The Galois image attached to the product of two generic Drinfeld modules is large, commensurable with the determinant-condition subgroup.
Moonen, An Introduction to Mumford–Tate Groups, lecture notes, available at https://www.math.ru.nl/ ~bmoonen/Lecturenotes/MTGps.pdf
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Galois representation of the product of two Drinfeld modules of generic characteristic
The Galois image attached to the product of two generic Drinfeld modules is large, commensurable with the determinant-condition subgroup.