The norm residue symbol for formal Drinfeld modules

TL;DR

This paper provides an explicit description of the Kummer pairing for formal Drinfeld modules with stable reduction, generalizing previous results for Carlitz modules and extending formulas to broader local field extensions.

Contribution

It introduces a detailed explicit formula for the Kummer pairing in the context of formal Drinfeld modules, expanding upon prior work for specific modules and field extensions.

Findings

Explicit description of the Kummer pairing using the logarithm of the Drinfeld module

Generalization of previous results for Carlitz modules and sign-normalized rank one Drinfeld modules

Extension of formulas to arbitrary finite extensions of local fields with torsion points

Abstract

In this paper, we study the Kummer pairing associated with formal Drinfeld modules having stable reduction of height one. We give an explicit description of the pairing \`a la Kolyvagin, in terms of the logarithm of the formal Drinfeld module, a certain derivation, torsion points and the trace. The results obtained give a generalization of the results of Angl\`es proved for Carlitz modules, and of Bars and Longhi proved for sign-normalized rank one Drinfeld modules. It also presents an extension of our previous formulas to arbitrary finite extensions of loca fields containing enough torsion points.