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Stickelberger elements and Kolyvagin systems

Published online by Cambridge University Press:  11 January 2016

Kâzim Büyükboduk
Kâzim Büyükboduk*
Affiliation:
Institut des Hautes Études Scientifiques, Route de Chartres, F-91440 Bures-sur-Yvette, France
*
Koç University, Mathematics, 34450 Sariyer, Istanbul, Turkeykbuyukboduk@ku.edu.tr
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Abstract

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In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is r > 1 (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

Keywords

11R2311R4211R27
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 203 , September 2011 , pp. 123 - 173
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[B1] Büyükboduk, K., Kolyvagin systems of Stark units, J. Reine Angew. Math. 631 (2009), 85–107.Google Scholar
[B2] Büyükboduk, K., Stark units and the main conjectures for totally real fields, Compos. Math. 145 (2009), 1163–1195.CrossRefGoogle Scholar
[B3] Büyükboduk, K., Λ-adic Kolyvagin systems, Int. Math. Res. Notices (2010), doi: 10:1093/imrn/rnq186.Google Scholar
[B4] Büyükboduk, K., On Euler systems of rank r and their Kolyvagin systems, Indiana Univ.Math. J. 59 (2010), 1245–1276.Google Scholar
[C] Colmez, P., Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571.Google Scholar
[DR] Deligne, P. and Ribet, K. A., Values of abelian L-functions at negative integers over totally real fields, Invent. Math. 59 (1980), 227–286.CrossRefGoogle Scholar
[dS] de, , Shalit, E., Iwasawa Theory of Elliptic Curves with Complex Multiplication, Perspect. Math. 3, Academic Press, Boston, 1987.Google Scholar
[G] Greenberg, R., “Trivial zeros of p-adic L-functions” in p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemp. Math. 165, Amer. Math. Soc., Providence, 1994, 149–174.Google Scholar
[Gr] Greither, C., Computing fitting ideals of Iwasawa modules, Math. Z. 246 (2004),733–767.Google Scholar
[Ka] Kato, K., Euler systems, Iwasawa theory, and Selmer groups, Kodai Math. J. 22 (1999), 313–372.Google Scholar
[Ku] Kurihara, M., On the structure of ideal class groups of CM-fields, Doc. Math. DMV Extra Vol (2003), 539–563.Google Scholar
[MR1] Mazur, B. and Rubin, K., Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), viii+96.Google Scholar
[MR2] Kurihara, M., Refined class number formulas and Kolyvagin systems, preprint, arXiv:0909.3916v1[math.NT].Google Scholar
[Mi] Milne, J. S., Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.Google Scholar
[P1] Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local, with an appendix by J.-M. Fontaine, Invent. Math. 115 (1994), 81–161.Google Scholar
[P2] Perrin-Riou, B., Systèmes d’Euler p-adiques et théorie d’Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), 1231–1307.CrossRefGoogle Scholar
[R1] Rubin, K., Stark units and Kolyvagin’s “Euler systems,” J. Reine Angew. Math. 425 (1992), 141–154.Google Scholar
[R2] Rubin, K., A Stark conjecture “over Z” for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), 33–62.Google Scholar
[R3] Rubin, K., Euler Systems, Ann. of Math. Stud. 147, Princeton University Press, Princeton, 2000.Google Scholar
[S] Siegel, C. L., über die Fourierschen Koeffizienten von Modulformen, Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl. II 1970, 15–56.Google Scholar
[T] Tate, J., Les conjectures de Stark sur les fonctions L d’Artin en s = 0, Progr. Math. 47, Birkh¨auser, Boston, 1984.Google Scholar
[W1] Wiles, A., On a conjecture of Brumer, Ann. of Math. (2) 131 (1990), 555–565.Google Scholar
[W2] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493–540.Google Scholar