Comptes Rendus
no. 9
Number Theory
Higher analogues of Stickelberger's theorem
[Analogues plus hauts du théorème de Stickelberger]

Présenté par : Christophe Soulé

Grzegorz Banaszak1
1 Department of Mathematics, Adam Mickiewicz University, Poznań, Poland
Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 575-580.

Soit l un nombre premier impair, soit F un corps de nombres totalement réel et soit E/F une extension abélienne de conducteur f, où E est un corps de nombres de type CM. Dans cette Note nous prouvons que S n (E/F,l)⊂ Ann ℤ l [G(E/F)] H 2 (𝒪 E [1/l];ℤ l (n+1)) pour tout entier impair n>0 et pour presque tout nombre premier l, où Sn(E/F,l) est l'idéal de Stickelberger (Ann. of Math. 135 (1992) 325–360 ; J. Coates, p-adic L-functions and Iwasawa's theory, in : Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). Si nous supposons la conjecture de Quillen–Lichtenbaum alors S n (E/F,l)⊂ Ann ℤ l [G(E/F)] K 2n (𝒪 E ) l .

Let l be an odd prime number, F denote any totally real number field and E/F be an Abelian CM extension of F of conductor f. In this paper we prove that for every n odd and almost all prime numbers l we have S n (E/F,l)⊂ Ann ℤ l [G(E/F)] H 2 (𝒪 E [1/l];ℤ l (n+1)) where Sn(E/F,l) is the Stickelberger ideal (Ann. of Math. 135 (1992) 325–360; J. Coates, p-adic L-functions and Iwasawa's theory, in: Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977). In addition if we assume the Quillen–Lichtenbaum conjecture then S n (E/F,l)⊂ Ann ℤ l [G(E/F)] K 2n (𝒪 E ) l .

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2003.09.019
Grzegorz Banaszak 1

1 Department of Mathematics, Adam Mickiewicz University, Poznań, Poland 
@article{CRMATH_2003__337_9_575_0,
     author = {Grzegorz Banaszak},
     title = {Higher analogues of {Stickelberger's} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {575--580},
     publisher = {Elsevier},
     volume = {337},
     number = {9},
     year = {2003},
     doi = {10.1016/j.crma.2003.09.019},
     language = {en},
}
TY  - JOUR
AU  - Grzegorz Banaszak
TI  - Higher analogues of Stickelberger's theorem
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 575
EP  - 580
VL  - 337
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2003.09.019
LA  - en
ID  - CRMATH_2003__337_9_575_0
ER  - 
%0 Journal Article
%A Grzegorz Banaszak
%T Higher analogues of Stickelberger's theorem
%J Comptes Rendus. Mathématique
%D 2003
%P 575-580
%V 337
%N 9
%I Elsevier
%R 10.1016/j.crma.2003.09.019
%G en
%F CRMATH_2003__337_9_575_0
Grzegorz Banaszak. Higher analogues of Stickelberger's theorem. Comptes Rendus. Mathématique, Volume 337 (2003) no. 9, pp. 575-580. doi : 10.1016/j.crma.2003.09.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.09.019/

[1] G. Banaszak Algebraic K-theory of number fields and rings of integers and the Stickelberger ideal, Ann. of Math., Volume 135 (1992), pp. 325-360

[2] G. Banaszak Generalization of the Moore exact sequence and the wild kernel for higher K-groups, Compositio Math., Volume 86 (1993), pp. 281-305

[3] G. Banaszak; W. Gajda; P. Krasoń; P. Zelewski A note on the Quillen–Lichtenbaum conjecture and the arithmetic of square rings, K-Theory, Volume 16 (1999) no. 3, pp. 229-243

[4] J. Coates p-adic L-functions and Iwasawa's theory in Algebraic Number Fields by A. Fröhlich, Academic Press, London, 1977

[5] J. Coates; W. Sinnott An analogue of Stickelberger's theorem for higher K-groups, Invent. Math., Volume 24 (1974), pp. 149-161

[6] P. Deligne; K. Ribet Values of Abelian L-functions at negative integers over totally real fields, Invent. Math., Volume 59 (1980), pp. 227-286

[7] W. Dwyer; E. Friedlander Algebraic and étale K-theory, Trans. Amer. Math. Soc., Volume 292 (1985), pp. 247-280

[8] T. Nguyen Quang Do Analogues supérieurs du noyau sauvage, Sém. Théor. Nombres Bordeaux (2), Volume 4 (1992), pp. 263-271

[9] C. Popescu, Stark's question and a strong form of Brumer's conjecture, Compositio Math., in press

[10] P. Schneider Über gewisse Galoiscohomologiegruppen, Math. Z., Volume 168 (1979), pp. 181-205

[11] V. Snaith, Relative K0, annihilators, Fitting ideals and the Stickelberger's phenomena, Preprint, 2003

[12] V. Snaith, Equivariant motivic phenomena, Preprint, 2003

[13] C. Soulé K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math., Volume 55 (1979), pp. 251-295

[14] J. Tate Relations between K2 and Galois cohomology, Invent. Math., Volume 36 (1976), pp. 257-274

[15] A. Wiles On a conjecture of Brumer, Ann. Math., Volume 131 (1990), pp. 555-565

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

On a Hasse principle for Mordell–Weil groups

Grzegorz Banaszak

C. R. Math (2009)

Une remarque sur l'annulateur du groupe des classes d'idéaux

Francesco Amoroso

C. R. Math (2006)