Truncated Euler systems over imaginary quadratic fields
Soogil Seo
Source: Nagoya Math. J. Volume 195
(2009), 97-111.
Abstract
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 $\textup{Cl}_{F}$ of $F$ is equal to the order of the quotient $U_{F}/El_{F}$ of the group of global units $U_{F}$ by the group of elliptic units $El_{F}$ of $F$. We introduce a filtration on $U_{F}/El_{F}$ 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.
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Permanent link to this document: http://projecteuclid.org/euclid.nmj/1252934373
Zentralblatt MATH identifier: 05611431
Mathematical Reviews number (MathSciNet): MR2552955
References
J.-R. Belliard, Sur la structure galoisienne des unités circulaires dans les $\mathbbZ_p$-extensions, J. Number Theory, 69 (1998), 16--49.
Mathematical Reviews (MathSciNet): MR1611081
Digital Object Identifier: doi:10.1006/jnth.1997.2200
R. Coleman, On an Archimedean characterization of the circular units, J. reine angew. Math., 356 (1985), 161--173.
Mathematical Reviews (MathSciNet): MR779380
E. de Shalit, Iwasawa theory of elliptic curves with complex multiplication. $p$-adic $L$ functions, Perspectives in Mathematics, vol. 3, Academic Press, 1987.
Mathematical Reviews (MathSciNet): MR917944
Zentralblatt MATH: 0674.12004
B. Ferrero and L. Washington, The Iwasawa invariant $\mu_p$ vanishes for abelian number fields, Ann. of Math., 109 (1979), 377--395.
Mathematical Reviews (MathSciNet): MR528968
Digital Object Identifier: doi:10.2307/1971116
JSTOR: links.jstor.org
R. Gillard, Unités elliptiques et unités cyclotomiques, Math. Ann., 243 (1979), 181--189.
Mathematical Reviews (MathSciNet): MR543728
Digital Object Identifier: doi:10.1007/BF01420425
B. Gross, On the factorization of $p$-adic $L$-series, Invent. Math., 57 (1980), 83--95.
Mathematical Reviews (MathSciNet): MR564185
Zentralblatt MATH: 0472.12011
Digital Object Identifier: doi:10.1007/BF01389819
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.
D. Kersey, Modular units inside cyclotomic units, Ann. of Math., 112 (1980), 361--380.
Mathematical Reviews (MathSciNet): MR592295
Digital Object Identifier: doi:10.2307/1971150
JSTOR: links.jstor.org
V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, vol. 2, Birkhäuser Verlag, 1990, pp. 435--483.
Mathematical Reviews (MathSciNet): MR1106906
D. Kubert and S. Lang, Modular units inside cyclotomic units, Bull. Soc. Math. France, 107 (1979), 161--178.
Mathematical Reviews (MathSciNet): MR545170
Zentralblatt MATH: 0409.12007
D. Kubert and S. Lang, Modular units, Grundlehren der Mathematischen Wissenschaften 244, Springer-Verlag, 1981.
Mathematical Reviews (MathSciNet): MR648603
B. Mazur and A. Wiles, Class fields of abelian extensions of $\mathbbQ$, Invent. Math., 76 (1984), 179--330.
Mathematical Reviews (MathSciNet): MR742853
Zentralblatt MATH: 0545.12005
Digital Object Identifier: doi:10.1007/BF01388599
K. Rubin, The main conjecture, Appendix to the second edition of S. Lang: Cyclotomic fields, Springer Verlag, 1990.
Mathematical Reviews (MathSciNet): MR1029028
Zentralblatt MATH: 0704.11038
K. Rubin, The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103 (1991), 25--68.
Mathematical Reviews (MathSciNet): MR1079839
Zentralblatt MATH: 0737.11030
Digital Object Identifier: doi:10.1007/BF01239508
K. Rubin, Euler Systems, Annals of Mathematics Studies, 147, Princeton University Press, 2000.
Mathematical Reviews (MathSciNet): MR1749177
S. Seo, Circular distributions and Euler systems, J. Number Theory, 88 (2001), 366--379.
Mathematical Reviews (MathSciNet): MR1832013
Zentralblatt MATH: 0995.11060
Digital Object Identifier: doi:10.1006/jnth.2000.2634
S. Seo, Circular distributions and Euler systems II, Compositio Math., 137 (2003), 91--98.
Mathematical Reviews (MathSciNet): MR1981938
Zentralblatt MATH: 1023.11056
Digital Object Identifier: doi:10.1023/A:1023644822410
S. Seo, On circular units over the cyclotomic $\mathbbZ_p$-Extension of an abelian field, Manuscripta Math., 115 (2004), 117--123.
Mathematical Reviews (MathSciNet): MR2093741
Zentralblatt MATH: 1081.11072
Digital Object Identifier: doi:10.1007/s00229-004-0490-9
S. Seo, Truncated Euler Systems, J. reine angew. Math., 614 (2008), 53--71.
Mathematical Reviews (MathSciNet): MR2376282
W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math., 108 (1978), 107--134.
Mathematical Reviews (MathSciNet): MR485778
Digital Object Identifier: doi:10.2307/1970932
JSTOR: links.jstor.org
W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Invent. Math., 62 (1980), 181--234.
Mathematical Reviews (MathSciNet): MR595586
Zentralblatt MATH: 0465.12001
Digital Object Identifier: doi:10.1007/BF01389158
F. Thaine, On the orders of ideal classes in prime cyclotomic fields, Math. Proc. Camb. Phil. Soc., 108 (1990), 197--201.
Mathematical Reviews (MathSciNet): MR1074708
Zentralblatt MATH: 0717.11046
Digital Object Identifier: doi:10.1017/S0305004100069073
A. Washington, The non-$p$-part of the class number in a cyclotomic $\mathbbZ_p$-extension, Invent. Math., 49 (1978), 87--97.
A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math., 131 (1990), 493--540.
Mathematical Reviews (MathSciNet): MR1053488
Digital Object Identifier: doi:10.2307/1971468
JSTOR: links.jstor.org
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