Tate sequences and Fitting ideals of Iwasawa modules
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- by C. Greither and M. Kurihara
- St. Petersburg Math. J. 27 (2016), 941-965
- DOI: https://doi.org/10.1090/spmj/1428
- 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.References
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Bibliographic Information
- C. Greither
- Affiliation: Institut für Theoretische Informatik und Mathematik, Universität der Bundeswehr, München, 85577 Neubiberg, Germany
- Email: cornelius.greither@unibw.de
- M. Kurihara
- Affiliation: Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
- Email: kurihara@math.keio.ac.jp
- Received by editor(s): June 15, 2015
- Published electronically: September 30, 2016
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 941-965
- MSC (2010): Primary 11R23, 11R29, 11R18
- DOI: https://doi.org/10.1090/spmj/1428
- MathSciNet review: 3589224
Dedicated: To our colleague and friend Sergeĭ V. Vostokov on the occasion of his seventieth birthday