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 Fc(X, Y) = X +Y +cXY be a polynomial formal group, which defines the formal module Fc( $$ \mathfrak{M} $$ ) 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.