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Let ϕ 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 ϕ is equal to A. Then we show that the closure of the analytic monodromy group of X in SLr ( \mathbbAFf \mathbb{A}_F^f  ) is open, where  \mathbbAFf \mathbb{A}_F^f  denotes the ring of finite adèles 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 étale fundamental group of X on the adèlic Tate module of ϕ is open in GLr ( \mathbbAFf \mathbb{A}_F^f  ). (2) Let ψ be a Drinfeld A-module of rank r defined over a finitely generated field extension of F, and suppose that ψ cannot be defined over a finite extension of F. Suppose again that the endomorphism ring of ψ is A. Then the image of the Galois representation on the adèlic Tate module of ψ is open in GLr ( \mathbbAFf \mathbb{A}_F^f  ).    Finally, we extend the above results to the case of arbitrary endomorphism rings.

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