Fesenko reciprocity map

Abstract

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.

Generalized Fesenko reciprocity map

“…is defined by Abelian local class field theory on the first component and by formulas (5.5) and (5.7) in [8] on the second component:…”

“…In [8], we studied the basic functorial and ramificationtheoretic properties of the reciprocity map of Fesenko. In this paper, which is a natural continuation and generalization of [1,2,3] and [8], we extend the theory of Fesenko to infinite APF Galois extensions L/K satisfying K ⊂ L ⊂ K ϕ d , where d is the residue-class degree [κ L : κ K ]. More precisely, for such extensions L/K, we construct a 1-cocycle,…”

“…§1. Preliminaries on the Fontaine−Wintenberger field of norms For a brief review of APF extensions and the Fontaine-Wintenberger field of norms, we refer the reader to [8], and for detailed proofs to [5,6,12].…”

“…Note that, in general, T/K is not a normal extension! We denote L [12] or by Lemma 3.3 in [8], L/L 0 is an infinite totally ramified APF Galois extension satisfying L 0 ⊆ L ⊆ (L 0 ) ϕ . Thus, the Fesenko theory developed in [1,2,3] and [8] …”

“…In Fesenko's theory, which was described in [1,2,3] and in detail in §5 of [8], an arrow φ (ϕ) M/K was defined for the extensions M/K, where M/K is a totally ramified APF Galois extension satisfying K ⊂ M ⊂ K ϕ . Now, as a first step, we generalize this arrow to infinite APF Galois extensions L of K having residue-class degree d and satisfying…”