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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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[1] Brown, K. S., Cohomology of groups , Graduate Texts in Math. 87 , Springer-Verlag, 1982.
[2] Cornacchia, P. and Greither, C., Fitting ideals of class groups of real fields with prime power conductor, J. Number Theory, 73 (1998), 459-471. · Zbl 0926.11085
[3] 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
[4] 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
[5] Greither, C., Computing Fitting ideals of Iwasawa modules, Math. Zeitschrift, 246 (2004), 733-767. · Zbl 1067.11067
[6] Greither C., Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compositio Math., 143 (2007), 1399-1426. · Zbl 1135.11059
[7] 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
[8] Iwasawa K., On \({\mathbf Z}_{\ell}\)-extensions of algebraic number fields, Ann. of Math., 98 (1973), 246-326. · Zbl 0285.12008
[9] Iwasawa K., Riemann-Hurwitz formula and, p -adic Galois representations for number fields, Tôhoku Math. J. 33 (1981), 263-288. · Zbl 0468.12004
[10] Kurihara, M., Iwasawa theory and Fitting ideals, J. reine angew. Math., 561 (2003), 39-86. · Zbl 1056.11063
[11] Kurihara, M., On the structure of ideal class groups of CM-fields, Documenta Mathematica, Extra Volume Kato (2003), 539-563., · Zbl 1135.11339
[12] 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
[13] Mazur, B. and Wiles, A., Class fields of abelian extensions of, Q , Invent. math. 76 (1984), 179-330. · Zbl 0545.12005
[14] Northcott, D. G., Finite free resolutions , Cambridge Univ. Press, Cambridge New York, 1976. · Zbl 0328.13010
[15] Serre, J.-P., Corps Locaux , Hermann, Paris, 1968 (troisième édition).
[16] Serre, J.-P., Sur le résidu de la fonction zêta \(p\)-adique, Comptes Rendus Acad. Sc. Paris, 287 (1978), Série A, 183-188.
[17] Washington, L., Introduction to cyclotomic fields , Graduate Texts in Math. 83 , Springer-Verlag, 1982. · Zbl 0484.12001
[18] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math., 131 (1990), 493-540. · Zbl 0719.11071
[19] Wingberg, K., Duality theorems for \(\Gamma\)-extensions of algebraic number fields, Compos. Math., 55 (1985), 333-381. · Zbl 0608.12012
[20] P. B. Bhattacharya, The Hilbert function of two ideals, Proc. Cambridge. Philos. Soc., 53 (1957), 568-575. · Zbl 0080.02903
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