action

Abstract

Suppose that $L/k$ is a finite and abelian extension such that $k$ is a totally real base field and $L$ is a CM-field. We regard the ideal class group $\mathrm{Cl}_{L}$ of $L$ as a $\mathrm{Gal}(L/k)$-module. As a sequel of the paper [8] by the first author, we study a problem whether the Stickelberger element for $L/k$ times the annihilator ideal of the roots of unity in $L$ is in the Fitting ideal of $\mathrm{Cl}_{L}$, and also a problem whether it is in the Fitting ideal of the Pontrjagin dual $(\mathrm{Cl}_{L})^{\vee}$. We systematically construct extensions $L/k$ for which these properties do not hold, and also give numerical examples.

Article information

Source
Tokyo J. Math., Volume 35, Number 2 (2012), 411-439.

Dates
First available in Project Euclid: 23 January 2013

https://projecteuclid.org/euclid.tjm/1358951328

Digital Object Identifier
doi:10.3836/tjm/1358951328

Mathematical Reviews number (MathSciNet)
MR3058716

Zentralblatt MATH identifier
1276.11178

Citation

KURIHARA, Masato; MIURA, Takashi. Ideal Class Groups of CM-fields with Non-cyclic Galois Action. Tokyo J. Math. 35 (2012), no. 2, 411--439. doi:10.3836/tjm/1358951328. https://projecteuclid.org/euclid.tjm/1358951328

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