Is the set of all meromorphic functions on the upper half plane which they are also meromorphic at infinity, a field?
I am trying to show that the set of all modular functions for $Sl_2(\mathbb{Z})$ is a field. But, I want to know that is it necessary to have that for all element $\gamma$ in $Sl_2(\mathbb{Z})$, for all $z\in H$, $f(\gamma z)=f(z)$?