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.

Publié le :
DOI : https://doi.org/10.5802/jtnb.512
Mots clés : Fitting ideals, Equivariant Main Conjecture
@article{JTNB_2005__17_2_643_0,
     author = {Thong Nguyen Quang Do},
     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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.512/}
}
Thong Nguyen Quang Do. 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. https://jtnb.centre-mersenne.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 1992015 | Zbl 02001021

[BG2] D. Burns & C. Greither, Equivariant Weierstrass Preparation and values of L-functions at negative integers. Doc. Math. (2003), Extra Vol., 157–185. | MR 2046598 | Zbl 02028834

[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 1951439 | Zbl 01910884

[CS] J. Coates & W. Sinnott, An analogue of Stickelberger’s theorem for the higher K-groups. Invent. Math. 24 (1974), 149–161. | Zbl 0282.12006

[DR] P. Deligne & K. Ribet, Values of abelian L-functions at negative integers. Invent. Math 59 (1980), 227–286. | MR 579702 | Zbl 0434.12009

[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 0919.11072

[G2] C. Greither, Some cases of Brumer’s conjecture. Math. Zeit. 233 (2000), 515–534. | Zbl 0965.11047

[G3] C. Greither, Computing Fitting ideals of Iwasawa modules. Math. Z. 246 (2004), no. 4, 733–767. | MR 2045837 | Zbl 1067.11067

[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 2002643 | Zbl 1044.11095

[HK2] A. Huber & G. Kings, Equivariant Bloch-Kato Conjecture and non abelian Iwasawa Main Conjecture. ICM 2002, vol. II, 149–162. | MR 1957029 | Zbl 1020.11067

[Ih] Y. Ihara, On Galois representations arising from towers of coverings of 1 {0,1,}. Invent. Math. 86 (1986), 427–459. | MR 860676 | Zbl 0595.14020

[Iw] K. Iwasawa, On -extensions of algebraic number fields. Annals of Math. 98 (1973), 246–326. | MR 349627 | Zbl 0285.12008

[J] U. Jannsen, Iwasawa modules up to isomorphism. Adv. Studies in Pure Math. 17 (1989), 171–207. | MR 1097615 | Zbl 0732.11061

[K1] M. Kurihara, Iwasawa theory and Fitting ideals. J. Reine Angew. Math. 561 (2003), 39–86. | MR 1998607 | Zbl 1056.11063

[K2] M. Kurihara, On the structure of ideal class groups of CM fields. Doc. Math. (2003), Extra Vol., 539–563. | MR 2046607 | Zbl 02028843

[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 1338860 | Zbl 0815.11051

[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 1408541 | Zbl 0863.19003

[LF] M. Le Floc’h, On Fitting ideals of certain étale K-groups. K-Theory 27 (2002), 281–292. | Zbl 1083.11073

[MW] B. Mazur & A. Wiles, Class fields of abelian extensions of . Invent. Math. 76 (1984), 179–330. | MR 742853 | Zbl 0545.12005

[N1] T. Nguyen Quang Do, Formations de classes et modules d’Iwasawa. Dans “Number Theory Noordwijkerhout”, Springer LNM 1068 (1984), 167–185. | Zbl 0543.12007

[N2] T. Nguyen Quang Do, Sur la p -torsion de certains modules galoisiens. Ann. Inst. Fourier 36 (1986), no. 2, 27–46. | Numdam | MR 850741 | Zbl 0576.12010

[N3] T. Nguyen Quang Do, Analogues supérieurs du noyau sauvage. J. Théorie des Nombres Bordeaux 4 (1992), 263–271. | Numdam | MR 1208865 | Zbl 0783.11042

[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 1737196 | Zbl 0948.11001

[R] K. Ribet, Report on p-adic L-functions over totally real fields. Astérisque 61 (1979), 177–192. | MR 556672 | Zbl 0408.12016

[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 1697095 | Zbl 0934.19001

[RW1] J. Ritter & A. Weiss, The Lifted Root Number Conjecture and Iwasawa theory. Memoirs AMS 157/748 (2002). | MR 1894887 | Zbl 1002.11082

[RW2] J. Ritter & A. Weiss, Towards equivariant Iwasawa theory. Manuscripta Math. 109 (2002), 131–146. | MR 1935024 | Zbl 1014.11066

[Sc] P. Schneider, Über gewisse Galoiscohomologiegruppen. Math. Zeit 168 (1979), 181–205. | MR 544704 | Zbl 0421.12024

[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 0393.12026

[Sn1] V. Snaith, “Algebraic K-groups as Galois modules”. Birkhauser, Progress in Math. 206 (2002). | Zbl 1011.11074

[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 0545.12009

[W1] A. Wiles, The Iwasawa conjecture for totally real fields. Annals of Math. 141 (1990), 493–540. | MR 1053488 | Zbl 0719.11071

[W2] A. Wiles, On a conjecture of Brumer. Annals of Math. 131 (1990), 555–565. | MR 1053490 | Zbl 0719.11082