Local Non Abelian Class Field Theory

Year 2024, Early Access, 1 - 20

Sevan Bedikyan

https://doi.org/10.15672/hujms.1391474

Abstract

The “local class field theory”, which can be defined as the description of the extensions of a given local field K in terms of the algebraic and analytic objects depending ony 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 some non-abelian extensions of the base field. On the other hand, Ikeda and Serbest extended Fesenko’s works to construct a non-abelian local class field theory. But there is a restriction for the base field K.
In this study, we extended Ikeda-Serbest’s construction of the local reciprocity map for a certain local field to any local field. Also we have shown that the extended map satisfies the certain functoriality and ramification theoretic properties.

Keywords

Local fields, class field theory, extensions of profinite groups

References

• [1] S. Bedikyan, Abelyen olmayan yerel sınıf cisim kuramı üzerine, PhD thesis, Mimar Sinan Fine Arts University, 2013.