Ramification theory in non-abelian local class field theory

Abstract

To the memory of I. M. Gelfand 1. Introduction. Let K be a local field, that is, a complete discrete valuation field with finite residue class field κ K of q = p f elements. For technical reasons, throughout the paper we shall assume that the multiplicative group µ µ µ p (K sep ) of all pth roots of unity in K sep satisfies µ µ µ p (K sep ) ⊂ K. Fix a Lubin-Tate splitting ϕ over K. That is, we fix an extension ϕ of the Frobenius automorphism of K nr to K sep (for details, cf. [Ko-dS]). In a sequence of papers [Ik-Se-1, Ik-Se-2, Ik-Se-3], following the idea of Fesenko developed in [Fes-1, Fes-2, Fes-3], we have constructed the non-abelian local reciprocity map Φ Φ Φ (ϕ) K for K, which is an isomorphism from the absolute Galois group G K of K onto a certain topological group ∇ (ϕ) K which depends on the choice of the Lubin-Tate splitting ϕ.The aim of the present paper is to study the ramification-theoretic properties of the map Φ Φ Φ (ϕ) K . We prove (in Theorems 4.15 and 4.16) that Φ Φ Φ (ϕ) K is compatible with the refined higher ramification "filtration" of the absolute Galois group G K of K (cf. 4.1) and the refined "filtration" of ∇ (ϕ) K (cf. 4.2). The organization of the paper is as follows. In Section 2, we collect the necessary results from the theory of local fields. In Section 3, we briefly review the main results of [Ik-Se-2] on the generalized Fesenko reciprocity map, and then sketch the construction of the non-abelian local reciprocity map Φ Φ Φ (ϕ) K following [Ik-Se-3]. In the last section, we first introduce the refined filtrations on G K and on ∇ (ϕ) K and then prove the main results of the paper, which are stated as Theorems 4.15 and 4.16.

Class field theory, its three main generalisations, and applications

“…Koch-de Shalit's metabelian local class field theory, which is of SCFT type, may be viewed as one of the developments related to this theorem. A related development is general arithmetic non-abelian CFT, which is of GCFT type, it includes local theory for arithmetically profinite extensions, with its existence theorem and compatibility with ramification theory (Fesenko [7], Ikeda-Serbest [23][24][25][26]), and some global theory (Ikeda [22]).…”

Fesenko reciprocity map

“…In a companion paper [10], we shall extend the construction of Fesenko to any Galois extension of K (in a fixed K sep ), and construct the non-abelian local class field theory. Thus, we feel that, the present paper together with [1,2,3] should be viewed as the technical and theoretical background, an introduction, as well as an appendix to our companion paper [10] on "generalized Fesenko theory".…”

Fesenko reciprocity map

“…A similar theory was announced by Laubie in [14], which is an extension of the paper [13] by Koch and de Shalit. The relationship of the Laubie theory with our generalized Fesenko theory will also be investigated in [10].…”

“…Thus, we feel that the present paper together with [1,2,3] should be viewed as the technical and theoretical background, an introduction, as well as an appendix to the paper [10] on "generalized Fesenko theory". A similar theory was announced by Laubie in [14], which is an extension of the paper [13] by Koch and de Shalit.…”

“…In a related paper (see [10]), we shall extend Fesenko's construction to any Galois extension of K (in a fixed K sep ), and construct the non-Abelian local class field theory. Thus, we feel that the present paper together with [1,2,3] should be viewed as the technical and theoretical background, an introduction, as well as an appendix to the paper [10] on "generalized Fesenko theory".…”