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The minimum of the levels of the proper factors of a holomorphic eta quotient

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Abstract

The levels of the factors of a holomorphic eta quotient f are bounded above with respect to the weight and the level of f. Unfortunately, this bound remains implicit due to the ineffectiveness of Mersmann’s finiteness theorem. On the other hand, for checking whether f is irreducible, it is essential to know at least an explicit upper bound for the minimum \(m_f\) among the levels of the proper factors of f. In the case where the level of f is a prime power, the least upper bound for \(m_f\) has been recently determined via construction of a special factor. However, this construction does not generalize unconditionally to arbitrary levels. Here, we provide an explicit upper bound \(M_N\) for the minimum of the levels of the proper factors of a holomorphic eta quotient f of level N.

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Notes

  1. 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, Eq. (3.12) makes sense for all possible orderings of the primes dividing N.

  2. Otherwise, the claim is trivial.

  3. Kronecker product of matrices is not commutative. However, since any given ordering of the primes dividing M induces a lexicographic ordering on \(\mathcal {D}_{M}\) and \(\mathcal {D}_N\) with which the entries of \(P_{M,N}\) are indexed, Equation (7.3) makes sense for all possible orderings of the primes dividing M.

  4. i. e. which are not products of other holomorphic eta quotients.

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Acknowledgements

I am very thankful to the Institute of Analysis and Number Theory of Graz University of Technology and in particular, Prof. Christoph Aistleitner for his kind support and hospitality during the composition of this paper. Also, I am grateful to the Indian Institute of Science Education and Research Kolkata for granting me an extraordinary leave for my academic visit to Graz, which led to the completion of this project. I would also like to thank the anonymous referees for their helpful comments.

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This work was supported by the Austrian Science Fund (FWF), project F-5512.

Appendices

Appendix: Irreducibility and factorizability of modular forms

The notions of irreducibility and factorizability make sense also for modular forms in general. For all \(k\in 2\mathbb {N}\), by \(E_k\) we denote the normalized Eisenstein series of weight k:

$$\begin{aligned} E_k(z):=1-\frac{2k}{B_k}\sum _{n=1}^\infty \sigma _{k-1}(n)q^n, \end{aligned}$$

where \(B_k\) denotes the kth Bernoulli number and the function \(\sigma _{r}:\mathbb {N}\rightarrow \mathbb {N}\) is given by

$$\begin{aligned} \sigma _{r}(n):=\sum _{d|n}d^{r}. \end{aligned}$$

For each even integer \(k>2\), \(E_k\) is a modular form of weight k on \({\text {SL}}_2(\mathbb {Z})\) (see [23]). In [6], we see that every modular form with the trivial multiplier system on \({\text {SL}}_2(\mathbb {Z})\) has a unique factorization of the form:

$$\begin{aligned} C_0E_4^aE_6^b\prod _{t\in \mathbb {C}^*}(E_4^3-tE_6^2)^{c_t}, \end{aligned}$$

for some \(C_0\in \mathbb {C}\) and some nonnegative integers \(a,b,c_t\), where \(c_t\) is zero for all but finitely many t. From [6], we also know that none of \(E_4\), \(E_6\) or \(E_4^3-tE_6^2\) for all \(t\in \mathbb {C}^*\) is factorizable on \({\text {SL}}_2(\mathbb {Z})\), whereas each of them has proper factors on \(\Gamma _0(4)\). So, the analog of Conjecture 1 fails for modular forms in general. In the following, we provide examples of modular forms with the trivial multiplier system on \(\Gamma _0(N)\) which are not factorizable on \(\Gamma _0(N)\) for an arbitrary integer \(N>1\).

