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On stronger versions of Brumer’s conjecture. (English) Zbl 1270.11117

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) \[ Ann_{\mathbb{Z} [G]}(\mu_L) \theta_S \subseteq \mathrm{Fitt}_{\mathbb Z [G]}(cl_L) \] might be true. It has been shown by C. Greither and the 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 C. Greither [Compos. Math. 143, No. 6, 1399–1426 (2007; Zbl 1135.11059)].
In the paper under review the author shows the existence of abelian CM-extensions for which neither (SB) nor (DSB) hold. Moreover, natural Iwasawa theoretic versions of (SB) and (DSB) are studied.

MSC:

11R29 Class numbers, class groups, discriminants
11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
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