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

Abstract

In this paper, we study the Galois representations attached to products of Drinfeld modules. As a function-field analogue of Serre's classical open image theorem for products of elliptic curves, we prove that if the two Drinfeld modules are not geometrically isogenous up to any Frobenius twist, then for any finite set of primes, the image of the associated product representation is sufficiently large. That is, the image group is commensurable with a subgroup defined by natural determinant compatibility conditions. Our approach combines Pink's minimal quasi-model theory for compact subgroups of linear algebraic groups over local fields with explicit reciprocity laws from global class field theory. As an arithmetic application of our main theorem, we establish a mutual torsion finiteness property for non-isogenous Drinfeld modules, mirroring the classical results of Ribet and Zarhin for abelian varieties.