#### Vol. 10, No. 2, 2016

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Kummer theory for Drinfeld modules

### Richard Pink

Vol. 10 (2016), No. 2, 215–234
##### 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 $M$.

Primary: 11G09
Secondary: 11R58