manuscripta math. 132, 19–49 (2010) © Springer-Verlag 2010 K. Ilhan Ikeda, Erol Serbest Non-abelian local reciprocity law Received: 2 April 2009 / Revised: 12 October 2009 Published online: 4 February 2010 Abstract. Following our work on the generalized Fesenko reciprocity map, we construct the non-abelian local reciprocity map Φ (ϕ) K of a local field K as a certain isomorphism from the absolute Galois group G K of K onto a topological group ∇ (ϕ) K ,Y whose definition involves Fontaine–Wintenberger theory of field of norms, and build the non-abelian local class field theory over K in the sense of Fesenko and Koch. 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, all through the text except the last section, we assume that the multiplicative group µ p ( K sep ) of all pth roots of unity in K sep further satisfies 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. [13]). In this paper, which is the natural continuation of [11] and [12], we construct the non-abelian local reciprocity map Φ (ϕ) K for K , which is an isomor- phism from the absolute Galois group G K of K onto a certain topological group ∇ (ϕ) K ,Y (cf. (6.11) and (6.17) in Sect. 6) and furthermore study the basic functorial properties of Φ (ϕ) K . Moreover, we sketch the construction of the non-abelian local reciprocity map Φ (ϕ) K of K , where K is any local field not necessarily satisfying µ p ( K sep ) ⊂ K via non-abelian Schreier theory in the profinite regime. The organization of the paper is as follows. In Sect. 2, we briefly review the main results of [12] on the generalized Fesenko reciprocity map, which will play To the memory of Cahit Arf K. I. Ikeda (B ): Department of Mathematics, Yeditepe University, 26 Aˇ gustos Yerle¸ simi, ˙ Inönü Mah., Kayı¸ sdaˇ gı Cad., 34755 Kadıköy, Istanbul, Turkey. e-mail: ilhan.ikeda@yeditepe.edu.tr E. Serbest: Gümü¸ s Pala Mahallesi, Gümü¸ s Sok., Öz Aksu Sitesi C-2/39, 34160 Avcılar, Istanbul, Turkey. e-mail: erols73@yahoo.com E. Serbest: (current address): Department of Mathematics, Yeditepe University, 26 Aˇ gustos Yerle¸ simi, ˙ Inönü Mah., Kayı¸ sdaˇ gı Cad., 34755 Kadıköy, Istanbul, Turkey. e-mail: erol.serbest@yeditepe.edu.tr Mathematics Subject Classification (2000): 11S37 DOI: 10.1007/s00229-010-0336-6