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Equivariant Main Conjecture, Fitting ideals and annihilators in Iwasawa theory. (Conjecture principale équivariante, idéaux de Fitting et annulateurs en théorie d’Iwasawa.) (French) Zbl 1098.11054
This article produces annihilators for so-called higher class groups, by exploiting the Equivariant Main Conjecture as proposed and proved by Ritter and Weiss. (The setting is as follows: \(K/k\) is an abelian extension of totally real fields, \(G\) is the Galois group and \(p\) is a fixed odd prime.)
In fact more is proved: the author determines in Theorem 4.3 the Fitting ideal of the \({\mathbb Z}_p[G]\)-module \(C(S,K,m):=H^2(G_S(K),{\mathbb Z}_p(m))\) for \(m>0\) positive and even, where \(S\) denotes the set of places that are over \(p\) or ramify in \(K/k\). From the Voevodsky-Rost theorem (see e.g. Theorem 70 in C. Weibel’s survey [Handbook of \(K\)-theory, Volume 2. Berlin: Springer, 139–190 (2005; Zbl 1097.19003)], we know that \(K_{2m-2}(O_K)\{p\}\) is isomorphic to \(C(K,m):=H^2(G_{S_p}(K),\mathbb{Z}_p(m))\) where \(S_p\) denotes the places over \(p\). Since \(C(K,m)\) maps injectively to \(C(S,K,m)\), it follows that the Fitting ideal of \(C(S,K,m)\) annihilates the group \(K_{2m-2}(O_K)\{p\}\). This is a refined version of the Coates-Sinnott conjecture. We should point out that the proof of the Equivariant Main Conjecture makes serious use of the nullity of the \(\mu\)-invariant of \(K\) at \(p\).
The author offers an interesting independent approach to the so-called envelope \(Y_\infty\) of the Iwasawa module in question and the resulting short exact sequence \(0 \to X_\infty \to Z_\infty \to z_\infty \to 0\), which leads to the definition of the invariant \(\mho_S\) of Ritter and Weiss and to the formulation of the Equivariant Main Conjecture. (The torsion module \(Z_\infty\) is an explicit quotient of \(Y_\infty\) and of finite projective dimension over \(\mathbb{Z}_p[G_\infty]\), where \(G_\infty\) is the Galois group of \(K_\infty\) over \(k\); the module \(z_\infty\) is an explicit quotient of \(\Delta G_\infty\).) This approach uses earlier work of the author [“Formations de classes et modules d’Iwasawa”, Number theory, Proc. Journ. Arith., Noordwijkerhout/Neth. 1983, Lect. Notes Math. 1068, 167–185 (1984; Zbl 0543.12007)]; a complete and succinct proof of the Equivariant Main Conjecture is given on p. 651f. For the proof of J. Ritter and A. Weiss, see [Manuscr. Math. 109, No. 2, 131–146 (2002; Zbl 1014.11066)].
The next steps in the paper under review are: calculation of the Fitting ideal of \(E^1(X_\infty)\) (the Iwasawa adjoint of \(X_\infty\)), twisting, and descent. (For a related approach to the calculation of Fitting ideals at infinite level, see the reviewer’s paper in [Math. Z. 246, No. 4, 733–767 (2004; Zbl 1067.11067)]). Under a fairly restrictive condition (G) (see p. 666), descent is possible in the untwisted situation, and one is able to show part of the Brumer conjecture. The main result (Thm. 4.3) was likewise proved by Burns and the reviewer (even a little bit more generally, allowing \(m\) to be odd and then considering minus parts), see Cor. 2 to Theorem 5.2 in [Doc. Math., J. DMV Extra Vol., 157–185 (2003; Zbl 1142.11371)],. One should also compare Cor. 12.5 and Remark 12.6 in M. Kurihara’s paper [J. Reine Angew. Math. 561, 39–86 (2003; Zbl 1056.11063)], which is concerned with the absolutely abelian case.

MSC:
11R23 Iwasawa theory
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R34 Galois cohomology
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References:
[1] D. Barsky, Sur la nullité du \(μ \)-invariant d’Iwasawa des corps totalement réels, prépublication (2005).
