Kummer Theory for Drinfeld Modules

TL;DR

This paper extends Kummer theory for Drinfeld modules by analyzing the Galois action on the prime-to-p0 division hulls of finitely generated A-submodules, generalizing previous results on torsion points.

Contribution

It introduces a framework for understanding Galois representations on extended modules associated with Drinfeld modules, broadening the scope of prior torsion point analyses.

Findings

Determines Galois image up to commensurability on division hulls of submodules.

Generalizes previous torsion point results to finitely generated submodules.

Provides explicit descriptions of Galois actions on extended Tate modules.

Abstract

Let {\phi} be a Drinfeld A-module of characteristic p0 over a finitely generated field K. Previous articles determined the image of the absolute Galois group of K up to commensurability in its action on all prime-to-p0 torsion points of {\phi}, or equivalently, on the prime-to-p0 adelic Tate module of {\phi}. In this article we consider in addition a finitely generated torsion free A-submodule M of K for the action of A through {\phi}. We determine the image of the absolute Galois group of K up to commensurability in its action on the prime-to-p0 division hull of M, or equivalently, on the extended prime-to-p0 adelic Tate module associated to {\phi} and M.