Equivariant Iwasawa theory and non-Abelian Stark-type conjectures (Q2840177)

This article offers attractive results, both at infinite level in Iwasawa towers (equivariant main conjectures) and at finite level: noncommutative versions of conjectures of Brumer and Stark, and of Coates and Sinnott. At infinite level, the setting is a \(\mathcal G\)-Galois extension \(\mathcal K/k\) such that \(\mathcal K\) is finite over the cyclotomic \(\mathbb Z_p\)-extension \(k_{\infty}\) and hence \(\mathcal G\) is an extension of \(\mathbb Z_p\) (usually written \(\Gamma\) in this context) by a finite group. At finite level, the setting is a finite Galois CM-extension \(K/k\). Most results assume the vanishing of Iwasawa's \(\mu\)-invariant.NEWLINENEWLINEThe author discusses three approaches to the Equivariant Main Conjecture EMC. The first is Ritter-Weiss theory, and the second comes from the work of \textit{J. Coates} et al. [Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)]. By establishing a certain isomorphism in the derived category of the category of \(\mathbb Z_p[[\mathcal G]]\)-modules, the author shows that these both formulations are in a certain precise sense equivalent. Ritter and Weiss themselves proved EMC in their formulation (note that this includes the non-abelian case), and Kakde's proof of the EMC is within the framework of Coates et al. [loc.cit.]. The author then discusses a third approach which is still more recent, due to Popescu and the reviewer (see, \url{arXiv:1103.3069} or J. Algebr. Geom., to appear). It has the potential advantage of being more explicit, but up to now it was restricted to the abelian situation. One of the merits of the paper under review is to extend this to the non-abelian case, and Nickel again proves that this third formulation is equivalent (in a precise sense) to the previously mentioned ones. Whilst there is already a thorough discussion by \textit{O. Venjakob} in [Springer Proc. Math. Stat. 29, 159--182 (2013; Zbl 1270.11112)] relating the first two approaches, the present treatment is simple and encompassing, taking care of all three approaches. This will be extremely useful to the community working in this area.NEWLINENEWLINENEWLINEThe two last sections \S4-5 are concerned with the non-abelian Brumer-Stark conjecture and the non-abelian Coates-Sinnott conjecture, respectively. For the statement of the former and the discussion of the fine points that distinguish the non-abelian case from the abelian one, we would like to refer to \textit{A. Nickel}'s earlier papers and their reviews, mainly to [J. Algebra 323, No. 10, 2756--2778 (2010; Zbl 1222.11132)]. The main result in \S4 is Theorem 4.5. It says that a certain equivariant L-value \(\theta_S^T\) is, roughly speaking, in the (noncommutative) Fitting ideal of the dual of the class group of \(K\). To be more precise, instead of the class group one takes the minus part of the \(p\)-part of the \(T\)-class group. Here \(T\) is a nonempty auxiliary set of places of \(K\) that pervades the whole theory. Its main role is to eliminate torsion in unit groups. The set \(T\) is also needed in the construction of \(\theta_S^T\). The set \(S\) is in a way canonical: it has to contain all ramified places. But there is another severe restriction (so-called imprimitivity): similarly as in work of Popescu and the reviewer one needs to have all \(p\)-adic places in \(S\). This is not an issue in \S5 where the focus switches, morally, to higher \(K\)-groups. In fact these only linger in the background. The ``Strong C-S conjecture'' says \(\theta^T_S(1-n)\) is in the Fitting ideal of \(H^2(C^T_S(\mathbb Z_p(n)))\). The complex \(C^T_S(\mathbb Z_p(n))\) is obtained from the cohomology complex of \(\mathbb Z_p(n)\) by a certain adjustment (cone construction) using the set \(T\), the point again being to get rid of torsion. The element \(\theta^T_S(1-n)\) is a twisted version of \(\theta_S^L\) (see above). After various reductions, the author establishes equivalent conditions for the Strong C-S conjecture and shows that the conjecture does hold for example if \(K\) is CM. As a consequence, the author proves a more traditional version without the auxiliary set \(T\) but involving also an \(H^1\) term. In the transition, he uses a result on the behaviour of Fitting ideals in 4-term sequences with c.t.~middle terms (see Prop.~5.3(ii) in Nickel's article mentioned earlier), whose abelian version goes back to the reviewer (Prop.~1 in [Math. Z. 246, No. 4, 733-767 (2004; Zbl 1067.11067)]).NEWLINENEWLINEThis is a nicely written, interesting and useful paper.