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On Modified Circular Units and Annihilation of real Classes

Published online by Cambridge University Press:  11 January 2016

Jean-Robert Belliard  and
Thống Nguyễn-Quang-Ðỗ
Jean-Robert Belliard
Affiliation:
Université de Franche-Comté, UMR 6623, 16, route de Gray, 25030 Besançon cedex, France, [email protected]
Thống Nguyễn-Quang-Ðỗ
Affiliation:
Université de Franche-Comté, UMR 6623, 16, route de Gray, 25030 Besançon cedex, France, [email protected]
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Abstract

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For an abelian totally real number field F and an odd prime number p which splits totally in F, we present a functorial approach to special “p-units” previously built by D. Solomon using “wild” Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the p-class group of F (in the particular context here), as well as related annihilation results and index formulae.

Keywords

11R2311R1811R20
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 177 , 2005 , pp. 77 - 115
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2005

References

[B1] Belliard, J.-R., Sur la structure galoisienne des unités circulaires dans les Zp-extensions, J. Number Theory, 69 (1998), no. 1,16–49.CrossRefGoogle Scholar
[B2] Belliard, J.-R., Sous-modules d’unités en théorie d’Iwasawa, Théorie des nombres, Années 1998/2001, Publ. Math. UFR Sci. Tech. Besançon, Univ. Franche-Comté, Besançon (2002), p. 12.Google Scholar
[BB1] Bley, W. and Burns, D., Equivariant Tamagawa numbers, Fitting ideals and Iwasawa theory, Compositio Math., 126 (2001), 213–247.CrossRefGoogle Scholar
[BB2] Bley, W. and Burns, D., Explicit units and the equivariant Tamagawa number conjecture, Amer. J. Math., 123 (2001), no. 5,931–949.CrossRefGoogle Scholar
[BG] Burns, D. and Greither, C., On the equivariant Tamagawa number conjecture for Tate motives, Invent. Math., 153 (2003), no. 2,303–359.CrossRefGoogle Scholar
[BN] Belliard, J.-R. et Nguyễen-Quang-Đỗ, T., Formules de classes pour les corps abéliens réels, Ann. Inst. Fourier (Grenoble), 51 (2001), no. 4,903–937.CrossRefGoogle Scholar
[CG] Cornacchia, P. and Greither, C., Fitting ideals of class groups of real fields with prime power conductor, J. Number Theory, 73 (1998), no. 2,459–471.CrossRefGoogle Scholar
[FGS] Federer, L. J. and Gross, B. H., Regulators and Iwasawa modules, Invent. Math., 62 (1981), no. 3,443–457. With an appendix by Warren Sinnott.Google Scholar
[Gi] Gillard, R., Unités cyclotomiques, unités semi-locales et Zι-extensions. II, Ann. Inst. Fourier (Grenoble), 29 (1979), no. 4,1–15.Google Scholar
[GK] Gold, R. and Kim, J.M., Bases for cyclotomic units, Compositio Math., 71 (1989), no. 1,13–27.Google Scholar
[Gr1] Greither, C., Class groups of abelian fields, and the Main Conjecture, Ann. Inst. Fourier (Grenoble), 42 (1992), no. 3,449–499.CrossRefGoogle Scholar
[Gr2] Greither, C., Uber relativ-invariante Kreiseinheiten und Stickelberger Elemente, Manuscripta Math., 80 (1993), no. 1,27–43.CrossRefGoogle Scholar
[Gr3] Greither, C., The structure of some minus class groups, and Chinburg’s third conjecture for abelian fields, Math. Z., 229 (1998), no. 1,107–136.CrossRefGoogle Scholar
[I] Iwasawa, K., On Zι-extensions of algebraic number fields, Ann. of Math. (2), 98 (1973), 246–326.CrossRefGoogle Scholar
[J] Jaulent, J.-F., Classes logarithmiques des corps de nombres, J. Théor. Nombres Bordeaux, 6 (1994), no. 2,301–325.CrossRefGoogle Scholar
[KN] Kučera, R. and Nekov´aˇr, J., Cyclotomic units in Zp-extensions, J. Algebra, 171 (1995), no. 2, 457–472.CrossRefGoogle Scholar
[Kuc] Kučera, R., A Note on Circular Units in Zp-extensions, J. Théor. Nombres Bordeaux, 15 (2003), no. 1,223–229.CrossRefGoogle Scholar
[Kuz1] Kuz′min, L. V., The Tate module of algebraic number fields, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 267–327.Google Scholar
[Kuz2] Kuz′min, L. V., On formulas for the class number of real abelian fields, Izv. Ross. Akad. Nauk Ser. Mat., 60 (1996), no. 4, 43–110.Google Scholar
[N1] Nguyễen-Quang-Đỗ, T., Formations de classes et modules d’Iwasawa, Number theory, Noordwijkerhout 1983, Springer, Berlin (1984), pp. 167–185.Google Scholar
[N2] Nguyễen-Quang-Đỗ, T., Sur la Zp-torsion de certains modules galoisiens, Ann. Inst. Fourier (Grenoble), 36 (1986), no. 2, 27–46.Google Scholar
[R] Rubin, K., Global units and ideal class groups, Invent. Math., 89 (1987), no. 3,511–526.CrossRefGoogle Scholar
[RW1] Ritter, J. and Weiss, A., The lifted root number conjecture and Iwasawa theory, Mem. Amer. Math. Soc., 157 (2002), no. 748,viii+90.Google Scholar
[RW2] Ritter, J. and Weiss, A., Representing Ω(l∞) for real abelian fields, J. Algebra Appl., 2 (2003), no. 3,237–276.CrossRefGoogle Scholar
[SG] Simon, P. and Garfunkel, A., Bridge over troubled water, Columbia Records, 1970.Google Scholar
[Si1] Sinnott, W., On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. of Math. (2), 108 (1978), no. 1,107–134.CrossRefGoogle Scholar
[Si2] Sinnott, W., On the Stickelberger ideal and the circular units of an abelian field, Invent. Math., 62 (1980/81), no. 2, 181–234.CrossRefGoogle Scholar
[So1] Solomon, D., On a construction of p-units in abelian fields, Invent. Math., 109 (1992), no. 2,329–350.CrossRefGoogle Scholar
[So2] Solomon, D., Galois relations for cyclotomic numbers and p-units, J. Number Theory, 46 (1994), no. 2,158–178.CrossRefGoogle Scholar
[Ta] Taya, H., On p-adic zeta functions and Zp -extensions of certain totally real number fields, Tohoku Math. J. (2), 51 (1999), no. 1,21–33.CrossRefGoogle Scholar
[Ts] Tsuji, T., Semi-local units modulo cyclotomic units, J. Number Theory, 78 (1999), no. 1,1–26.CrossRefGoogle Scholar
[W] Washington, L. C., Introduction to cyclotomic fields, second edition, Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
[Wi] Wingberg, K., Duality theorems for Γ-extensions of algebraic number fields, Compositio Math., 55 (1985), no. 3,333–381.Google Scholar