Skip to main content
SpringerLink
Log in

Toward equivariant Iwasawa theory, III

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

Let l be an odd prime number and K ∞/k a Galois extension of totally real number fields, with and K ∞/k ∞ finite, where k ∞ is the cyclotomic -extension of k. The ``main conjecture'' of equivariant Iwasawa theory, as formulated in [RW2], is, up to its uniqueness statement, reduced to the existence of a nonabelian pseudomeasure whenever G ∞=G(K ∞/k) is an l-group and Iwasawa's μ-invariant vanishes. This follows from combining the validity of the conjecture in the maximal order case with special congruences. The main tool of proof is a generalization of the Taylor-Oliver integral group logarithm so that it applies to the setting of Iwasawa theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Burns, D., Greither, C.: On the Equivariant Tamagawa Number Conjecture for Tate motives. Invent. math. 153, 303–359 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Burns, D., Greither, C.: Equivariant Weierstraß preparation and values of L-functions at negative integers. Documenta Math., extra volume Kato 157–185 (2003)

  3. Boltje, R.: A canonical Brauer induction formula. Astérisque 181–182, 31–59 (1990)

  4. Curtis, C.W., Reiner, I.: Methods of Representation Theory, vol. I, II. John Wiley & Sons 1981,1987

  5. Deligne, P., Ribet, K.: Values of abelian L-functions at negative integers over totally real fields. Invent. Math. 59, 227–286 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fröhlich, A.: Galois Module Structure of Algebraic Integers. Springer-Verlag (1983)

  7. Iwasawa, K.: On the μ-invariants of -extensions. Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo 1–11 (1973)

  8. Oliver, R.: Whitehead Groups of Finite Groups. LMS Lecture Notes Series 132, Cambridge (1988)

  9. Reiner, I.: Maximal Orders. Academic Press (1975)

  10. Ritter, J., Weiss, A.: Toward equivariant Iwasawa theory. Manuscripta Math. 109, 131–146 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ritter, J., Weiss, A.: Toward equivariant Iwasawa theory, II. Indagationes Mathematicae 15, 549–572 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Serre, J.-P., Sur le résidu de la fonction zêta p-adique d'un corps de nombres. C.R.Acad.Sci. Paris 287, série A, 183–188 (1978)

    Google Scholar 

  13. Snaith, V.P.: Explicit Brauer Induction. Cambridge University Press, 1994

  14. Snaith, V.P.: Algebraic K-groups as Galois Modules. Birkhäuser (2002); Progress in Mathematics 206

  15. Taylor, M.J.: Classgroups of Group Rings. LMS Lecture Notes Series 91, Cambridge 1984

  16. Wiles, A.: The Iwasawa conjecture for totally real fields. Annals of Math. 131, 493–540 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Augsburg, 86135, Augsburg, Germany

    Jürgen Ritter

  2. Department of Mathematics, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Alfred Weiss

Authors
  1. Jürgen Ritter
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alfred Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürgen Ritter.

Additional information

We acknowledge financial support provided by NSERC and the University of Augsburg.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ritter, J., Weiss, A. Toward equivariant Iwasawa theory, III. Math. Ann. 336, 27–49 (2006). https://doi.org/10.1007/s00208-006-0773-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-006-0773-4

Keywords

Search

Navigation