The article under review concerns the non-commutative Iwasawa main conjecture over totally real fields for the Tate motive. Historically, the first version of this main conjecture was stated for the cyclotomic \(\mathbb{Z}_p\)-extension of a totally real number field and proved by A. Wiles [“The Iwasawa conjecture for totally real fields”, Ann. Math. 131, No. 3, 493–540 (1990; Zbl 0719.11071)]. Based on recent work [J. H. Coates et al., Publ. Math., Inst. Hautes Étud. Sci. 101, 163–208 (2005; Zbl 1108.11081)] and [T. Fukaya and K. Kato, “A formulation of conjectures on \(p\)-adic zeta functions in non-commutative Iwasawa theory”, Proceedings of the St. Petersburg Mathematical Society, N. N. Uraltseva (ed.), Vol. 12, Transl., Ser. 2. Am. Math. Soc. 219, 1–85 (2006; Zbl 1238.11105)] its formulation has been extended (also for more general types of motives) to admissible \(p\)-adic Lie-extensions \(k_\infty\) of a number field \(k\) (i.e.\(k_\infty/k\) is an (infinite) Galois extension, in which at most finitely many places of \(k\) ramify, containing the cyclotomic \(\mathbb{Z}_p\)-extension of \(k\), and whose Galois group is a \(p\)-adic Lie group).
At the moment of the reviewing the article is already outdated. Indeed, it is a precursor of the article [”The main conjecture of Iwasawa theory for totally real fields”, preprint, arXiv:1008.0142] by the same author. While the paper under discussion only proves the conjecture in special cases (concerning the type of admissible \(p\)-adic Lie extensions), the recent one contains a full proof of it for all admissible \(p\)-adic Lie extensions. Practically simultaneously, J. Ritter and A. Weiss [J. Am. Math. Soc. 24, No. 4, 1015–1050 (2011; Zbl 1228.11165)] have given a proof of the same conjecture for one-dimensional \(p\)-adic Lie-extensions, which can be combined with Burns’ insight in [D. Burns, “On main conjectures in non-commutative Iwasawa theory and related conjectures”, Preprint (2011)] based on Fukaya and Kato’s result Prop.1.5.1 in [Zbl 1238.11105] on the compatibility of \(K_1\) with certain projective limits to obtain the conjecture for general (admissible) \(p\)-adic Lie-extensions. See also the reviewer’s work [“On the work of Ritter and Weiss in comparison with Kakde’s approach,” preprint, arXiv:1110.6366] for a survey and comparison. In all cases the vanishing of Iwasawa’s \(\mu\)-invariant has to be assumed. Various other special cases had been known by the work of (in alphabetical order) [T. Hara, J. Number Theory 130, No. 4, 1068–1097 (2010; Zbl 1196.11148); Duke Math. J. 158, No. 2, 247–305 (2011; Zbl 1238.11100)], K. Kato [Iwasawa theory of totally real fields for Galois extensions of Heisenberg type, preprint (2007) per bibl.], J. Ritter and A. Weiss e.g., [“Non-abelian pseudomeasures and congruences between abelian Iwasa \(L\)-functions”, Pure Appl. Math. Q. 4, No. 4, 1085–1106 (2008; Zbl 1193.11104)].
In his article Kakde first reviews the commutative case of the main conjecture which is a consequence of Wiles’ work mentioned above. Then he explains the strategy of Burns and Kato how to derive the validity of the non-commutative main conjecture (for extensions of dimension \(1\)) from the validity of the commutative main conjecture plus the verification of new types of congruences between Serre’s \(p\)-adic pseudomeasures associated to various intermediate abelian \(p\)-adic Lie extensions. These congruences arise from an purely algebraic determination of the first algebraic \(K\)-group of the non-commutative Iwasawa algebra (and its canonical localisation) in terms of commutative Iwasawa algebras; this part involves technically the integral group logarithm of (independently) B. Oliver and M. Taylor. In order to check the congruences the \(q\)-expansion principle of Deligne and Ribet is used.