Equivariant Main Conjecture, Fitting ideals and annihilators in Iwasawa theory
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 \textit{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 \textit{J. Ritter} and \textit{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 \textit{M. Kurihara's} paper [J. Reine Angew. Math. 561, 39--86 (2003; Zbl 1056.11063)], which is concerned with the absolutely abelian case.
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