The integral logarithm in Iwasawa theory : an exercise

Ritter, Jürgen; Weiss, Alfred

doi:10.5802/jtnb.711

The integral logarithm in Iwasawa theory : an exercise

Ritter, Jürgen ¹ ; Weiss, Alfred ²

¹ Schnurbeinstraße 14 86391 Deuringen, Germany
² Department of Mathematics University of Alberta Edmonton, AB, Canada T6C 2G1

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 197-207.

Résumé
Abstract

Soient $l$ un nombre premier impair et $H$ un groupe fini abélien. Nous décrivons le groupe d’unités de $Λ_{\land} [H]$ (la complétion du localisé de $ℤ_{l} [[T]] [H]$ en $l$ ) ainsi que le noyau et le conoyau du logarithme intégral $L : Λ_{\land} {[H]}^{\times} \to Λ_{\land} [H]$ , qui apparaît dans la théorie d’Iwasawa non-commutative.

Let $l$ be an odd prime number and $H$ a finite abelian $l$ -group. We describe the unit group of $Λ_{\land} [H]$ (the completion of the localization at $l$ of $ℤ_{l} [[T]] [H]$ ) as well as the kernel and cokernel of the integral logarithm $L : Λ_{\land} {[H]}^{\times} \to Λ_{\land} [H]$ , which appears in non-commutative Iwasawa theory.

MR Zbl

DOI : 10.5802/jtnb.711

Affiliations des auteurs :

Ritter, Jürgen ¹ ; Weiss, Alfred ²

¹ Schnurbeinstraße 14 86391 Deuringen, Germany
² Department of Mathematics University of Alberta Edmonton, AB, Canada T6C 2G1

@article{JTNB_2010__22_1_197_0,
     author = {Ritter, J\"urgen and Weiss, Alfred},
     title = {The integral logarithm in {Iwasawa} theory~: an exercise},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {197--207},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.711},
     zbl = {1214.11124},
     mrnumber = {2675880},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.711/}
}

TY  - JOUR
AU  - Ritter, Jürgen
AU  - Weiss, Alfred
TI  - The integral logarithm in Iwasawa theory : an exercise
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2010
SP  - 197
EP  - 207
VL  - 22
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.711/
DO  - 10.5802/jtnb.711
LA  - en
ID  - JTNB_2010__22_1_197_0
ER  -

%0 Journal Article
%A Ritter, Jürgen
%A Weiss, Alfred
%T The integral logarithm in Iwasawa theory : an exercise
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 197-207
%V 22
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.711/
%R 10.5802/jtnb.711
%G en
%F JTNB_2010__22_1_197_0

Ritter, Jürgen; Weiss, Alfred. The integral logarithm in Iwasawa theory : an exercise. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 197-207. doi : 10.5802/jtnb.711. http://www.numdam.org/articles/10.5802/jtnb.711/

Bibliographie
Cité par

[1] C.W. Curtis and I. Reiner, Methods of Representation Theory, I,II. John Wiley & Sons, 1981, 1987. | Zbl

[2] B. Coleman, Local units modulo circular units. Proc. AMS 89 (1983), 1–7. | MR | Zbl

[3] I. Fesenko and M. Kurihara, Invitation to higher local fields. Geometry & Topology Monographs 3 (2000), ISSN 1464-8997 (online). | MR | Zbl

[4] A. Fröhlich, Galois Module Structure of Algebraic Integers. Springer-Verlag, 1983. | MR | Zbl

[5] T. Fukaya and K. Kato, A formulation of conjectures on $p$ -adic zeta functions in non-commutative Iwasawa theory. Proc. St. Petersburg Math. Soc. 11 (2005). | MR

[6] K. Kato, Iwasawa theory of totally real fields for Galois extensions of Heisenberg type. Preprint (‘Very preliminary version’ , 2006)

[7] S. Lang, Cylotomic Fields I-II. Springer GTM 121 (1990). | MR | Zbl

[8] R. Oliver, Whitehead Groups of Finite Groups. LMS Lecture Notes Series 132, Cambridge (1988). | MR | Zbl

[9] J. Ritter and A. Weiss, Toward equivariant Iwasawa theory, II. Indagationes Mathematicae 15 (2004), 549–572. | MR | Zbl

[10] J. Ritter and A. Weiss, Toward equivariant Iwasawa theory, III. Mathematische Annalen 336 (2006), 27–49. | MR | Zbl

[11] J. Ritter and A. Weiss, Non-abelian pseudomeasures and congruences between abelian Iwasawa $L$ -functions. Pure and Applied Mathematics Quarterly 4 (2008), 1085–1106. | MR

[12] J. Ritter and A. Weiss, Congruences between abelian pseudomeasures. Math. Res. Lett. 15 (2008), 715–725. | MR | Zbl

[13] J. Ritter and A. Weiss, Equivariant Iwasawa theory : an example. Documenta Mathematica 13 (2008), 117–129. | MR

Cité par Sources :