I have seen in the literature solutions to modular differential equations which include rational powers of the $j$-invariant (also called $j$-function), e.g., $j(\tau)^{1/p}$ for some $p \in \mathbb Z \backslash\{0\}$.
I am curious how this is defined, because not only is there an issue of which branch to choose, a priori I would think monodromy may enter the picture too, and so I would question whether $j(\tau)^{1/p}$ is single-valued for a particular $p$.
Is there any source that discusses this or could someone provide clarification? It seems to pop up often without discussion so I presume there might be some conventions I'm unaware of.