Title: The norm residue symbol for higher local fields,
Abstract: Since the development of higher local class field theory, several explicit
reciprocity laws have been constructed. In particular, there are formulas
describing the higher-dimensional Hilbert symbol given, among others, by M.
Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for
the higher-dimensional Kummer pairing associated to certain (one-dimensional)
$p$-divisible groups.
In this paper we construct an explicit reciprocity law describing the Kummer
pairing associated to any (one-dimensional) formal group. The formulas are a
generalization to higher-dimensional local fields of Kolyvagin's reciprocity
laws. The formulas obtained describe the values of the pairing in terms of
multidimensional $p$-adic differentiation, the logarithm of the formal group,
the generalized trace and the norm on Milnor K-groups.
In the second part of this paper, we will apply the results obtained here to
give explicit formulas for the generalized Hilbert symbol and the Kummer
pairing associated to a Lubin-Tate formal group. The results obtained in the
second paper constitute a generalization to higher local fields, of the
formulas of Artin-Hasse, K. Iwasawa and A. Wiles.