The integral logarithm in Iwasawa theory : an exercise
Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 197-207.

Soient l un nombre premier impair et H un groupe fini abélien. Nous décrivons le groupe d’unités de Λ [H] (la complétion du localisé de l [[T]][H] en l) ainsi que le noyau et le conoyau du logarithme intégral L:Λ [H] × Λ [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 [H] (the completion of the localization at l of l [[T]][H]) as well as the kernel and cokernel of the integral logarithm L:Λ [H] × Λ [H], which appears in non-commutative Iwasawa theory.

Reçu le :
Révisé le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.711
@article{JTNB_2010__22_1_197_0,
     author = {J\"urgen Ritter and Alfred Weiss},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.711/}
}
Jürgen Ritter; Alfred Weiss. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.711/

[1] C.W. Curtis and I. Reiner, Methods of Representation Theory, I,II. John Wiley & Sons, 1981, 1987. | Zbl 0616.20001

[2] B. Coleman, Local units modulo circular units. Proc. AMS 89 (1983), 1–7. | MR 706497 | Zbl 0528.12005

[3] I. Fesenko and M. Kurihara, Invitation to higher local fields. Geometry & Topology Monographs 3 (2000), ISSN 1464-8997 (online). | MR 1804915 | Zbl 0954.00026

[4] A. Fröhlich, Galois Module Structure of Algebraic Integers. Springer-Verlag, 1983. | MR 717033 | Zbl 0501.12012

[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 2276851 | Zbl pre05620988

[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 1029028 | Zbl 0704.11038

[8] R. Oliver, Whitehead Groups of Finite Groups. LMS Lecture Notes Series 132, Cambridge (1988). | MR 933091 | Zbl 0636.18001

[9] J. Ritter and A. Weiss, Toward equivariant Iwasawa theory, II. Indagationes Mathematicae 15 (2004), 549–572. | MR 2114937 | Zbl 1142.11369

[10] J. Ritter and A. Weiss, Toward equivariant Iwasawa theory, III. Mathematische Annalen 336 (2006), 27–49. | MR 2242618 | Zbl 1154.11038

[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 2441694 | Zbl pre05380356

[12] J. Ritter and A. Weiss, Congruences between abelian pseudomeasures. Math. Res. Lett. 15 (2008), 715–725. | MR 2424908 | Zbl 1158.11047

[13] J. Ritter and A. Weiss, Equivariant Iwasawa theory : an example. Documenta Mathematica 13 (2008), 117–129. | MR 2420909 | Zbl pre05291240