Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-07-25T14:05:25.773Z Has data issue: false hasContentIssue false
  • English
  • Français

Truncated Euler Systems over Imaginary Quadratic Fields

Published online by Cambridge University Press:  11 January 2016

Soogil Seo
Soogil Seo*
Affiliation:
Department of MathematicsYonsei University, 134 Sinchon-Dong Seodaemun-Gu Seoul 120-749, South Koreasgseo@yonsei.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

Keywords

11R2311R2711R29
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 195 , 2009 , pp. 97 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Belliard, J.-R., Sur la structure galoisienne des unités circulaires dans les Zp-extensions, J. Number Theory, 69 (1998), 16–49.CrossRefGoogle Scholar
[2] Coleman, R., On an Archimedean characterization of the circular units, J. reine angew. Math., 356 (1985), 161–173.Google Scholar
[3] de, , Shalit, E., Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions, Perspectives in Mathematics, vol. 3, Academic Press, 1987.Google Scholar
[4] Ferrero, B. and Washington, L., The Iwasawa invariant μp vanishes for abelian number fields, Ann. of Math., 109 (1979), 377–395.Google Scholar
[5] Gillard, R., Unités elliptiques et unités cyclotomiques, Math. Ann., 243 (1979), 181–189.Google Scholar
[6] Gross, B., On the factorization of p-adic L-series, Invent. Math., 57 (1980), 83–95.Google Scholar
[7] Johnson-Leung, J. and Kings, G., On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields, available at: http://front.math. ucdavis.edu/0804.2828.Google Scholar
[8] Kersey, D., Modular units inside cyclotomic units, Ann. of Math., 112 (1980), 361–380.Google Scholar
[9] Kolyvagin, V. A., Euler systems, The Grothendieck Festschrift, vol. 2, Birkh¨auser Verlag, 1990, pp. 435–483.Google Scholar
[10] Kubert, D. and Lang, S., Modular units inside cyclotomic units, Bull. Soc. Math. France, 107 (1979), 161–178.Google Scholar
[11] Kubert, D. and Lang, S., Modular units, Grundlehren der Mathematischen Wis-senschaften 244, Springer-Verlag, 1981.Google Scholar
[12] Mazur, B. and Wiles, A., Class fields of abelian extensions of Q, Invent. Math., 76 (1984), 179–330.Google Scholar
[13] Rubin, K., The main conjecture, Appendix to the second edition of S. Lang: Cyclotomic fields, Springer Verlag, 1990.Google Scholar
[14] Rubin, K., The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103 (1991), 25–68.Google Scholar
[15] Rubin, K., Euler Systems, Annals of Mathematics Studies, 147, Princeton University Press, 2000.Google Scholar
[16] Seo, S., Circular distributions and Euler systems, J. Number Theory, 88 (2001), 366–379.Google Scholar
[17] Seo, S., Circular distributions and Euler systems II, Compositio Math., 137 (2003), 91–98.Google Scholar
[18] Seo, S., On circular units over the cyclotomic Zp-Extension of an abelian field, Manuscripta Math., 115 (2004), 117–123.CrossRefGoogle Scholar
[19] Seo, S., Truncated Euler Systems, J. reine angew. Math., 614 (2008), 53–71.Google Scholar
[20] Sinnott, W., On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math., 108 (1978), 107–134.Google Scholar
[21] Sinnott, W., On the Stickelberger ideal and the circular units of a cyclotomic field, Invent. Math., 62 (1980), 181–234.Google Scholar
[22] Thaine, F., On the orders of ideal classes in prime cyclotomic fields, Math. Proc. Camb. Phil. Soc, 108 (1990), 197–201.Google Scholar
[23] Washington, A., The non-p-part of the class number in a cyclotomic Zp-extension, Invent. Math., 49 (1978), 87–97.CrossRefGoogle Scholar
[24] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math., 131 (1990), 493–540.Google Scholar