A norm pairing in formal modules

A pairing of the multiplicative group of a local field (a finite extension of the field of $p$-adic numbers $\mathbf Q_p$) with the group of points of a Lubin–Tate formal group is defined explicitly. The values of the pairing are roots of an isogeny of the formal group. The main properties of this pairing are established: bilinearity, invariance under the choice of a local uniformizing element, and independence of the method of expanding elements into series with respect to this uniformizing element.
These properties of the pairing are used to prove that it agrees with the generalized Hilbert norm residue symbol when the field over whose ring of integers the formal group is defined is totally ramified over $\mathbf Q_p$. This yields an explicit expression for the generalized Hilbert symbol on the group of points of the formal group.
Bibliography: 12 titles.