Cogalois theory and Drinfeld modules

TL;DR

This paper extends cogalois theory to rank one Drinfeld modules with class number one, demonstrating limitations for higher ranks or class numbers, and analyzing torsion in specific Carlitz module extensions.

Contribution

It generalizes cogalois theory to certain Drinfeld modules and identifies its limitations for higher ranks or class numbers.

Findings

Cogalois theory applies to rank one Drinfeld modules with class number one.

No cogalois theory exists for Drinfeld modules of higher rank or class number.

Analyzes torsion points in specific Carlitz module extensions.

Abstract

In this paper we generalize the results of \cite{sanchez} to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension ${\mathbb{F}}_{q}(T)(\Lambda_{P^{n}}) /{\mathbb{F}}_{q}(T)(\Lambda_{P})$.