On the metabelian local Artin map. I: Galois conjugation law (Q2703139)

Let \(K\) be a non-archemidean local field. The work examines the action of an automorphism \(\widetilde\sigma\) of \(K^{sep}\) on a metabelian local Artin map as constructed by \textit{H. Koch} and \textit{E. de Shalit} [J. Reine Angew. Math. 478, 85-106 (1996; Zbl 0858.11061)]. More precisely, \(K^{sep}\) is a separable closure of \(K\) and the latter metabelian local Artin map is from a topological group \({\mathfrak G}(K,\varphi)\) to the Galois group of the maximal metabelian extension over \(K\). The group \({\mathfrak G}(K,\varphi)\) depends on \(K\) and a Lubin-Tate splitting \(\varphi\) of the exact sequence \(0\to I_K\to G_K\to\widehat \mathbb{Z}\to 0\). The automorphism \(\widetilde\sigma\) shifts \(K\) within \(K^{sep}\) and acts on \(\varphi\) by conjugacy. NEWLINENEWLINENEWLINEThe group \({\mathfrak G}(K,\varphi)\) is constructed by means of Coleman's power series [\textit{R. F. Coleman}, Invent. Math. 124, 215-241 (1996; Zbl 0851.11030)] and the authors examine the consequences of the shift of \(K\) for those power series in all detail. The author also announces to introduce a transfer law for the metabelian Artin map that is simular to the abelian one. NEWLINENEWLINENEWLINEHe remarks that it seems to be possible to recover non-abelian local class field theory by inductively extending the results of Koch and de Shalit to \(n\)-abelian extensions of \(K\). Indeed a thesis of \textit{Alexander Gurevich} [Description of Galois groups of local fields with aid of power series, Berlin (1997)] is about this induction step.