Counting Drinfeld modules with prescribed local conditions

We give asymptotics for the number of isomorphism classes of Drinfeld \({\mathbb {F}}_q[T]\) ${\mathbb{F}}_q[T]$ -modules over \({\mathbb {F}}_q(T)\) ${\mathbb{F}}_q(T)$ of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective stacks. Our results for counting points of bounded height on weighted projective stacks over global function fields may be of independent interest.