Hilbert pairing on Lorentz formal group

Abstract

In this paper, we construct explicit pairing in Cartier series for formal Lorentz groups of the form: (X + Y + XY )/(1 + c2XY ), where c is unit of the ring of integers of the local field. At the same time, except for trivial properties such as bilinearity we prove important property invariance, i. e. independence from selecting a variable. These properties, with the aid of pairing above, allow us to explicitly construct the generalized Hilbert symbol for formal Lorentz groups over rings of integers of local fields. In the first section of the paper are the basic notation and auxiliary results on endomorphisms of formal module, formal logarithm and exponent are constructed. In the second section, we build the main function, which plays an important role in the construction of the pairing an Artin Hasse function. In the last section we define Hilbert pairing on modules of Cartier curves explicitly, also we check bilinearity of pairing (see lemma 3.1) and variable substitution formula for Artin Hasse function (lemma 3.2). Further in this section we prove invariance of the constructed paring regarding variable substitution X = g(Y ). Refs 15.