By \(X_0(N)\), we denote the compact modular curve \(\Gamma _0(N)\backslash (\mathfrak {H}\cup \mathbb {P}^1(\mathbb {Q}))\). Since for a modular form g on \(\Gamma _0(N)\), the order of vanishing of g at any two \(\Gamma _0(N)\)-equivalent elements of \(\mathfrak {H}\cup \mathbb {P}^1(\mathbb {Q})\) are the same (see [7, 18]), the order of vanishing of g at a point in \( X_0(N)\) makes sense. Given \(x_0\in X_0(N)\), if there exists a modular form \(f_{x_0,N}\) on \(\Gamma _0(N)\) which vanishes nowhere on \(X_0(N)\) except at \(x_0\) such that \(f_{x_0,N}\) has the least order of vanishing among all the modular forms on \(\Gamma _0(N)\) which vanishes only at \(x_0\), then clearly, \(f_{x_0,N}\) is not factorizable on \(\Gamma _0(N)\). In particular, from the invertibility of the order matrix (see (3.9)), it follows that both \(f_{0,N}\) and \(f_{\infty ,N}\) are given by eta quotients for all \(N\in \mathbb {N}\). More precisely, \(f_{0,N}\) (resp. \(f_{\infty ,N}\)) is the least positive integer power of \(\eta ^{A^{-1}_N(\underline{},1)}\) (resp. \(\eta ^{A^{-1}_N(\underline{}, N)}\)) that yields an eta quotient with integer exponents which satisfies Newman’s criteria (see [14, 15] or [20]). Proceeding thus or by an easy generalization a result from [21], we obtain that for all \(n\in \mathbb {N}\) and for all primes \(p\ge 5\), we have

$$\begin{aligned} f_{0,p^n}=\left( \frac{\eta ^p}{\eta _p}\right) ^2 \ \text { and } \ f_{\infty ,p^n}=\left( \frac{\eta _{p^n}^p}{\eta _{p^{n-1}}}\right) ^2. \end{aligned}$$

In particular, for each prime \(p\ge 5\), the modular form \(({\eta ^p}/{\eta _p})^2\) of weight \(p-1\) is not factorizable on \(\Gamma _0(p^n)\) for all \(n\in \mathbb {N}\). For \(N\in \mathbb {N}\), by \(B_N\in \mathbb {Z}^{\mathcal {D}_N}\times \mathbb {Z}^{\mathcal {D}_N}\), we denote the matrix whose every column is the least possible integer multiple of the corresponding column of \(A^{-1}_N\in \mathbb {Q}^{\mathcal {D}_N}\times \mathbb {Q}^{\mathcal {D}_N}\) (see Section 4 in [6]). Let \(\mu \) denote the Möbius function. It follows by a similar argument as above that for all \(N\in \mathbb {N}\) which have at least two distinct odd prime divisors and for \(B_N\) as defined above, the modular form

$$\begin{aligned} f_{0,N}=\eta ^{B_N(\underline{},1)}=\prod _{d\in \mathcal {D}_N}\eta _d^{B_N(d,1)}=\prod _{d|N}\eta _d^{\mu (d)\mathrm {rad}(N)/d} \end{aligned}$$
(7.5)

has trivial multiplier system on \(\Gamma _0(N)\) and it is not factorizable on \(\Gamma _0(M)\) for any multiple M of N, whose set of prime divisors is the same as that of N. However, the lack of an analogous holomorphy preserving homomorphism (as in Lemma 2: From the graded ring of weakly holomorphic modular forms of level M to that of level N, where \(N\Vert M\)) prevents us from concluding that the above modular form is irreducible.

Since the least possible weight of which nonzero modular forms with the trivial multiplier system exist is 2, any modular form of weight 2 (e.g. \(NE_2(Nz)-E_2(z)\) on \(\Gamma _0(N)\) for all \(N>1\)) is irreducible. Instances of occurrence of irreducible modular forms of higher weights are not known. Indeed, their existence is questionable:

Open question 1

Are there irreducible modular forms with the trivial multiplier system and of weights greater than two?

In contrast, we note that it follows from Theorem 3 in [6] that there exist irreducible holomorphic eta quotientsFootnote 4 of arbitrarily large weights.

A conjecture

For hundred randomly chosen levels N which have at most three distinct prime divisiors and which divide

$$\begin{aligned} 2^8\cdot 3^3\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23, \end{aligned}$$

we checked using Theorem A (see Sect. 1) and Zagier’s list (see Theorem 1 in [5]) that each reducible holomorphic eta quotient of weight 1 and of level N is a product of weight half holomorphic eta quotients whose levels divide N. With this numerical evidence, we speculate the following:

Conjecture 1

Let \(N>1\) be an integer and let f be a reducible holomorphic eta quotient of level N. Let M be the least positive integer such that f is factorizable on \(\Gamma _0(M)\). Then \(M=N\).

In [2], we see that the above conjecture holds in the case where N is a prime power.

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Bhattacharya, S. The minimum of the levels of the proper factors of a holomorphic eta quotient. Res Math Sci 8, 33 (2021). https://doi.org/10.1007/s40687-021-00267-2

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