Abstract. In recent papers, Fesenko has defined the non-Abelian local reciprocity map for every totally ramified arithmetically profinite (APF) Galois extension of a given local field K, by extending the work of Hazewinkel and Neukirch-Iwasawa. The theory of Fesenko extends the previous non-Abelian generalizations of local class field theory given by Koch-de Shalit and by A. Gurevich. In this paper, which is research-expository in nature, we give a detailed account of Fesenko's work, including all the skipped proofs.

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.

No abstract

Abstract. The paper is a natural continuation and generalization of the works of Fesenko and of the authors. Fesenko's theory is carried over to infinite APF Galois extensions L over a local field K with a finite residue-class field is constructed, and its functorial and ramification-theoretic properties are studied. The case of d = 1 recovers the theory of Fesenko.

In this paper, which is the natural continuation and generalization of [1, 2, 3] and [8], we extend the theory of Fesenko to infinite AP F -Galois extensions L over a local field K, with finite residue-class field κK of q = p f elements, satisfying µ µ µp(K sep ) ⊂ K and K ⊂ L ⊂ K ϕ d where the residue-class degree [κL : κK ] = d. More precisely, for such extensions L/K, fixing a Lubin-Tate splitting ϕ over K, we construct a 1-cocycle, Φ Φ Φ (ϕ)