Local non-abelian class field theory

Abstract

The “local class field theory”, which can be defined as the description of the extensions of a given local field K with finite residue field of q = pf elements in terms of the algebraic and analytic objects depending only on the base K is one of the central problems of modern number theory. The theory developed for the abelian extensions, around the fundamental works of Artin and Hasse in the first quarter of the 20th century. It is natural to ask if one could construct this theory including the non-abelian extensions of the base field. There are two approches to this problem. One approach is based on the ideas of Langlands, and the other on Koch. Koch’s method was later generalized by Fesenko and Koch-de Shalit for specific type of non-abelian extensions of the base field. Laubie extended Koch-de Shalit’s work and constructed a local non-abelian class field theory for K. On the other hand, İkeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory for K, containing a pth root of unity. In this study, we extended İkeda-Serbest’s construction of the local reciprocity map for K containing a pth root of unity to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties. © 2025, Hacettepe University. All rights reserved.