zbMATH — the first resource for mathematics

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.

11R29 Class numbers, class groups, discriminants
11R23 Iwasawa theory
11R42 Zeta functions and \(L\)-functions of number fields
Full Text: DOI
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.