×

zbMATH — the first resource for mathematics

Truncated Euler systems over imaginary quadratic fields. (English) Zbl 1226.11115
Let \(F\) be an abelian extension of an imaginary quadratic field. The author introduces a filtration on the group of units modulo elliptic units of \(F\) and conjectures that the associated graded Galois module is isomorphic to the ideal class group of \(F\). This is shown to be compatible with various facts from Iwasawa theory. The present work is the analogue of earlier work of the author [J. Reine Angew. Math. 614, 53–71 (2008; Zbl 1204.11170)], where the case of real abelian extensions of the rationals and units modulo cyclotomic units was considered.
MSC:
11R23 Iwasawa theory
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11G16 Elliptic and modular units
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] J.-R. Belliard, Sur la structure galoisienne des unités circulaires dans les \(\mathbbZ_p\)-extensions , J. Number Theory, 69 (1998), 16–49. · Zbl 0911.11051 · doi:10.1006/jnth.1997.2200
[2] R. Coleman, On an Archimedean characterization of the circular units , J. reine angew. Math., 356 (1985), 161–173. · Zbl 0548.12003 · doi:10.1515/crll.1985.356.161 · crelle:GDZPPN002202204 · eudml:152702
[3] E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication. \(p\)-adic \(L\) functions, Perspectives in Mathematics, vol. 3, Academic Press, 1987. · Zbl 0674.12004
[4] B. Ferrero and L. Washington, The Iwasawa invariant \(\mu_p\) vanishes for abelian number fields , Ann. of Math., 109 (1979), 377–395. JSTOR: · Zbl 0443.12001 · doi:10.2307/1971116 · links.jstor.org
[5] R. Gillard, Unités elliptiques et unités cyclotomiques , Math. Ann., 243 (1979), 181–189. · Zbl 0396.12005 · doi:10.1007/BF01420425 · eudml:182813
[6] B. Gross, On the factorization of \(p\)-adic \(L\)-series , Invent. Math., 57 (1980), 83–95. · Zbl 0472.12011 · doi:10.1007/BF01389819 · eudml:142702
[7] J. Johnson-Leung and G. Kings, On the equivariant and the non-equivariant main conjecture for imaginary quadratic fields , available at: http://front.math. ucdavis.edu/0804.2828. · Zbl 1230.11135
[8] D. Kersey, Modular units inside cyclotomic units , Ann. of Math., 112 (1980), 361–380. JSTOR: · Zbl 0451.12005 · doi:10.2307/1971150 · links.jstor.org
[9] V. A. Kolyvagin, Euler systems , The Grothendieck Festschrift, vol. 2, Birkhäuser Verlag, 1990, pp. 435–483. · Zbl 0742.14017
[10] D. Kubert and S. Lang, Modular units inside cyclotomic units , Bull. Soc. Math. France, 107 (1979), 161–178. · Zbl 0409.12007 · numdam:BSMF_1979__107__161_0 · eudml:87342
[11] D. Kubert and S. Lang, Modular units, Grundlehren der Mathematischen Wissenschaften 244, Springer-Verlag, 1981. · Zbl 0492.12002
[12] B. Mazur and A. Wiles, Class fields of abelian extensions of \(\mathbbQ\) , Invent. Math., 76 (1984), 179–330. · Zbl 0545.12005 · doi:10.1007/BF01388599 · eudml:143124
[13] K. Rubin, The main conjecture, Appendix to the second edition of S. Lang: Cyclotomic fields, Springer Verlag, 1990. · Zbl 0704.11038
[14] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields , Invent. Math., 103 (1991), 25–68. · Zbl 0737.11030 · doi:10.1007/BF01239508 · eudml:143852
[15] K. Rubin, Euler Systems, Annals of Mathematics Studies, 147, Princeton University Press, 2000.
[16] S. Seo, Circular distributions and Euler systems , J. Number Theory, 88 (2001), 366–379. · Zbl 0995.11060 · doi:10.1006/jnth.2000.2634
[17] S. Seo, Circular distributions and Euler systems II , Compositio Math., 137 (2003), 91–98. · Zbl 1023.11056 · doi:10.1023/A:1023644822410
[18] S. Seo, On circular units over the cyclotomic \(\mathbbZ_p\)-Extension of an abelian field , Manuscripta Math., 115 (2004), 117–123. · Zbl 1081.11072 · doi:10.1007/s00229-004-0490-9
[19] S. Seo, Truncated Euler Systems , J. reine angew. Math., 614 (2008), 53–71.
[20] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field , Ann. of Math., 108 (1978), 107–134. JSTOR: · Zbl 0395.12014 · doi:10.2307/1970932 · links.jstor.org
[21] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field , Invent. Math., 62 (1980), 181–234. · Zbl 0465.12001 · doi:10.1007/BF01389158 · eudml:142770
[22] F. Thaine, On the orders of ideal classes in prime cyclotomic fields , Math. Proc. Camb. Phil. Soc., 108 (1990), 197–201. · Zbl 0717.11046 · doi:10.1017/S0305004100069073
[23] A. Washington, The non-\(p\)-part of the class number in a cyclotomic \(\mathbbZ_p\)-extension , Invent. Math., 49 (1978), 87–97. · Zbl 0403.12007 · doi:10.1007/BF01399512 · eudml:142595
[24] A. Wiles, The Iwasawa conjecture for totally real fields , Ann. of Math., 131 (1990), 493–540. JSTOR: · Zbl 0719.11071 · doi:10.2307/1971468 · links.jstor.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.