An Explicit Form of the Hilbert Symbol for Polynomial Formal Groups Over a Multidimensional Local Field. I

Let K be a multidimensional local field with characteristic different from the characteristic of its residue field, c be a unit of K, and F c (X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module F c ( 𝔐 ) over the maximal ideal of the ring of integers in K. Assume that K contains the group of roots of the isogeny [ p m ] c (X), which we denote by μ F c , m . Let be the multiplicative group of Cartier curves and c be the formal analog of the module Fc( 𝔐 ). In the present paper, the formal symbol ·, · c : K n ( )× c → μ F c , m is constructed and its basic properties are checked. This is the first step in the construction of an explicit formula for the Hilbert symbol.