Taylor coefficients of Anderson generating functions and Drinfeld torsion extensions

TL;DR

This paper extends the understanding of torsion extensions of Drinfeld modules by describing their structure via Anderson generating functions and hyperderivatives, and analyzes the associated Galois representations in a broad setting.

Contribution

It generalizes previous work on Carlitz modules to arbitrary rank Drinfeld modules, providing new descriptions of torsion extensions and Galois representations.

Findings

Torsion extensions characterized by evaluations of Anderson generating functions.

Galois representation image lies in the motivic Galois group.

Generalization of Chang and Papanikolas' results to the $rak{p}$-adic case.

Abstract

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the $\mathfrak{p}$-adic Tate module lies in the $\mathfrak{p}$-adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the $t$-adic case.