[2] D. Burns & C. Greither, On the Equivariant Tamagawa Number Conjecture for Tate motives. Invent. Math. 153 (2003), no. 2, 303-359. · Zbl 1142.11076
[3] D. Burns & C. Greither, Equivariant Weierstrass Preparation and values of \(L\)-functions at negative integers. Doc. Math. (2003), Extra Vol., 157-185. · Zbl 1142.11371
[4] D. Benois & T. Nguyen Quang Do. Les nombres de Tamagawa locaux et la conjecture de Bloch et Kato pour les motifs \({\mathbb{Q}}(m)\) sur un corps abélien. Ann. Sci. ENS 35 (2002), 641-672. · Zbl 1125.11351
[5] J. Coates & W. Sinnott, An analogue of Stickelberger’s theorem for the higher \(K\)-groups. Invent. Math. 24 (1974), 149-161. · Zbl 0282.12006
[6] P. Deligne & K. Ribet, Values of abelian \(L\)-functions at negative integers. Invent. Math 59 (1980), 227-286. · Zbl 0434.12009
[7] 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
[8] C. Greither, Some cases of Brumer’s conjecture. Math. Zeit. 233 (2000), 515-534. · Zbl 0965.11047
[9] C. Greither, Computing Fitting ideals of Iwasawa modules. Math. Z. 246 (2004), no. 4, 733-767. · Zbl 1067.11067
[10] 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. · Zbl 1044.11095
[11] A. Huber & G. Kings, Equivariant Bloch-Kato Conjecture and non abelian Iwasawa Main Conjecture. ICM 2002, vol. II, 149-162. · Zbl 1020.11067
[12] Y. Ihara, On Galois representations arising from towers of coverings of \({\mathbb{P}}^1 \ \lbrace 0, 1, ∞ \rbrace \). Invent. Math. 86 (1986), 427-459. · Zbl 0595.14020
[13] K. Iwasawa, On \({\mathbb{Z}}_{ℓ }\)-extensions of algebraic number fields. Annals of Math. 98 (1973), 246-326. · Zbl 0285.12008
[14] U. Jannsen, Iwasawa modules up to isomorphism. Adv. Studies in Pure Math. 17 (1989), 171-207. · Zbl 0732.11061
[15] K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil \(L\)-functions via \(B_{\rm dR}\). I. Arithmetic algebraic geometry (Trento, 1991). 50-163, Lecture Notes in Math., 1553, Springer, Berlin, 1993. · Zbl 0815.11051
[16] M. Kurihara, Iwasawa theory and Fitting ideals. J. Reine Angew. Math. 561 (2003), 39-86. · Zbl 1056.11063
[17] M. Kurihara, On the structure of ideal class groups of \(CM\) fields. Doc. Math. (2003), Extra Vol., 539-563. · Zbl 1135.11339
[18] 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. · Zbl 0863.19003
[19] M. Le Floc’h, On Fitting ideals of certain étale \(K\)-groups. K-Theory 27 (2002), 281-292. · Zbl 1083.11073
[20] B. Mazur & A. Wiles, Class fields of abelian extensions of \({\mathbb{Q}}\). Invent. Math. 76 (1984), 179-330. · Zbl 0545.12005
[21] T. Nguyen Quang Do, Formations de classes et modules d’Iwasawa. Dans “Number Theory Noordwijkerhout”, Springer LNM 1068 (1984), 167-185. · Zbl 0543.12007
[22] T. Nguyen Quang Do, Sur la \({\mathbb{Z}}_p\)-torsion de certains modules galoisiens. Ann. Inst. Fourier 36 (1986), no. 2, 27-46. · Zbl 0576.12010
[23] T. Nguyen Quang Do, Analogues supérieurs du noyau sauvage. J. Théorie des Nombres Bordeaux 4 (1992), 263-271. · Zbl 0783.11042
[24] T. Nguyen Quang Do, Quelques applications de la Conjecture Principale Equivariante, lettre à M. Kurihara (15/02/02). · Zbl 1098.11054
[25] J. Neukirch, A. Schmidt & K. Wingberg, Cohomology of Number Fields. Grundlehren 323, Springer, 2000. · Zbl 0948.11001
[26] K. Ribet, Report on \(p\)-adic \(L\)-functions over totally real fields. Astérisque 61 (1979), 177-192. · Zbl 0408.12016
[27] J. Ritter & A. Weiss, The Lifted Root Number Conjecture and Iwasawa theory. Memoirs AMS 157/748 (2002). · Zbl 1002.11082
[28] J. Ritter & A. Weiss, Towards equivariant Iwasawa theory. Manuscripta Math. 109 (2002), 131-146. · Zbl 1014.11066
[29] J. Rognes & C.A. Weibel, Two-primary algebraic \(K\)-theory of rings of integers in number fields. J. AMS (1) 13 (2000), 1-54. · Zbl 0934.19001
[30] P. Schneider, Über gewisse Galoiscohomologiegruppen. Math. Zeit 168 (1979), 181-205. · Zbl 0421.12024
[31] 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
[32] V. Snaith, “Algebraic \(K\)-groups as Galois modules”. Birkhauser, Progress in Math. 206 (2002). · Zbl 1011.11074
[33] V. Snaith, Relative \(K_0,\) Fitting ideals and the Stickelberger phenomena, preprint (2002). · Zbl 1108.19001
[34] J. Tate, “Les conjectures de Stark sur les fonctions \(L\) d’Artin en \(s=0\)”. Birkhauser, Progress in Math. 47 (1984). · Zbl 0545.12009
[35] A. Wiles, The Iwasawa conjecture for totally real fields. Annals of Math. 141 (1990), 493-540. · Zbl 0719.11071
[36] A. Wiles, On a conjecture of Brumer. Annals of Math. 131 (1990), 555-565. · Zbl 0719.11082
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