Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa
Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 643-668.

Pour un nombre premier impair p et une extension abélienne K/k de corps de nombres totalement réels, nous utilisons la Conjecture Principale Équivariante démontrée par Ritter et Weiss (modulo la nullité de l’invariant μ p ) pour calculer l’idéal de Fitting d’un certain module d’Iwasawa sur l’algèbre complète p [[G ]],G =Gal(K /k) et K est la p -extension cyclotomique de K. Par descente, nous en déduisons la p-partie de la version cohomologique de la conjecture de Coates-Sinnott, ainsi qu’une forme faible de la p-partie de la conjecture de Brumer

For an odd prime number p and an abelian extension of totally real number fields K/k, we use the Equivariant Main Conjecture proved by Ritter and Weiss (modulo the vanishing of the μ p invariant) to compute the Fitting ideal of a certain Iwasawa module over the complete group algebra p [[G ]], where G =Gal(K /k), K being the cyclotomic p -extension of K. By descent, this gives the p-part of (a cohomological version of) the Coates-Sinnott conjecture, as well as a weak form of the p-part of the Brumer conjecture.

DOI : 10.5802/jtnb.512
Mots clés : Fitting ideals, Equivariant Main Conjecture
Nguyen Quang Do, Thong 1

1 UMR 6623 CNRS Université de Franche-Comté 16, Route de Gray 25030 Besançon Cedex - France
@article{JTNB_2005__17_2_643_0,
     author = {Nguyen Quang Do, Thong},
     title = {Conjecture principale \'equivariante, id\'eaux de {Fitting} et annulateurs en th\'eorie {d{\textquoteright}Iwasawa}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {643--668},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {2},
     year = {2005},
     doi = {10.5802/jtnb.512},
     zbl = {1098.11054},
     mrnumber = {2211312},
     language = {fr},
     url = {http://www.numdam.org/articles/10.5802/jtnb.512/}
}
TY  - JOUR
AU  - Nguyen Quang Do, Thong
TI  - Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2005
SP  - 643
EP  - 668
VL  - 17
IS  - 2
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.512/
DO  - 10.5802/jtnb.512
LA  - fr
ID  - JTNB_2005__17_2_643_0
ER  - 
%0 Journal Article
%A Nguyen Quang Do, Thong
%T Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa
%J Journal de théorie des nombres de Bordeaux
%D 2005
%P 643-668
%V 17
%N 2
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.512/
%R 10.5802/jtnb.512
%G fr
%F JTNB_2005__17_2_643_0
Nguyen Quang Do, Thong. Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 643-668. doi : 10.5802/jtnb.512. http://www.numdam.org/articles/10.5802/jtnb.512/

[B] D. Barsky, Sur la nullité du μ-invariant d’Iwasawa des corps totalement réels, prépublication (2005).

[BG1] D. Burns & C. Greither, On the Equivariant Tamagawa Number Conjecture for Tate motives. Invent. Math. 153 (2003), no. 2, 303–359. | MR | Zbl

[BG2] D. Burns & C. Greither, Equivariant Weierstrass Preparation and values of L-functions at negative integers. Doc. Math. (2003), Extra Vol., 157–185. | MR | Zbl

[BN] D. Benois & T. Nguyen Quang Do. Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs (m) sur un corps abélien. Ann. Sci. ENS 35 (2002), 641–672. | Numdam | MR | Zbl

[CS] J. Coates & W. Sinnott, An analogue of Stickelberger’s theorem for the higher K-groups. Invent. Math. 24 (1974), 149–161. | Zbl

[DR] P. Deligne & K. Ribet, Values of abelian L-functions at negative integers. Invent. Math 59 (1980), 227–286. | MR | Zbl

[G1] C. Greither, The structure of some minus class groups, and Chinburg’s third conjecture for abelian fields. Math. Zeit. 229 (1998), 107–136. | Zbl

[G2] C. Greither, Some cases of Brumer’s conjecture. Math. Zeit. 233 (2000), 515–534. | Zbl

[G3] C. Greither, Computing Fitting ideals of Iwasawa modules. Math. Z. 246 (2004), no. 4, 733–767. | MR | Zbl

[HK1] A. Huber & G. Kings, Bloch-Kato Conjecture and Main Conjecture of Iwasawa theory for Dirichlet characters. Duke Math. J. 119 (2003), no. 3, 393–464. | MR | Zbl

