Bloch-Kato exponential maps for local fields with imperfect residue fields

TL;DR

This paper extends the Bloch-Kato exponential map to certain local fields with imperfect residue fields and proves an explicit reciprocity law, advancing the understanding of Galois cohomology and p-adic representations in number theory.

Contribution

It generalizes the Bloch-Kato exponential map to fields with imperfect residue fields and establishes an explicit reciprocity law in this broader context.

Findings

Extended Bloch-Kato exponential map to new class of local fields

Proved explicit reciprocity law for these fields

Calculated Galois cohomology via (phi,G)-modules

Abstract

In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module.