Recognizing if a power series is a $q$-expansion of a modular form

Question

Given a power series in $q$, is it possible to tell if it is the $q$-expansion of a modular form (of level $N$ say)?

I don't need to show results of this sort, but it has come up enough that I'm curious how one might approach it. As for an example I have in mind:

After playing around with SAGE, in level 2, the modular form $\Delta$ factors into $\delta\gamma^2$, where $\delta$ and $\gamma$ are modular forms of level 2 (this I think I can verify).

However, it also appears that $\delta$ and $\gamma$ have the following $q$-expansions: \begin{align*} \delta &= q\prod_{n\geq 1}{(1+a(n)q^{n})^{8}}, \end{align*} where \begin{align*} a(n) = \begin{cases} 1 &\mbox{if } n \equiv 1\pmod{2} \\ 0 & \mbox{if } n \equiv 2\pmod{4}\\ -1 & \mbox{if } n\equiv 0\pmod{4} \end{cases} \end{align*} and \begin{align*} \gamma &= \prod_{n\geq 1}{(1-q^n)^{16b(n)}}, \end{align*} where \begin{align*} b(n) = \begin{cases} 1 &\mbox{if } n \equiv 1\pmod{2} \\ \frac{1}{2} & \mbox{if } n \equiv 0\pmod{2}.\end{cases} \end{align*} Finally, if you plug in the product expansions for $\delta$, $\gamma$ in $\Delta = \delta\gamma^2$ and rearrange terms, you get the product expansion for $\Delta$.

Now, if we knew that the product formulas were actually modular forms, then the rest should quickly follow. However, how might one show that a power series is from a modular form?

(The only thing I know about this is that in Koblitz's book, he derives the product formula from the transformation law for the eta function, which came from the corresponding law for $E_2$. I also know from googling of the phrase "Bocherds products" but at first glance they don't seem to be exactly what I'm looking for, though probably very interesting.

I also hope I am not posting too many questions on this forum about the same topics, but it's much more useful for me to ask around about something then to keep trying to imagine what the answer might be.)

Answer 1

This certainly isn't a complete answer, but I'd like to point out that recognizing whether a given power series is the $q$-expansion of a modular form tends to be extremely difficult. For example, if $E$ is an elliptic curve over $\mathbb Q$, one defines the $L$-function of $E$ by $$ L(E,s) =\prod_p \frac{1}{\det(1-\rho_{E,\ell}(\operatorname{Frob}_p)p^{-s},(T_\ell E)^{I_p})} $$ where $\rho_{E,\ell}:G_{\mathbb Q}\to GL(2,\mathbb Z_\ell)$ comes from the Tate module of $E$ and $I_p\subset G_{\mathbb Q}$ is the inertia group of $p$.

You can expand $L(E,s)$ as $$ L(E,s) = \sum_{n\geqslant 1} \frac{a_n}{n^s} $$ A natural question is: is the $q$-expansion $$ \sum_{n\geqslant 1} a_n q^n $$ a modular form? The answer is yes, precisely when $E$ is modular. So in this example, proving that a certain class of $q$-expansions consists of modular forms is equivalent to proving the Taniyama-Shimura conjecture, which was enormously difficult.