## Explicit reciprocity laws.(English)Zbl 0358.12001

Astérisque 41-42, 7-17 (1977).
This paper is written in the spirit of the important work of K. Iwasawa [J. Math. Soc. Japan 16, 42–82 (1964; Zbl 0125.29207)]. The authors study an analogue of the Hilbert norm residue symbol for the fields $$\Phi_n=\mathbb Q_p(G_{\pi^{n+1}})$$ where $$(G_{\pi^{n+1}}$$ is the kernel of $$[\pi^{n+1}]_G$$, $$G$$ is some Lubin-Tate formal group associated with $$\pi$$, which is a fixed local parameter for $$\mathbb Z_p$$. The authors introduce analogues of the maps $$\Psi_n$$, first used by Iwasawa in the cyclotomic case. Some of the explicit reciprocity laws when $$n=0$$ are given. The computation of $$\Psi_0$$ gives an analogue of Artin-Hasse law for $$n=0$$.
For the entire collection see [Zbl 0341.00005].
Reviewer: Gr. Todorova

### MSC:

 11S31 Class field theory; $$p$$-adic formal groups

Zbl 0125.29207