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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Tate sequences and Fitting ideals of Iwasawa modules

Authors: C. Greither and M. Kurihara
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 941-965
MSC (2010): Primary 11R23, 11R29, 11R18
Published electronically: September 30, 2016
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Abstract: We consider Abelian CM extensions $ L/k$ of a totally real field $ k$, and we essentially determine the Fitting ideal of the dualized Iwasawa module studied by the second author in the case where only places above $ p$ ramify. In doing so we recover and generalize the results mentioned above. Remarkably, our explicit description of the Fitting ideal, apart from the contribution of the usual Stickelberger element $ \dot \Theta $ at infinity, only depends on the group structure of the Galois group $ \mathrm {Gal}(L/k)$ and not on the specific extension $ L$. From our computation it is then easy to deduce that $ \dot T \dot \Theta $ is not in the Fitting ideal as soon as the $ p$-part of $ \mathrm {Gal}(L/k)$ is not cyclic. We need a lot of technical preparations: resolutions of the trivial module $ \mathbb{Z}$ over a group ring, discussion of the minors of certain big matrices that arise in this context, and auxiliary results about the behavior of Fitting ideals in short exact sequences.

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  • [CG] P. Cornacchia and C. Greither, Fitting ideals of class groups of real fields with prime power conductor, J. Number Theory 73 (1998), no. 2, 459-471. MR 1658000
  • [Gr1] C. Greither, Computing Fitting ideals of Iwasawa modules, Math. Z. 246 (2004), no. 4, 733-767. MR 2045837
  • [Gr2] -, Determining Fitting ideals of minus class groups via the equivariant Tamagawa number conjecture, Compos. Math. 143 (2007), no. 6, 1399-1426. MR 2371374
  • [GK] C. Greither and M. Kurihara, Stickelberger elements, Fitting ideals of class groups of CM fields, and dualisation, Math. Z. 260 (2008), no. 4, 905-930. MR 2443336
  • [Ku1] M. Kurihara, Iwasawa theory and Fitting ideals, J. Reine Angew. Math. 561 (2003), 39-86. MR 1998607
  • [Ku2] -, On the structure of ideal class groups of CM fields, Doc. Math. 2003, Extra vol., 539-563. MR 2046607
  • [Ku3] -, On stronger versions of Brumer's conjecture, Tokyo J. Math. 34 (2011), no. 2, 407-428. MR 2918914
  • [KM1] M. Kurihara and T. Miura, Stickelberger ideals and Fitting ideals of class groups for abelian number fields, Math. Ann. 350 (2011), no. 3, 549-575. MR 2805636
  • [KM2] -, Ideal class groups of CM-fields with non-cyclic Galois action, Tokyo J. Math. 35 (2012), no. 2, 411-439. MR 3058716
  • [MW] B. Mazur and A. Wiles, Class fields of abelian extensions of $ \mathbb{Q}$, Invent. Math. 76 (1984), no. 2, 179-330. MR 742853
  • [Wi] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493-540. MR 1053488

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Additional Information

C. Greither
Affiliation: Institut für Theoretische Informatik und Mathematik, Universität der Bundeswehr, München, 85577 Neubiberg, Germany

M. Kurihara
Affiliation: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

Keywords: Tate sequences, class groups, cohomology, totally real fields, CM-fields
Received by editor(s): June 15, 2015
Published electronically: September 30, 2016
Dedicated: To our colleague and friend Sergeĭ V. Vostokov on the occasion of his seventieth birthday
Article copyright: © Copyright 2016 American Mathematical Society

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