Applications and motivation of $\eta$-quotient generators and algorithms

Question

So initially Dummit, Kisilevsky and McKay found all Dedekind $\eta$-products which are newforms. This result was subsequently generalized by Martin. Further, Ono and Martin provide an exhaustive list of modular elliptic curves whose associated modular forms is expressible as an $\eta$-quotient.

Also, it is known that $$ E_4(z) = \dfrac{\eta(z)^{16}}{\eta(2z)^8} + 2^8 \cdot \dfrac{\eta(2z)^{16}}{\eta(z)^8} $$ and $$ E_6(z) = \dfrac{\eta(z)^{24}}{\eta(2z)^{12}} - 2^5 \cdot 3 \cdot 5 \cdot \eta(2z)^{12} - 2^9 \cdot 3 \cdot 11 \cdot \dfrac{\eta(2z)^{12}\eta(4z)^8}{\eta(z)^8} + 2^{13} \cdot \dfrac{\eta(4z)^{24}}{\eta(2z)^{12}}, $$ leading Ono to pose the question in his book "Web of Modularity", which spaces of modular forms are generated by eta-quotients. This question was addressed somewhat in this paper (http://arxiv.org/pdf/math/0701478v1.pdf) by Kilford.

My question is, what is the motivation for this question by Ono as most of these expressions are rather "messy". Is this just a question of interest in itself or are there further consequences to this open question? Why are spaces generated by $\eta$-quotients of interest as opposed to some other modular form such as the Eisenstein series?

Also is there an algorithm that given a modular form that can be expressed as a linear combination (or even rational function), can compute the corresponding expression?

Answer 1

I am not a specialist of that subject, but one of the interest of the $\eta$-function is its direct relation with the partition function $p(n)$, the number of ways to write $n$ as a sum of positive integers disregarding order. There are a long line of works since Ramanujan on the arithmetic property of $p(n)$ with still many beautiful open questions. The relation of $\eta$ with $p(n)$ is simply, as you probably know well, that $$\eta(z) = q^{1/24} \prod_{n \geq 1} (1-q^n) = q^{1/24} (\sum_n p(n) q^n)^{-1}$$ with $q=e^{2 i \pi z}$. Relating $\eta$ with eigenforms of integral weights (basic example: $\eta^{24}=\Delta$), a kind of object for which we have a lot of tools, in particular the existence of a Galois representation attached to it, may if cleverly used give some information about the partition function (and I know that some results of Ahlgren, Nicolas, Serre, Boylan, Ono, and many others, that use this method. I am just quoting here at random papers that I happen to have looked at recently).

Now, this is not to say that this is the only and even main motivation to look for such kind of identities, nor that all such identities are useful in the study of the partition function, but at least that is one motivation I can understand.