Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-28T08:16:33.632Z Has data issue: false hasContentIssue false
  • English
  • Français

Norm index formula for the Tate kernels and applications

Published online by Cambridge University Press:  24 May 2011

J. Assim  and
A. Movahhedi
J. Assim
Affiliation:
Université Moulay Ismail, Mathématiques et informatique, B.P 11201 Zitoune Meknès, Meknès 50000, [email protected]
A. Movahhedi
Affiliation:
XLIM UMR 6172 CNRS/Univ. de Limoges, Mathématiques et informatique, 123, Avenue A. Thomas, 87060 Limoges, [email protected]
Get access

Abstract

Let p be an odd prime and L/F a p-extension of number fields with Galois group G. The aim of this paper is to provide answers to a question of Kahn concerning lower bounds for the order of the kernel and cokernel of the functorial map K2F → K2LG. To this end, we first determine a norm index formula for generalized Tate kernels and then express our lower bounds in terms of the ramification in L/F.

Keywords

CapitulationTate kernelIwasawa theoryK-groupCohomology
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 2 , April 2012 , pp. 359 - 383
Copyright
Copyright © ISOPP 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

As95.Assim, J.: Codescente en K-théorie étale et corps de nombres. Manuscripta Math. 86 (1995), no. 4, 499–518.CrossRefGoogle Scholar
AM04.Assim, J., Movahhedi, A.: Bounds for étale capitulation kernels. K-Theory 33 (2004), no. 3, 199–213.CrossRefGoogle Scholar
Bo77.Borel, A.: Cohomologie de SLn et valeurs de fonctions zeta aux points entiers. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636.Google Scholar
CE56.Cartan, H., Eilenberg, S.: Homological algebra. Princeton University Press, Princeton, N. J., 1956.Google Scholar
CKPS98.Chinburg, T.; Kolster, M.; Pappas, G.; Snaith, V.: Galois structure of K-groups of rings of integers. K-Theory 14 (1998), no. 4, 319–369.CrossRefGoogle Scholar
DF85.Dwyer, W., Friedlander, E.: Algebraic and étale K-theory. Trans. Amer. Math. Soc. 292 (1985), no. 1, 247–280.Google Scholar
Gr86.Gras, G.: Remarks on K2 of number fields. J. Number Theory 23 (1986), no. 3, 322–335.CrossRefGoogle Scholar
GJ89.Gras, G., Jaulent, J.-F.: Sur les corps de nombres réguliers. Math. Z. 202 (1989), no. 3, 343–365.CrossRefGoogle Scholar
Gr78.Greenberg, R.: A note on K2 and the theory of Zp-extensions. Amer. J. Math. 100 (1978), no. 6, 1235–1245.CrossRefGoogle Scholar
Ha78.Haberland, K.: Galois cohomology of algebraic number fields. With two appendices by Helmut Koch and Thomas Zink. Berlin: VEB Deutscher Verlag der Wissenschaften (1978), 145 p.Google Scholar
Hu05.Hutchinson, K.: Tate Kernels, Étale K-Theory and The Gross Kernel. Preprint (2005).Google Scholar
Iw73.Iwasawa, K.: On Zℓ-extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326.CrossRefGoogle Scholar
Ka93.Kahn, B.: Descente galoisienne et K2 des corps de nombres, K-Theory 7 (1993), no. 1, 55–100.CrossRefGoogle Scholar
Ko91.Kolster, M.: An idelic approach to the wild kernel. Invent. Math. 103 (1991), no. 1, 9–24.CrossRefGoogle Scholar
Ko93.Kolster, M.: Remarks on étale K-theory and Leopoldt's conjecture. Séminaire de Théorie des nombres, Paris 1991–92, 37-62, Progr. Math. 116, Birkhäuser, Boston, Boston, MA, 1993.Google Scholar
Ko02.Kolster, M.: Higher relative class number formulae. Math. Ann. 323 (2002), no. 4, 667–692.CrossRefGoogle Scholar
Ko04.Kolster, M.: K-theory and arithmetic. Contemporary developments in algebraic K-theory, 191–258 (electronic), ICTP Lect. Notes, XV, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004.Google Scholar
KM00.Kolster, M., Movahhedi, A.: Galois co-descent for étale wild kernels and capitulation, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 1, 35–65.CrossRefGoogle Scholar
KN98.Kolster, M., Nguyen Quang Do, T.: Syntomic regulators and special values of padic L-functions, Invent. Math. 133 (1998), no. 2, 417–447.CrossRefGoogle Scholar
KNF96.Kolster, M., Nguyen Quang Do, T., Fleckinger, V.: Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J. 84 (1996), no. 3, 679–717.CrossRefGoogle Scholar
LMN05.Le Floc'h, M., Movahhedi, A., Nguyen Quang Do, T.: On capitulation cokernels in Iwasawa theory. Amer. J. Math. 127 (2005), no. 4, 851–877.CrossRefGoogle Scholar
Mi71.Milnor, John W.: Introduction to algebraic K-theory. Annals of Mathematics Studies 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. xiii + 184 pp.Google Scholar
Mo88.Movahhedi, A.: Sur les p-extensions des corps p-rationnels, Thèse Paris 7 (1988).Google Scholar
Mo90.Movahhedi, A.: Sur les p-extensions des corps p-rationnels, Math. Nachr. 149 (1990), 163–176.CrossRefGoogle Scholar
MN90.Movahhedi, A., Nguyen Quang Do, T.: Sur l'arithmétique des corps de nombres p-rationnels, Séminaire de Théorie des nombres, Paris 1987–88, 155–200, Progr. Math. 81, Birkhäuser, Boston, MA, 1990.Google Scholar
Ng86.Nguyen Quang Do, T.: Sur la Zp-torsion de certains modules galoisiens. Ann. Inst. Fourier (Grenoble) 36 (1986), no. 2, 27–46.CrossRefGoogle Scholar
Ng92.Nguyen Quang Do, T.: Analogues supérieurs du noyau sauvage. Sém. Théor. Nombres Bordeaux (2) 4 (1992), no. 2, 263–271CrossRefGoogle Scholar
Sc79.Schneider, P.: Über gewisse Galoiscohomologiegruppen. Math. Z. 168 (1979), no. 2, 181–205.CrossRefGoogle Scholar
So79.Soulé, C.: K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale. Invent. Math. 55 (1979), no. 3, 251–295.CrossRefGoogle Scholar
Ta66.Tate, J.: On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire Bourbaki 1965/66, Exp. no. 306, 415–440 (1966).Google Scholar
Ta73.Tate, J.: Letter from Tate to Iwasawa on a relation between K2 and Galois cohomology. Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972), Lecture Notes in Math. 342, Springer, Berlin, (1973) 524–527.Google Scholar
Ta76.Tate, J.: Relations between K2 and Galois cohomology. Invent. Math. 36 (1976), 257–274.CrossRefGoogle Scholar
Va08.Vauclair, D.: Noyaux de Tate et capitulation. J. Number Theory 128 (2008), no. 3, 619–638.CrossRefGoogle Scholar
Wa97.Washington, Lawrence C.: Introduction to cyclotomic fields. Second edition. Graduate Texts in Mathematics 83. Springer-Verlag, New York, 1997.Google Scholar