Explicit formulae for the Hilbert symbol of a formal group over the Witt vectors

Abstract

In this paper an explicit formula is obtained for a generalisation of the Hilbert symbol, associated with an arbitrary formal group of finite height, defined over the ring of Witt vectors with coefficients in a perfect field of characteristic p>0. This formula becomes the Bruckner-Vostokov formula in the case of a multiplicative formal group. The proof is based on an application of Fontaine's theory of p-adic periods of formal groups, the Fontaine-Wintenberg field-of-norms functor, and Witt's explicit reciprocity law in characteristic p.