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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.)

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This is probably an inefficient method in practice, but you may want to look up converse theorems, which says that a $q$-series is a modular form if the associated L series satisfies many functional equations. –  Sanchez Jun 29 '13 at 6:46
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If you only want to go up to a fixed level, then the relevant space of modular forms is finite-dimensional, so checking whether a power series belongs in this form reduces to checking finitely many of its initial terms against a known basis of the relevant space of modular forms. –  Qiaochu Yuan Jun 29 '13 at 7:42
    
@Sanchez Thanks, I didn't realize such theorems existed –  Dtseng Jun 30 '13 at 0:38
    
@Qiaochu So from what you're saying, if one of the product expansions are indeed a modular form of level 2 and weight 4, say, then we can narrow our search to just one modular form (namely the only one whose initial terms agrees). This makes me feel better (as it is something useful that I should have seen), though comparing the coefficients might still be difficult (as I don't know of a way showing the product expansion for the delta function agrees with what you get from (E_4^3-E_6^2)/1728 by just expanding and comparing coefficients). –  Dtseng Jun 30 '13 at 0:42
    
@Dtseng If you are interested in modular forms, maybe you would like to have a glance at the book by André Weil. It is about elliptic functions, but certainly serves for helping you to manipulate and compare the coefficients of modular forms. –  awllower Jun 30 '13 at 3:50
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1 Answer

up vote 2 down vote accepted

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.

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