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.