I have been reading Serre's 'A Course in Arithmetic'. In section $2.2$ he shows that lattice functions $F$ of weight $2k$ are in one-to-one correspondence with modular functions of the same weight, by defining for each lattice function $F$, the function $f: \mathbb{H} \longrightarrow \mathbb{C}$ given by $f(\frac{\omega_1}{\omega_2}) := \omega_2^{2k}F(\omega_1, \omega_2) = F(\Gamma(\omega_1, \omega_2))$. While it is clear (by the defining property $F(\lambda \Gamma) = \lambda^{-2k} F(\Gamma) \cdots \hspace{1.0mm}(1)$ of the lattice function $F$) that $f$ is well-defined and the modularity property of $f$ also follows by the invariance of $F$ under $SL_2(\mathbb{Z})$-action (a consequence of the well-definedness of $F$ as a lattice function), it is not clear to me why $f$ must be meromorphic on $\mathbb{H} \cup \{\infty\}$ (for that is how a modular function has been defined by him), since all we know about $F$ (which is the only thing used to defined $f$) is the property $(1)$. Is this a consequence of some property of lattice functions? Thanks in advance.



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