Fesenko Reciprocity Map

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. In a series of very interesting papers [1, 2, 3], Fesenko 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 [8] and Neukirch–Iwasawa [15]. “Fesenko theory” extends the previous non-Abelian generalizations of local class field theory given by Koch and de Shalit in [13] and by A. Gurevich in [7]. In this paper, which is research-expository in nature, we give a very detailed account of Fesenko’s work [1, 2, 3], thereby complementing those papers by including all the proofs. Let us describe how our paper is organized. In the first Section, we briefly review the Abelian local class field theory and the construction of the local Artin reciprocity map, following the Hazewinkel method and the Neukirch–Iwasawa method. In Sections 3 and 4, we follow [4, 5, 6], and [17] to review the theory of APF extensions over K, and sketch the construction of Fontaine–Wintenberger’s field of norms X(L/K) attached to an APF extension L/K. In order to do so, we briefly review the ramification theory of K in Section 2. Finally, in Section 5, we give a detailed construction of the Fesenko reciprocity map Φ L/K defined for any totally ramified and APF Galois extension L over K under the assumption that μp(K) ⊂ K, where p = char(κK), and investigate the functorial and ramification-theoretic properties of the Fesenko reciprocity maps defined for totally ramified and APF Galois extensions over K. In a related paper (see [10]), we shall extend Fesenko’s construction to any Galois extension of K (in a fixed K), 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”. 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]. Notation. Throughout this work, K denotes a local field (a complete discrete valuation field) with finite residue field OK/pK =: κK of qK = q = p elements with p a prime 2000 Mathematics Subject Classification. Primary 11S20, 11S31.