The higher Hilbert pairing via (phi,G)-modules

TL;DR

This paper establishes Tate duality for higher local fields of mixed characteristic using (phi,G)-modules, and defines a non-degenerate higher Hilbert pairing, re-deriving known formulas.

Contribution

It introduces a new approach to higher Hilbert pairings via (phi,G)-modules, extending Tate duality to higher local fields.

Findings

Proves Tate duality for higher local fields of mixed characteristic

Defines a non-degenerate higher Hilbert pairing

Recovers formulas of Brueckner and Vostokov

Abstract

We prove the Tate duality for higher dimensional local fields of mixed characteristic (0,p), when p is an odd prime, using the theory of higher fields of norms. Assuming that p is not ramified in the basefield, we then use this construction to define the higher Hilbert pairing. In particular, we show that the Hilbert pairing is non-degenerate, and we re-discover the formulae of Brueckner and Vostokov.