Kummer Theory for Drinfeld 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.