I am studying Mock Modular forms in the book Harmonic Maass Forms and Mock Modular Forms: Theory and Applications and i have problems with a theorem.
In Theorem 6.4 it says
If $Z_E^+ $ has poles in $\mathbb{H}$ then there is a canonical modular function $M_E$ on $\Gamma_0(N_E)$ which has algebraic coefficients for which $Z_E^+ - M_E$ is holomorphic on $\mathbb{H}$.
$Z_E(\tau)$ is the modified Weierstrass $\zeta$-Function evaluated for the Eichler integral $\mathcal{E}_E(\tau)$.
The book provides an explanation with saying it follows from using standard algebraic facts for modular functions but i fail to understand it. So i would be glad if someone could elaborate on this more. Thanks alot in advance.