Explicit reciprocity laws. (English) Zbl 0358.12001

Coates, J.; Wiles, A.

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

Keywords:

analogue of the Hilbert norm residue symbol; Lubin-Tate formal group; explicit reciprocity laws; analogue of Artin-Hasse law