On a norm property of Hilbert symbol over polynomial formal module in multidimensional local field

Abstract

A polynomial formal group Fc(X, Y ) = X+Y +cXY over a two-dimensional local field K, which contains p-th root of unity, is considered. It acts naturally on the maximal ideal M of the ring of integers OK. There is a constructive Hilbert pairing {·, ·}c : K2(K)× Fc(M) → ξ c, where ξ c stands for a module of roots of [p]c (p-th degree isogeny of Fc) with respect to the formal summation. There is also a norm map of Milnor groups Norm: K2(L) → K2(K) for a two-dimensional local fields extension L/K. Elements of its image are called norms in K2(L). The main result of this work proves, in a constructive way, the so-called norm property of the pairing {·, ·}c: {x, β}c = 0 ⇐⇒ x is a norm in K2(K([p] −1 c (β))), where [p] −1 c (β) denotes roots of the equation [p]c = β. Refs 6.