Tokyo Journal of Mathematics

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

Masato KURIHARA and Takashi MIURA

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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.

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

First available in Project Euclid: 23 January 2013

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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.

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