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

Abstract

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(\( \mathfrak{M} \)) 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_Fc,_m. Let be the multiplicative group of Cartier curves and _c be the formal analog of the module Fc(\( \mathfrak{M} \)). In the present paper, the formal symbol { ·, · }c : K_n()×_c → μ_Fc,_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.