From the classical to the noncommutative Iwasawa theory (for totally real number fields) (Q2900374)

This article is an introduction to the main conjecture of noncommutative Iwasawa theory for totally real number fields.NEWLINENEWLINELet \(F\) be a totally real number field and let \(p\) be an odd prime. Let \(F_{\infty} / F\) be a totally real extension of \(F\) that contains the cyclotomic \(\mathbb Z_p\)-extension of \(F\) and such that only finitely many primes ramify in \(F_{\infty}\). Let \(\Sigma\) be a finite set of places of \(F\) containing all these primes and the archimedean places. Assume further that \(G := Gal(F_{\infty}/F)\) is a \(p\)-adic Lie group (i.e. the extension \(F_{\infty}/F\) is ``admissible''). Let \(X_{\Sigma}\) be the Galois group of the extension \(M_{\Sigma} / F_{\infty}\), where \(M_{\Sigma}\) denotes the maximal abelian pro-\(p\)-extension of \(F_{\infty}\) unramified outside \(\Sigma\). The aim of Iwasawa theory in this context is to understand \(X_{\Sigma}\) as a module over the Iwasawa algebra \(\Lambda(G) := \mathbb Z_p[[G]]\).NEWLINENEWLINEThe article starts with the classical result of \textit{A. Wiles} [Ann. Math. (2) 131, No. 3, 493--540 (1990; Zbl 0719.11071)], where \(G = H \times \Gamma\) is abelian, \(H\) is finite and \(p \nmid |H|\) (and \(\Gamma \simeq \mathbb Z_p\) is the Galois group of the cyclotomic extension). In this case the main conjecture of Iwasawa theory can be stated as an equality of two ideals attached to each irreducible character \(\chi\) of \(H\). The author then explains how to generalize this to the more general setting above. For this he introduces the required facts on \(K\)-theory of Iwasawa algebras and the (noncommutative) determinant functor following the (comparatively) elementary construction of \textit{T. Fukaya} and \textit{K. Kato} [Proceedings of the St. Petersburg Mathematical Society, Vol. 12, Transl., Ser. 2. Am. Math. Soc. 219, 1--85 (2006; Zbl 1238.11105)]. The Iwasawa module \(X_{\Sigma}\) is then replaced with a perfect complex of \(\Lambda(G)\)-modules. The author does a good job in explaining the relationship between the classical and the modern formulation of the main conjecture.NEWLINENEWLINEHe finally quotes the recent achievements in the area, namely the author's result [Invent. Math. 193, No. 3, 539--626 (2013; Zbl 1300.11112)] on the main conjecture if Iwasawa's \(\mu\)-invariant vanishes. Note that in the case of \(p\)-adic Lie groups of dimension \(1\), this has independently been established by \textit{J. Ritter} and \textit{A. Weiss} [J. Am. Math. Soc. 24, No. 4, 1015--1050 (2011; Zbl 1228.11165)].NEWLINENEWLINEFor the entire collection see [Zbl 1237.11001].