# The norm residue symbol for higher local fields

## TL;DR

This paper develops explicit reciprocity laws for higher local fields, generalizing classical formulas by connecting Kummer pairings, multidimensional p-adic differentiation, and formal groups, with applications to Hilbert symbols and Lubin-Tate groups.

## Contribution

It constructs a general explicit reciprocity law for the Kummer pairing associated to any one-dimensional formal group over higher local fields, extending previous formulas.

## Key findings

- Formulas for Kummer pairings in higher local fields.
- Explicit descriptions of the generalized Hilbert symbol.
- Applications to Lubin-Tate formal groups.

## 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.