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

Summary

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 SL_r ($\mathbb{A}_F^f$) is open, where $\mathbb{A}_F^f$ 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 GL_r ($\mathbb{A}_F^f$). (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 GL_r ($\mathbb{A}_F^f$).

Finally, we extend the above results to the case of arbitrary endomorphism rings.