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

Nguyen Quang Do, Thong

J. Théor. Nombres Bordx. 17, No. 2, 643-668 (2005).

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.

Reviewer: Cornelius Greither (Neubiberg)

MSC:

11R23	Iwasawa theory
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R34	Galois cohomology

Keywords:

Main conjecture; Fitting ideals; cohomology

Citations:

Zbl 0543.12007; Zbl 1014.11066; Zbl 1067.11067; Zbl 1056.11063; Zbl 1097.19003; Zbl 1142.11371

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