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

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 fundamental group of X in SLr(A f F ) is open, where 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(AfF ). (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(AfF ). Finally, we extend the above results to the case of arbitrary endomorphism rings. Mathematics Subject Classification: 11F80, 11G09, 14D05.