Monodromy Groups associated to Non-Isotrivial Drinfeld Modules in Generic Characteristic

Abstract

Let $\phi$ be a non-isotrivial family of Drinfeld A-modules of rank r in generic characteristic with a suitable level structure over a connected smooth algebraic variety X. Suppose that the endomorphism ring of $\phi$ is equal to A. Then we show that the closure of the analytic fundamental group of X in $SL_r(\mathbb{A}_F^f)$ is open, where $\mathbb{A}_F^f$ denotes the ring of finite adeles of the quotient field F of A. From this we deduce two further results: (1) If X is defined over a finitely generated field extension of F, the image of the arithmetic \'etale fundamental group of X on the adelic Tate module of $\phi$ is open in $GL_r(\mathbb{A}_F^f)$. (2) Let $\psi$ be a Drinfeld A-module of rank r defined over a finitely generated field extension of F, and suppose that $\psi$ cannot be defined over a finite extension of F. Suppose again that the endomorphism ring of $\psi$ is A. Then the image of the Galois representation on the adelic Tate module of $\psi$ is open in $GL_r(\mathbb{A}_F^f)$. Finally, we extend the above results to the case of arbitrary endomorphism rings.