Generalized Fesenko Reciprocity Map

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 κK of q = p f elements, satisfying μp(K) ⊂ K and K ⊂ L ⊂ Kφd , where the residue-class degree [κL : κK ] is equal to d. More precisely, for such extensions L/K and a fixed Lubin–Tate splitting φ over K, a 1-cocycle Φ (φ) L/K : Gal(L/K) → K/NL0/KL × 0 × U X̃(L/K)/YL/L0 , where L0 = L ∩ Knr, is constructed, and its functorial and ramification-theoretic properties are studied. The case of d = 1 recovers the theory of Fesenko. Let K be a local field (that is, a complete discrete valuation field) with a finite residueclass field κK of q = p elements. Assume that μp(K) ⊂ K. We fix a Lubin–Tate splitting φ over K (see [10]). In [1, 2, 3], Fesenko introduced a very general non-Abelian local reciprocity map Φ L/K : Gal(L/K) → U X̃(L/K) /YL/K defined for any totally ramified infinite APF Galois extension L/K satisfying K ⊂ L ⊂ Kφ, which generalized the earlier non-Abelian local class field theories of Koch and de Shalit [10] and Gurevich [7]. 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 ⊂ Kφd , where d is the residue-class degree [κL : κK ]. More precisely, for such extensions L/K, we construct a 1-cocycle, Φ L/K : Gal(L/K) → K /NL0/KL × 0 × U X̃(L/K)/YL/L0 , where L0 = L∩K, and study its functorial and ramification-theoretic properties. Note that the case where d = 1 recovers the theory of Fesenko. The organization of this paper is as follows. In the first section, we briefly review the necessary background material from the Fontaine–Wintenberger theory of fields of norms. In the second section, we introduce the generalized Fesenko reciprocity map Φ L/K of an extension L/K that is an infinite APF Galois extension satisfying K ⊂ L ⊂ Kφd , where the residue-class degree [κL : κK ] is equal to d, and study its functorial and ramification-theoretic properties. The material and results of this paper play a fundamental role in our construction of non-Abelian local class field theory [9], which generalizes also the Laubie theory [11]. 2000 Mathematics Subject Classification. Primary 11S37.