Abstract
We show that for any positive integer N, there are only finitely many holomorphic eta quotients of level N, none of which is a product of two holomorphic eta quotients other than 1 and itself. This result is an analog of Zagier’s conjecture/Mersmann’s theorem which states that of any given weight, there are only finitely many irreducible holomorphic eta quotients, none of which is an integral rescaling of another eta quotient. We construct such eta quotients for all cubefree levels. In particular, our construction demonstrates the existence of irreducible holomorphic eta quotients of arbitrarily large weights.
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Notes
Unlike the general case, irreducibility of a holomorphic eta quotient of weight 1 is rather easy to determine, because a holomorphic eta quotient of weight 1 and level N is irreducible if and only if it is not factorizable on \(\Gamma _0({\text {lcm}}(N,12))\) (see Lemma 1 in [4]). In particular, the irreducibility of the holomorphic eta quotients listed in Appendix A in [7] could be easily verified.
Kronecker product of matrices is not commutative. However, since any given ordering of the primes dividing N induces a lexicographic ordering on \(\mathcal {D}_N\) with which the entries of \(A_N\) are indexed, Equation (3.16) makes sense for all possible orderings of the primes dividing N.
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Acknowledgements
I am thankful to Sander Zwegers, who asked during my talk at Cologne whether a Mersmann type finiteness theorem holds if we keep the level of the eta quotients fixed instead of their weight. Corollary 1 is precisely an answer to his question. I would like to thank Don Zagier, Christian Weiß, Danylo Radchenko, Armin Straub, Nadim Rustom, and Christian Kaiser for their comments. I made the computations for the tables using \(\mathtt {PARI/GP}\) [22] and \(\mathtt {Normaliz}\) [21] which I learnt to use from Don and Danylo. I am grateful to them for acquainting me with these very useful computational tools. In particular, Danylo computed \({k_{\max }}(24)\), \({k_{\max }}(28)\) and \({k_{\max }}(30)\) for Table 2. I am grateful to the Max Planck Institute for Mathematics in Bonn and to CIRM : FBK (International Center for Mathematical Research of the Bruno Kessler Foundation) in Trento for providing me with an office space and supporting me with a fellowship during the preparation of this article.
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Appendix: Comparison of the weights
Appendix: Comparison of the weights
By \({k_{\max }}(N)/2\), we denote the maximum of the weights of level N which are not factorizable on \(\Gamma _0(N)\). Let p be a prime. From the discussion about holomorphic eta quotients on \(\Gamma _0(p)\) in Sect. 1, it follows that \({k_{\max }}(p)=p-1\). Also, from Theorem 6.4 in [7], we know \({k_{\max }}(p^2)=(p-1)^2\). With the support of a huge amount of experimental data, we make the following conjecture:
Conjecture 2
(a) For each prime number p, we have \({k_{\max }}(p^3)=(p-1)^2\).
(b) For each odd prime p and for all integers \(n>3\), we have
where \(r_n\in \{0,1\}\) is the residue of n modulo 2.
For all odd primes p and for all integers \(n>3\), in [6] we see examples of irreducible holomorphic eta quotients of level \(p^n\) and of the same weight as in (6.1) (see Corollary 1 and (2.1) in [6]). However, the catch of the above problem is to show that any holomorphic eta quotient of level \(p^n\) whose weight is greater than the quantity given in (6.1) must be reducible (see Conjecture 1 in [6] and Theorem 2 in [4]).
In Table 2, we compare \({k_{\max }}(N)\) with \(\kappa (N)\) for several \(N\in \mathbb {N}\), where \(\kappa (N)/2\) is the weight of the eta quotient \(F_N\)which we defined in Theorem 2 (see also 4.2). Since we have already discussed above the cases of odd prime powers as well as those of \(2^n\) for \(n\le 3\), we omit such levels from the following table.
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Bhattacharya, S. Finiteness of irreducible holomorphic eta quotients of a given level. Ramanujan J 48, 423–443 (2019). https://doi.org/10.1007/s11139-017-9982-6
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DOI: https://doi.org/10.1007/s11139-017-9982-6