An explicit formula for the Hilbert symbol of a formal group

TL;DR

This paper extends Abrashkin's formula for the Hilbert symbol of a formal group by removing the roots of unity assumption, using ($phi, Gamma$)-modules and cohomological methods.

Contribution

It introduces a new approach to compute the Hilbert symbol without the roots of unity assumption, generalizing existing formulas with explicit cohomological tools.

Findings

Derived explicit formulas for the cup-product and Kummer map.

Generalized Herr complex for false Tate curve extension.

Removed the roots of unity assumption in Abrashkin's formula.

Abstract

Abrashkin established the Bruckner-Vostokov formula for the Hilbert symbol of a formal group under the assumption that roots of unity belong to the base field. The main motivation of this work is to remove this hypothesis. It is obtained by combining methods of ($\varphi, \Gamma$)-modules and a cohomological interpretation of Abrashkin's technique. To do this, we build ($\varphi, \Gamma$)-modules adapted to the false Tate curve extension and generalize some related tools like the Herr complex with explicit formulas for the cup-product and the Kummer map.