Non-abelian local reciprocity law
This paper is the continuation of previous work of the authors [Algebra Anal. 20, No. 3, 112--162 (2008); translation in St. Petersburg Math. J. 20, No. 3, 407--445 (2009; Zbl 1206.11138); Algebra Anal. 20, No. 4, 118--159 (2008); translation in St. Petersburg Math. J. 20, No. 4, 593--624 (2009; Zbl 1206.11140)]. Let \(K\) be a field complete under a discrete valuation with finite residue field \(k\) of \(q=p^r\) elements. Fix an extension \(\varphi\) of the Frobenius automorphism of the maximal unramified extension \(K^{\text{un}}\) of \(K\) to the separable closure \(K^{\text{sep}}\) of \(K\). The authors construct the non-abelian reciprocity map \(\Phi_ K^{(\varphi)}\) for \(K\), which is an isomorphism of the absolute Galois group \(G_ K\) of \(K\) onto a topological group \(\Delta_{K,Y} ^ {(\varphi)}\) described in Section 6 of the paper. This is achieved under the assumption that the multiplicative group \(\mu_ p(K^{\text{sep}})\) of the \(p\)--th roots of unity of \(K^ {\text{sep}}\) are already contained in \(K\). It is also sketched the construction of the non-abelian local reciprocity map \(\Phi_ K^{(\varphi)}\) of \(K\) when \(K\) does not necessarily satisfy that \(\mu_ p(K^ {\text{sep}}) \subseteq K\) via non-abelian Schreier theory. After reviewing the main results on Fesenko reciprocity map and studying APF Galois extensions \(\Gamma _ d^ {(n)}/K\), the authors give the construction of \(\Phi_ K ^ {(\varphi)}\) in terms of the generalized reciprocity maps \(\Phi_ {\Gamma_ d^ {(n)}/K}^ {(\varphi)}\) and study the basic properties of this map. In the final section it is sketched the construction of the non-abelian local reciprocity map \(\Phi_ K^ {(\varphi_ 0)}\) for any local field by generating the non-abelian reciprocity map \(\Phi _ {K_ 0}^{(\varphi_ 0)}\) of \(K_ 0\) and the local Artin map \(\roman{Art}_ {K_ 0/K}\) of \(K_ 0/K\), where \(\zeta _ p\in K^ {\text{sep}}\) is a fixed primitive root of unity and \(K_ 0=K(\zeta _ p)\).
totally ramified arithmetically profinite (APF) Galois extension
scientific article; zbMATH DE number 5702983
scientific article