Nonabelian local reciprocity maps. (English) Zbl 1039.11085

Fesenko, Ivan

Miyake, Katsuya (ed.), Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3–12, 1998. Tokyo: Mathematical Society of Japan (ISBN 4-931469-11-6/hbk). Adv. Stud. Pure Math. 30, 63-78 (2001).

The author constructs non-Abelian reciprocity maps for arithmetically profinite Galois extensions \(L/F\) of local fields. The Neukirch-Iwasawa norm map \({\mathcal N}:\text{Gal}(L/F)\to U^\lozenge\), where \(U^\lozenge\) is a certain subquotient of the unit group of the field of norms, and the Hazelwinkel homomorphism is generalized to give an explicit inverse \({\mathcal H}\) of \({\mathcal N}\). These constructions encompass both the metabelian class field theory of Koch and de Shalit, and the soluble theory of Gurevich.
For the entire collection see [Zbl 0968.00031].

Reviewer: M. E. Keating (London)

MSC:

11S31

Class field theory; \(p\)-adic formal groups

Keywords:

non-Abelian reciprocity; arithmetically profinite extension; local field