The Norm Residue Symbol for Formal Drinfeld Modules

Abstract

We study the Kummer pairing associated with formal Drinfeld modules of stable reduction and height one over local fields. Building on the explicit reciprocity framework initiated by Kolyvagin and later developed for Drinfeld modules, we develop a derivation-based approach to the Kummer pairing and establish an explicit reciprocity law in this setting. The resulting characterization reduces the study of the pairing to the computation of the associated derivation at a uniformizer. This yields an explicit reciprocity law valid for arbitrary finite separable extensions containing sufficient torsion. In the setting considered in our previous work, the general construction specializes to explicit formulas and leads to congruences for local norm maps associated with torsion points of formal Drinfeld modules.