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Metabelian local class field theory. (English) Zbl 0858.11061

The paper is a continuation of the first author’s paper [Proc. 2nd Gauss Symp., Symposia Gaussiana, 287-300 (1995; Zbl 0851.11066)]. A new tool which comes into play is R. F. Coleman’s power series [Invent. Math. 124, No. 1-3, 215-241 (1966; Zbl 0851.11030)]. Let \(K\) be a local field with finite residue field \(k\). Let \(\overline k\) be the separable closure of \(k\). Fix a lifting \(\varphi \in G_K\) of the Frobenius automorphism. To a finite extension \(L\) of \(K\) on which \(\varphi\) acts trivially one can attach a unique Lubin-Tate formal group \(F_L\). Denote by \(\{u\} (X)\) the residue series of the endomorphism of \(F_L\) associated to a unit \(u\) of \(K\). Let \(G_d\) be the set of all pairs \((\pi^nu,h(X)) \in K^* \times k[[X]]^*\) satisfying \(\varphi^d h(X)=h(X)\{u\} (X)/X\), where \(\varphi\) acts as the Frobenius automorphism on the coefficients and trivially on \(X\). Denote by \(G(L)\) the inverse limit of \(G_d\), \(d\in \mathbb{N}\), with respect to a natural transition map \(G_{dd'}\to G_d\). Using Coleman’s power series, the authors define a metabelian norm map \(N_{K'/K} : G(K')\to G(K)\) for a finite extension \(K'\) of \(K\), \(K'\subset L\).
The main result is that the correspondence \(L\to N_{L/K} G(L)\) is one-to-one between all finite metabelian extensions of \(K\) and all open subgroups of finite index in \(G(K)\) and there is a canonical isomorphism \(G(K)/N_{L/K} G(L) \simeq G(L/K)\) which is compatible with field extensions.
The paper is excellently written. It would be important to find an exposition of the theory without using formal groups and generalize it to the case of perfect residue fields.

MSC:

11S31 Class field theory; \(p\)-adic formal groups
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