Proof of Meromorphicity of Eisenstein Series in Serre's 'A Course in Arithmetic'

Question

I was reading the chapter on Modular Forms from Serre's 'A Course in Arithmetic'. In the first line of his proof that the Eisenstein Series $G_k(z)$ is a modular form, he says, "The above arguments show that $G_k(z)$ is weakly modular of weight $2k$." However, all he has shown in the previous paragraphs is that the infinite series defining $G_k$ on complex lattices is absolutely convergent. In other words all that has been shown till that point is that the series $G_k(z) := \sum^{'}_{m,n} \frac{1}{(mz+n)^{2k}} \cdots (1)$ is absolutely convergent at every $z \in \mathbb{H}$. While each summand $\frac{1}{(mz+n)^{2k}}$ is meromorphic on $\mathbb{H}$, it is not clear to me why an infinite sum of meromorphic functions must be meromorphic. He has shown (later in the proof) that the series $(1)$ converges normally (hence uniformly) but that too only on the fundamental domain $D$ but I did not find any result which applies in this situation to show that the infinite sum in $(1)$ is meromorphic (for instance, I think I read somewhere on this site that one could apply Montel's Theorem to show that a series $F(z):=\sum_{n=0}^{\infty} f_n(z)$ of meromorphic functions $f_n$ is meromorphic if $F$ has no essential singularity and the set of poles is a discrete subset of $\mathbb{C}$, but here the set of poles in $\mathbb{H}$ is $\left\{-\frac{n}{m} | m,n \in \mathbb{Z}, mn<0 \right\}$ which is a dense subset of $\mathbb{C}$). How does the claimed meromorphicity follow in this context?

Edit: Apologies, I realized that the poles of $G_k$ are not in the upper half plane $\mathbb{H}$ but on the real line itself, and in fact Montel does aply. Is this reasoning correct? Also is there a simpler way of seeing this? (For Serre does not say anything on the matter of meromorphicity, so maybe there is a much more obvious or elementary explanation for this?)

Answer 1

The defining series of $G_{2k}$ converges locally uniformly and is analytic on $\Bbb{C-R}$ and nowhere else. Same for meromorphic.

Then we look at the expansion at the cusp $i\infty$ of the modular curve $SL_2(\Bbb{Z})\setminus\Bbb{H}$

The idea is that $G_{2k}$ being $1$-periodic and analytic on $\Bbb{H}$ we get that $G_{2k}(\frac{\log s}{2i\pi})$ is well-defined (not depending on the branch of $\log$) and analytic on $0<|s|<1$.

We find that it is also analytic and non-zero at $s=0$.

We say that $G_{2k}$ is analytic and non-zero at $i\infty$

Let $f$ be another weight $2k$-modular form with a zero of order $m$ at $i\infty$. Then $G_{2k}/f$ is meromorphic from the compact Riemann surface $SL_2(\Bbb{Z})\setminus(\Bbb{H}\cup \Bbb{Q}\cup i\infty)$ to $\Bbb{C}$, with a pole of order $m$ at $i\infty$.