On the finiteness of simple Eta-quotients of fixed weight or fixed level

Abstract

This Bachelor's thesis is concerned with the (holomorphic) modular forms one can obtain from rescaling Dedekind's $\eta$-function and taking quotients of these rescalings, so-called $\eta$-quotients. Eta-quotients are of interest because they are explicit examples of holomorphic modular forms to some congruence subgroup with some multiplier system. They even provide examples of holomorphic modular forms of any weight $k \in \frac{1}{2}\mathbb{Z}_{>0}$. The motivation for this thesis was the study of the following theorem by Mersmann:\\ The following fourteen $\eta$-quotients are the only simple $\eta$-quotients of weight $1/2$: \begin{equation*} \begin{split} \eta, \frac{\eta^2}{\eta_2},\ \frac{\eta_2^2}{\eta},\ \frac{\eta_2^3}{\eta\eta_4},\ \frac{\eta_2^5}{\eta^2\eta_4^2},\ \frac{\eta\eta_4}{\eta_2},\ \frac{\eta\eta_6^2}{\eta_2\eta_3},\ \frac{\eta^2\eta_6}{\eta_2\eta_3},\ % \frac{\eta_2^2\eta_3}{\eta\eta_6},\ \\ \frac{\eta_2\eta_3^2}{\eta\eta_6},\ \frac{\eta_2^2\eta_3\eta_{12}}{\eta\eta_4\eta_6},\ \frac{\eta_2^5\eta_3\eta_{12}}{\eta^2\eta_4^2\eta_6^2},\ \frac{\eta \eta_4\eta_6^2}{\eta_2\eta_3\eta_{12}},\ % \frac{\eta\eta_4\eta_6^5}{\eta_2^2\eta_3^2\eta_{12}^2}. \end{split} \end{equation*} The attribute ``simple'' of an $\eta$-quotient combines multiple properties that cannot be left out if one wishes to obtain a finiteness--result on $\eta$-quotients as in the above theorem. The original goal of this thesis was to find a similar list of simple $\eta$-quotients of weight $1$. After writing and running an algorithm that detected simple $\eta$-quotients, it turned out that the list of simple $\eta$-quotients of weight one would be substantially longer than the list above. Hence, the focus of the thesis shifted to proving the finiteness of simple $\eta$-quotients of fixed weight. Indeed, we have the following theorem: \begin{Thm*}[Mersmann's First Theorem] There exist only finitely many simple eta-quotients of any fixed weight. \end{Thm*} This result had already been established by G. Mersmann in a brilliant Master's thesis and had been reproven by S. Bhattacharya in his dissertation. The author tried to understand the idea of Bhattacharya's proof of Mersmann's First Theorem, noticed and corrected the mistakes that appeared and filled the gaps the proof had, to obtain a complete proof of Mersmann's First Theorem following Bhattacharya's idea. After establishing Mersmann's First Theorem, a similar result is proven in the case that the level is fixed: \begin{Thm*} [Finiteness of quasi-irreducible \etaqs of fixed level] There exist only finitely many quasi-irreducible \etaqs to any given level $N \in \BZ_{>0}$. \end{Thm*} In this case, the author also analysed the proof to obtain an explicit bound on the number of quasi-irreducible \etaqs of level $N$. Ultimately, this thesis prompts three additional follow-up questions which are still open to the best knowledge of the author: What explicit bound(s) on the number of simple $\eta$-quotients of weight $k$ can we find? What is the list of all simple $\eta$-quotients of weight $k$? Can we improve the bound on the number of quasi-irreducible $\eta$-quotients of level $N$ established in the last subsection? Notice that establishing an explicit bound on the number of simple $\eta$-quotients of fixed weight $k$ would also be the decisive step towards the full list of $\eta$-quotients of weight $k$ since this would give a termination condition to an algorithm that finds all simple $\eta$-quotients of given level $N$ and weight $k$ which has already been implemented by the author.