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Stark units and the main conjectures for totally real fields

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  23 October 2009

Kâzım Büyükboduk
Kâzım Büyükboduk*
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305, USA (email: [email protected])
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Abstract

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The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

Keywords

Iwasawa theoryKolyvagin systemsStark conjectures

MSC classification

Secondary: 11R23: Iwasawa theory 11R27: Units and factorization 11R29: Class numbers, class groups, discriminants 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1163 - 1195
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Benois, D., On Iwasawa theory of crystalline representations, Duke Math. J. 104 (2000), 211–267.CrossRefGoogle Scholar
[2]Büyükboduk, K., Λ-adic Kolyvagin systems, Preprint (2007), http://arxiv.org/abs/0706.0377v1.Google Scholar
[3]Büyükboduk, K., Stickelberger elements and Kolyvagin systems, Preprint (2008), http://arxiv.org/abs/0808.2588.Google Scholar
[4]Büyükboduk, K., Kolyvagin systems of Stark units, J. Reine Angew. Math. 631 (2009), 85–107.Google Scholar
[5]Cherbonnier, F. and Colmez, P., Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), 241–268.CrossRefGoogle Scholar
[6]Coleman, R. F., Division values in local fields, Invent. Math. 53 (1979), 91–116.Google Scholar
[7]Colmez, P., Théorie d’Iwasawa des représentations de de Rham d’un corps local, Ann. of Math. (2) 148 (1998), 485–571.CrossRefGoogle Scholar
[8]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
[9]de Shalit, E., Iwasawa theory of elliptic curves with complex multiplication: p-adic L functions, Perspectives in Mathematics, vol. 3 (Academic Press, Boston, MA, 1987).Google Scholar
[10]Fontaine, J.-M., Représentations p-adiques des corps locaux. I, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, MA, 1990), 249–309.Google Scholar
[11]Greenberg, R., Trivial zeros of p-adic L-functions, in p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemporary Mathematics, vol. 165 (American Mathematical Society, Providence, RI, 1994), 149–174.CrossRefGoogle Scholar
[12]Herr, L., Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563–600.Google Scholar
[13]Herr, L., Une approche nouvelle de la dualité locale de Tate, Math. Ann. 320 (2001), 307–337.Google Scholar
[14]Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964), 42–82.CrossRefGoogle Scholar
[15]Krasner, M., Sur la représentation exponentielle dans les corps relativement galoisiens de nombers 𝔭-adiques, Acta Arith. 3 (1939), 133–173.Google Scholar
[16]Milne, J. S., Arithmetic duality theorems, Perspectives in Mathematics, vol. 1 (Academic Press, Boston, MA, 1986).Google Scholar
[17]Mazur, B. and Rubin, K., Kolyvagin systems, Mem. Amer. Math. Soc. 168 (2004), no. 799.Google Scholar
[18]Mazur, B., Tate, J. and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.Google Scholar
[19]Perrin-Riou, B., La fonction L p-adique de Kubota-Leopoldt, in Arithmetic geometry (Tempe, AZ, 1993), Contemporary Mathematics, vol. 174 (American Mathematical Society, Providence, RI, 1994), 65–93.Google Scholar
[20]Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local (With an appendix by Jean-Marc Fontaine), Invent. Math. 115 (1994), 81–161.CrossRefGoogle Scholar
[21]Perrin-Riou, B., Fonctions L p-adiques des représentations p-adiques, Astérisque, vol. 229 (Société Mathématique de France, Paris, 1995).Google Scholar
[22]Perrin-Riou, B., Systèmes d’Euler p-adiques et théorie d’Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998), 1231–1307.Google Scholar
[23]Rubin, K., Stark units and Kolyvagin’s ‘Euler systems’, J. reine angew. Math. 425 (1992), 141–154.Google Scholar
[24]Rubin, K., A Stark conjecture ‘over Z’ for abelian L-functions with multiple zeros, Ann. Inst. Fourier (Grenoble) 46 (1996), 33–62.Google Scholar
[25]Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
[26]Solomon, D., On p-adic abelian Stark conjectures at s=1, Ann. Inst. Fourier (Grenoble) 52 (2002), 379–417.Google Scholar
[27]Solomon, D., Abelian conjectures of Stark type in ℤp-extensions of totally real fields, in Stark’s conjectures: recent work and new directions, Contemporary Mathematics, vol. 358 (American Mathematical Society, Providence, RI, 2004), 143–178.CrossRefGoogle Scholar
[28]Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493–540.Google Scholar