Explicit Reciprocity Laws for Formal Drinfeld Modules

TL;DR

This paper establishes explicit reciprocity laws for a class of formal Drinfeld modules with stable reduction of height one, extending previous results for Carlitz and rank one Drinfeld modules using formal power series techniques.

Contribution

It provides explicit formulas for the Kummer pairing in formal Drinfeld modules over local fields of positive characteristic, generalizing prior work to a broader class of modules.

Findings

Explicit formulas for the Kummer pairing in formal Drinfeld modules.

Extension of reciprocity laws to formal power series modules.

Methodology adapts previous approaches to the formal power series setting.

Abstract

In this paper, we prove explicit reciprocity laws for a class of formal Drinfeld modules having stable reduction of height one, in the spirit of those existing in characteristic zero (cf. the work of Wiles). We begin by defining the Kummer pairing in the language of formal Drinfeld modules defined over local fields of positive characteristic. We then prove explicit formulas for this pairing in terms of the logarithm of the formal Drinfeld module, a certain Coleman power series, torsion points and the trace. Our results extend the explicit formulas already proved by Angl\`es for Carlitz modules, and by Bars and Longhi for sign-normalized rank one Drinfeld modules. The approach followed is similar to the ones followed in the previously mentioned papers, taking into account the subtleties derived from the fact that the formal Drinfeld modules considered are formal power series, and are no longer polynomials.