[HK2] A. Huber & G. Kings, Equivariant Bloch-Kato Conjecture and non abelian Iwasawa Main Conjecture. ICM 2002, vol. II, 149–162. | MR | Zbl

[Ih] Y. Ihara, On Galois representations arising from towers of coverings of 1 {0,1,}. Invent. Math. 86 (1986), 427–459. | MR | Zbl

[Iw] K. Iwasawa, On -extensions of algebraic number fields. Annals of Math. 98 (1973), 246–326. | MR | Zbl

[J] U. Jannsen, Iwasawa modules up to isomorphism. Adv. Studies in Pure Math. 17 (1989), 171–207. | MR | Zbl

[K] K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B dR . I. Arithmetic algebraic geometry (Trento, 1991). 50–163, Lecture Notes in Math., 1553, Springer, Berlin, 1993. | MR | Zbl

[K1] M. Kurihara, Iwasawa theory and Fitting ideals. J. Reine Angew. Math. 561 (2003), 39–86. | MR | Zbl

[K2] M. Kurihara, On the structure of ideal class groups of CM fields. Doc. Math. (2003), Extra Vol., 539–563. | MR | Zbl

[KNF] M. Kolster, T. Nguyen Quang Do & V. Fleckinger, Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures. Duke Math. J. 84 (1996), no. 3, 679–717. | MR | Zbl

[LF] M. Le Floc’h, On Fitting ideals of certain étale K-groups. K-Theory 27 (2002), 281–292. | Zbl

[MW] B. Mazur & A. Wiles, Class fields of abelian extensions of . Invent. Math. 76 (1984), 179–330. | MR | Zbl

[N1] T. Nguyen Quang Do, Formations de classes et modules d’Iwasawa. Dans “Number Theory Noordwijkerhout”, Springer LNM 1068 (1984), 167–185. | Zbl

[N2] T. Nguyen Quang Do, Sur la p -torsion de certains modules galoisiens. Ann. Inst. Fourier 36 (1986), no. 2, 27–46. | Numdam | MR | Zbl

[N3] T. Nguyen Quang Do, Analogues supérieurs du noyau sauvage. J. Théorie des Nombres Bordeaux 4 (1992), 263–271. | Numdam | MR | Zbl

[N4] T. Nguyen Quang Do, Quelques applications de la Conjecture Principale Equivariante, lettre à M. Kurihara (15/02/02).

[NSW] J. Neukirch, A. Schmidt & K. Wingberg, Cohomology of Number Fields. Grundlehren 323, Springer, 2000. | MR | Zbl

[R] K. Ribet, Report on p-adic L-functions over totally real fields. Astérisque 61 (1979), 177–192. | Numdam | MR | Zbl

[RW1] J. Ritter & A. Weiss, The Lifted Root Number Conjecture and Iwasawa theory. Memoirs AMS 157/748 (2002). | MR | Zbl

[RW2] J. Ritter & A. Weiss, Towards equivariant Iwasawa theory. Manuscripta Math. 109 (2002), 131–146. | MR | Zbl

[Ro-W] J. Rognes & C.A. Weibel, Two-primary algebraic K-theory of rings of integers in number fields. J. AMS (1) 13 (2000), 1–54. | MR | Zbl

[Sc] P. Schneider, Über gewisse Galoiscohomologiegruppen. Math. Zeit 168 (1979), 181–205. | MR | Zbl

[Se] J.-P. Serre, Sur le résidu de la fonction zêta p-adique d’un corps de nombres. CRAS Paris 287, A (1978), 183–188. | Zbl

[Sn1] V. Snaith, “Algebraic K-groups as Galois modules”. Birkhauser, Progress in Math. 206 (2002). | Zbl

[Sn2] V. Snaith, Relative K 0 , Fitting ideals and the Stickelberger phenomena, preprint (2002).

[T] J. Tate, “Les conjectures de Stark sur les fonctions L d’Artin en s=0. Birkhauser, Progress in Math. 47 (1984). | Zbl

[W1] A. Wiles, The Iwasawa conjecture for totally real fields. Annals of Math. 141 (1990), 493–540. | MR | Zbl

[W2] A. Wiles, On a conjecture of Brumer. Annals of Math. 131 (1990), 555–565. | MR | Zbl

Cité par Sources :