Kummer theory for Drinfeld modules

Abstract

Let $\varphi$ be a Drinfeld $A$-module of characteristic ${\mathfrak{p}}_0$ 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-${\mathfrak{p}}_0$ torsion points of $\varphi$, or equivalently, on the prime-to-${\mathfrak{p}}_0$ adelic Tate module of $\varphi$. In this article we consider in addition a finitely generated torsion free $A$-submodule $M$ of $K$ for the action of $A$ through $\varphi$. We determine the image of the absolute Galois group of $K$ up to commensurability in its action on the prime-to-${\mathfrak{p}}_0$ division hull of $M$, or equivalently, on the extended prime-to-${\mathfrak{p}}_0$ adelic Tate module associated to $\varphi$ and $M$.