Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T01:43:22.641Z Has data issue: false hasContentIssue false
  • English
  • Français

On λ-invariants of number fields and étale cohomology

Published online by Cambridge University Press:  05 July 2013

Manfred Kolster  and
Abbas Movahhedi
Manfred Kolster
Affiliation:
Department of Mathematics, McMaster University, 1280 Main St. West Hamilton, Ontario L8S 4K1, [email protected]
Abbas Movahhedi
Affiliation:
Université de Limoges, XLIM UMR 7252 CNRS, 123 Av. Albert Thomas, 87060 Limoges Cedex, [email protected]
Get access

Abstract

For an odd prime p we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over F(μp∞), the field obtained from a totally real number field F by adjoining all p-power roots of unity. We use a new approach based on the relationship between eigenspaces and étale cohomology groups over the cyclotomic ℤp-extension F∞ of F. The systematic use of étale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces.

Keywords

Iwasawa theoryétale cohomologyλ-invariants11R2311R3419F27
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 1: Nanjing Special Issue on K-theory, number theory and geometry , August 2013 , pp. 167 - 181
Copyright
Copyright © ISOPP 2013 

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

1.Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1977), 613–636.Google Scholar
2.Coates, J., On K 2 and some classical conjectures in algebraic number theory, Ann. of Math. 95 (1972), 99–116.Google Scholar
3.Curtis, C. and Reiner, I., Representation theory of finite groups and associative algebras, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962.Google Scholar
4.Ferrero, B. and Washington, L., The Iwasawa invariant μp vanishes for abelian number fields, Ann. of Math. 109 (1979), 195–228.Google Scholar
5.Gold, R. and Madan, M., Iwasawa invariants, Comm. Algebra 13(7) (1985), 1559–1578.Google Scholar
6.Hutchinson, K., Étale wild kernels of exceptional number fields, J. of Number Th. 120 (2006), 47–71.CrossRefGoogle Scholar
7.Iwasawa, K., On ℤl-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246–326.Google Scholar
8.Iwasawa, K., Riemann-Hurwitz formula and p-adic Galois representations for number fields, Tohoku Math. J. 33 (1981), 263–288.CrossRefGoogle Scholar
9.Kida, Y., l-extensions of CM-fields and cyclotomic invariants, J. of Number Th. 12 (1980), 519–528.CrossRefGoogle Scholar
10.Kolster, M. and Movahhedi, A., Galois co-descent for étale wild kernels and capitulation, Ann.Inst. Fourier 50 (2000), 35–65.CrossRefGoogle Scholar
11.Kolster, M., K-Theory and Arithmetic, Contemporary Developments in Algebraic K-Theory, ICTP Lecture Notes 15 (2003), 191–258.Google Scholar
12.Milne, J. S., Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, N.J, 1980.Google Scholar
13.Quang Do, T. Nguyen, K 3 et formules de Riemann-Hurwitz p-adiques, K-Theory 7 (1993), 429–441.Google Scholar
14.Quang Do, T. Nguyen, Analogues supérieurs du noyau sauvage, Sém. de Théorie des Nombres Bordeaux 4(2) (1992), 263–271.Google Scholar
15.Schneider, P., Über gewisse Galoiscohomologiegruppen, Math. Z. 168 (1979), 181–205.Google Scholar
16.Serre, J.-P., Cohomologie Galoisienne, Lecture Notes in Mathematics 5, Springer-Verlag, 1964.Google Scholar
17.Sinnott, W. M., On p-adic L-functions and the Riemann-Hurwitz genus formula, Compositio Math. 53 (1984), 3–17.Google Scholar
18.Soulé, C., K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1984), 251–295.Google Scholar
19.Tate, J., Relations between K 2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.Google Scholar
20.Washington, L.C., Introduction to Cyclotomic Fields, Graduate Texts in Mathematics 83, Springer Verlag, 1982.Google Scholar
21.Wingberg, K., Duality theorems for Γ-extensions of algebraic number fields with restricted ramification, Comp. Math. 55 (1985), 333–381.Google Scholar