In Farkas Kra riemann surfaces textbook, it defines a meromorphic $q$-differential as follows:
Let $q$ be an integer. By a (meromorphic) $q$-differential $\omega$ on $M$ we mean an assignment of a meromorphic function $f$ to each local coordinate $z$ on $M$ so that $$f(z)dz^q$$ is invariantly defined.
It's not clear (to me) how this should be interpreted. What is $dz^q$ here?