Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

On the ``main conjecture'' of equivariant Iwasawa theory


Authors: Jürgen Ritter and Alfred Weiss
Journal: J. Amer. Math. Soc. 24 (2011), 1015-1050
MSC (2010): Primary 11R23, 11R42, 11S40
DOI: https://doi.org/10.1090/S0894-0347-2011-00704-2
Published electronically: May 13, 2011
MathSciNet review: 2813337
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the ``main conjecture'' of equivariant Iwasawa theory when $ \mu=0$.


References [Enhancements On Off] (What's this?)

  • [Ca] Cassou-Noguès, P., Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $ p$-adiques. Invent. Math. 51 (1979), 29-59. MR 524276 (80h:12009b)
  • [CR] Curtis, C. W. and Reiner, I., Methods of Representation Theory, vol. 2. John Wiley & Sons (1987). MR 892316 (88f:20002)
  • [DR] Deligne, P., and Ribet, K., Values of abelian $ L$-functions at negative integers over totally real fields. Invent. Math. 59 (1980), 227-286. MR 579702 (81m:12019)
  • [FK] Fukaya, T., and Kato, K., A formulation of conjectures on $ p$-adic zeta functions in noncommutative Iwasawa theory. Proceedings of the St. Petersburg Mathematical Society, vol. XII (N. N. Uraltseva, ed.), AMS Translations - Series 2, 219 (2006), 1-86. MR 2276851 (2007k:11200)
  • [Gr] Greenberg, R., On $ p$-adic Artin $ L$-functions. Nagoya Math. J. 89 (1983), 77-87. MR 692344 (85b:11104)
  • [Ha1] Hara, T. Iwasawa theory of totally real fields for certain non-commutative $ p$-extensions. Journal of Number Theory 130 (2010), 1068-1097. MR 2600423 (2011c:11166)
  • [Ha2] -, Inductive construction of the $ p$-adic zeta functions for non-commutative $ p$-extensions of totally real fields with exponent $ p$. arXiv:0908.2178v1 [math.NT].
  • [HIO] Hawkes, T., Isaacs, I. M., and Özaydin, M., On the Möbius function of a finite group. Rocky Mountain J. of Mathematics 19 (1989), 1003-1033. MR 1039540 (90k:20046)
  • [Kk] Kakde, M., Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases. J. Algebraic Geometry, posted on April 5, 2011, PII S 1056-3911(2011)00539-0 (to appear in print).
  • [Kt1] Kato, K., Iwasawa theory and generalizations. Proc. ICM, Madrid, Spain, 2006; European Math. Soc. (2007), 335-357. MR 2334196 (2008e:11133)
  • [Kt2] -, Iwasawa theory of totally real fields for Galois extensions of Heisenberg type. Preprint (``Very preliminary version'', 2007).
  • [La] Lau, I., Algebraic contributions to equivariant Iwasawa theory. Ph.D. thesis, Universität Augsburg (2010).
  • [NSW] Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields. Springer Grundlehren der math. Wiss.  323 (2000). MR 1737196 (2000j:11168)
  • [RWt] Ritter, J., and Weiss, A., The Lifted Root Number Conjecture and Iwasawa theory. Memoirs of the AMS 157/748 (2002). MR 1894887 (2003d:11164)
  • [RW1] -, Towards equivariant Iwasawa theory. Manuscripta Math. 109 (2002), 131-146. MR 1935024 (2003i:11161)
  • [RW2] -, Towards equivariant Iwasawa theory, II. Indag. Mathemat. 15 (2004), 549-572. MR 2114937 (2006d:11132)
  • [RW3] -, Towards equivariant Iwasawa theory, III. Math. Ann. 336 (2006), 27-49. MR 2242618 (2007d:11123)
  • [RW4] -, Towards equivariant Iwasawa theory, IV. Homology, Homotopy and Applications 7 (2005), 155-171. MR 2205173 (2006j:11151)
  • [RW5] -, Non-abelian pseudomeasures and congruences between abelian Iwasawa $ L$-functions. Pure and Applied Math. Quarterly 4 (2008), 1085-1106. MR 2441694 (2010b:11149)
  • [RW6] -, The integral logarithm in Iwasawa theory: An exercise. Journal de Théorie des Nombres de Bordeaux 22 (2010), 197-207. MR 2675880 (2011f:11144)
  • [RW7] -, Congruences between abelian pseudomeasures. Math. Res. Lett. 15 (2008), 715-725. MR 2424908 (2009m:11183)
  • [RW8] -, Equivariant Iwasawa Theory: An Example. Documenta Math. 13 (2008), 117-129. MR 2420909 (2009h:11183)
  • [RoW] Roblot, X.-F., and Weiss, A., Numerical evidence toward a 2-adic equivariant ``main conjecture''. Experimental Mathematics 20 (2011), 169-176.
  • [Se] Serre, J.-P., Sur le résidu de la fonction zêta $ p$-adique d'un corps de nombres., C. R. Acad. Sci. Paris 287 (1978), Série A, 183-188. MR 0506177 (58:22024)
  • [Wi] Wiles, A., The Iwasawa conjecture for totally real fields. Annals of Math. 131 (1990), 493-540. MR 1053488 (91i:11163)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11R23, 11R42, 11S40

Retrieve articles in all journals with MSC (2010): 11R23, 11R42, 11S40


Additional Information

Jürgen Ritter
Affiliation: Schnurbeinstrasse 14, 86391 Deuringen, Germany
Email: jr@ritter-maths.de

Alfred Weiss
Affiliation: Department of Mathematics, University of Alberta, Edmonton, AB, Canada T6G2G1
Email: weissa@ualberta.ca

DOI: https://doi.org/10.1090/S0894-0347-2011-00704-2
Received by editor(s): June 29, 2010
Received by editor(s) in revised form: March 28, 2011, and April 11, 2011
Published electronically: May 13, 2011
Additional Notes: The authors acknowledge financial support provided by DFG and NSERC
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society