×

Ideal class groups of CM-fields with non-cyclic Galois action. (English) Zbl 1276.11178

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 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 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.

MSC:

11R29 Class numbers, class groups, discriminants
11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Brown, K. S., Cohomology of groups , Graduate Texts in Math. 87 , Springer-Verlag (1982).
[2] Deligne, P. and Ribet, K., Values of abelian \(L\)-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227-286. · Zbl 0434.12009
[3] Greither, C., Arithmetic annihilators and Stark-type conjectures, Stark’s Conjectures: Recent Work and New Directions , Contemporary Math. 358 (2004), 55-78. · Zbl 1072.11083
[4] Greither, C., Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math. 143 (2007), 1399-1426. · Zbl 1135.11059
[5] Greither, C. and Kurihara, M., Stickelberger elements, Fitting ideals of class groups of CM fields, and dualisation, Math. Zeitschrift 260 (2008), 905-930. · Zbl 1159.11042
[6] Kurihara, M., On the ideal class groups of the maximal real subfields of number fields with all roots of unity, J. Europ. Math. Soc. 1 (1999), 35-49. · Zbl 0949.11055
[7] Kurihara, M., Iwasawa theory and Fitting ideals, J. reine angew. Math. 561 (2003), 39-86. · Zbl 1056.11063
[8] Kurihara, M., On stronger versions of Brumer’s conjecture, Tokyo Journal of Math. 34 (2011), 407-428. · Zbl 1270.11117
[9] Kurihara, M. and Miura, T., Stickelberger ideals and Fitting ideals of class groups for abelian number fields, Math. Annalen 350 (2011), 549-575. · Zbl 1235.11099
[10] Mazur, B. and Wiles, A., Class fields of abelian extensions of Q , Invent. math. 76 (1984), 179-330. · Zbl 0545.12005
[11] Nickel, A., On the equivariant Tamagawa number conjecture in tame CM-extensions, Math. Zeitschrift 268 (2011), 1-35. · Zbl 1222.11133
[12] Northcott, D. G., Finite free resolutions , Cambridge Univ. Press, Cambridge New York, 1976. · Zbl 0328.13010
[13] Popescu, C., Stark’s question and a refinement of Brumer’s conjecture extrapolated to the function field case, Compos. Math. 140 (2004), 631-646. · Zbl 1059.11069
[14] Serre, J.-P., Corps Locaux , Hermann, Paris 1968 (troisième édition).
[15] Tate, J., Les conjectures de Stark sur les Fonctions \(L\) d’Artin en \(s = 0\) , Progress in Math. 47 , Birkhäuser, 1984. · Zbl 0545.12009
[16] Washington, L., Introduction to cyclotomic fields , Graduate Texts in Math. 83 , Springer-Verlag, 1982. · Zbl 0484.12001
[17] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493-540. · Zbl 0719.11071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.