Galois representation of the product of two Drinfeld modules of generic characteristic

In this paper, we study the Galois representations attached to products of Drinfeld modules. As an analogue of Serre’s classical result on the images of Galois representations associated with products of elliptic curves, we prove that for any finite set of primes, the image of the corresponding product representation is sufficiently large, in the sense that it is commensurable with a subgroup defined by a natural determinant condition. Our approach combines Pink’s minimal model theory for compact subgroups of linear groups over local fields with explicit reciprocity laws for global function fields.