Ideal class groups of CM-fields with non-cyclic Galois action
Let \(L/k\) be a finite abelian CM-extension of number fields with Galois group \(G\). Let \(\mu_L\) denote the roots of unity in \(L\) and \(cl_L\) the class group of \(L\). Then Brumer's conjecture asserts that \[ \mathrm{Ann}_{\mathbb{Z}[G]} (\mu_L) \theta_S \subseteq \mathrm{Ann}_{ \mathbb Z[G]} (cl_L), \] where \(S\) is a finite set of places of \(L\) containing all archimedean places and all that ramify in \(L/k\); here, \(\theta_S\) denotes the Stickelberger element which is defined via values of Artin L-series at zero. It is natural to ask if the stronger statement (SB) \[ \mathrm{Ann}_{\mathbb Z[G]} (\mu_L )\theta_S \subseteq \mathrm{Fitt}_{\mathbb Z[G]} (cl_L) \] might be true (here, \(\mathrm{Fitt}_{\mathbb Z[G]} (cl_L)\) denotes the (zeroth) Fitting ideal of the class group). It is shown by \textit{C. Greither} and the first author [Math. Z. 260, No. 4, 905--930 (2008; Zbl 1159.11042)] that (SB) does not hold in general. However, the dual version (DSB) of (SB), where \(cl_L\) is replaced with its Pontryagin dual, seems to be more likely to hold. For instance, its \(p\)-part (for odd \(p\)) is implied by the (appropriate special case of the) equivariant Tamagawa number conjecture if the \(p\)-part of the roots of unity in \(L\) is cohomologically trivial by a result of \textit{C. Greither} [Compos. Math. 143, No. 6, 1399--1426 (2007; Zbl 1135.11059)]. The first author [Tokyo J. Math. 34, No. 2, 407--428 (2011; Zbl 1270.11117)] has shown the existence of abelian CM-extensions for which neither (SB) nor (DSB) hold. However, explicit numerical examples for which (DSB) does not hold were not given. In the paper under review the authors give explicit conditions under which (DSB) does not hold and come up with explicit numerical examples for which neither (SB) nor (DSB) hold. In contrast to the aforementioned article of the first author the approach is not of Iwasawa-theoretic nature; the finite abelian extensions are studied directly. The authors deal with the following set-up which explains the title of the paper: \(L/k\) is a finite abelian CM-extension, \(p\) is an odd prime such that \(K := k(\zeta_p)\) is a subfield of \(L\) and the extension \(L/K\) is supposed to be not cyclic.
scientific article; zbMATH DE number 6146579
scientific article