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# Infinite product exponents for modular forms

- Asra Ali
^{1}and - Nitya Mani
^{2}Email author

**Received: **22 June 2016

**Accepted: **14 July 2016

**Published: **1 November 2016

## Abstract

Recently, Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over \(\Gamma _0(N)\). Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve \(X_0(N)\) is 0 or 1.

## Keywords

- Modular Form
- Fourier Expansion
- Cusp Form
- Congruence Subgroup
- Index Subgroup

## 1 Introduction

*N*is through eta-quotients, expressions of the form

*d*. Some holomorphic modular forms can be expressed as an eta-quotient, giving the values

*c*(

*m*) in their infinite product expansion. However, only finitely many weight 2 newforms can be expressed as an eta-quotient, completely characterized by Ono and Martin (see [9]). Clearly, in this case, the values

*c*(

*m*) are bounded.

However, *c*(*m*) is not a repeating sequence (or even bounded) for generic modular forms *f*(*z*). Understanding the function *c*(*m*) can help determine the divisors of the associated modular forms. These appear in the theory of Borcherds products, which examines the rare case where the divisors of a form are supported at Heegner points. Our work makes use of divisors in general context. Bruinier et al. first derived an expression for *c*(*m*) as a function of *m* for an associated meromorphic modular form defined over \({{\mathrm{SL}}}_2(\mathbb {Z})\) (see [3]). Later work by Ahlgren [2] and Choi [4] generalized this result to meromorphic modular forms defined over \(\Gamma _0(N)\). Related work by Movasati and Nikdelan also motivated this study. Further, Kohnen (see [7]) established a bound on the growth of *c*(*m*) independent of these formulations if *f*(*z*) has no zeros or poles on the upper half plane. He showed that given this condition, if *f*(*z*) is a modular form for any finite index subgroup of \({{\mathrm{SL}}}_2(\mathbb {Z})\), \(c(m) \ll _f \log \log n \cdot \log n\) and that if *f*(*z*) is a modular form for any congruence subgroup of \({{\mathrm{SL}}}_2(\mathbb {Z})\), \(c(m) \ll _f (\log \log n)^2\). Here we give tight bounds on *c*(*m*) for some infinite classes of modular forms and an upper bound on the growth of *c*(*m*) for any holomorphic modular form *f*(*z*).

*f*(

*z*) is written as

### Theorem 1.1

*f*(

*z*) (with infinite product expansion as in (1.1)) in a fundamental domain \(\mathcal {F}_N\) is \(\{z_j = x_j + iy_j \}_{j = 1, \ldots , r}\) with \(y_1 \le \cdots \le y_r\) and \(r \ge 1\). Then we have that

If the genus of \(X_0(N)\) is 0 or 1, then we also obtain a lower bound on *c*(*m*). In particular, we obtain an \(\Omega \) bound, defined as follows. Given two arithmetic functions *f*, *g* defined on the natural numbers, \(f = \Omega (g)\) implies that there exists some positive constant *c* such that for all \(n_0 \in \mathbb {N}\) there exists infinitely many \(n > n_0\) so that \(f(n) \ge c\cdot g(n)\).

### Theorem 1.2

*f*(

*z*) in \(\mathcal {F}_N\) is \(\{z_j = x_j + iy_j \}_{j = 1, \ldots , r}\) with \(y_1 \le \cdots \le y_r\) and \(r \ge 1\). If

*f*(

*z*) is a modular form for \(\Gamma _0(N)\) such that the genus of \(X_0(N)\) is 0 or 1, then we have that

From the above results, we can also obtain a similar (slightly weaker) bound to that in [7] stated above.

### Corollary 1.3

Suppose that *f*(*z*) is a modular form for \(\Gamma _0(N)\) with no zeros or poles on the upper half plane. Then, we obtain \(c_f(m) \ll \log m \cdot \log \log m\).

## 2 Preliminaries

*f*(

*z*) with Fourier expansion \(f(z) = \sum _{n = h}^{\infty } a(n) q^n\) with \(a(h) = 1\) for a congruence subgroup

*N*,

*f*(

*z*) at

*z*on \(X_0(N)\). Now, we will define a meromorphic modular form of weight 2 for \(\Gamma _0(N)\) that frequently appears in these studies (see [2, 3]):

### Proposition 2.1

*f*(

*z*) be a modular form for \(\Gamma _0(N)\). Define

*j*th powers of divisors of

*n*. Then \(f_\theta (z)\) is a meromorphic modular form of weight 2 for \(\Gamma _0(N)\).

*m*Poincaré series (Theorem 1 in [10]) for all \(z \in \mathcal {H}\) and \(s \in \mathbb {C}\) with \({{\mathrm{Re}}}(s) > 1\):

*I*-Bessel function of order \(\nu \).

### Proposition 2.2

(§1 in [4]) Define \(j_{N, m}(z)\) to be the analytic continuation of \(F_{N, m}(z, s)\) (where \({{\mathrm{Re}}}(s) \le 1\)) as \(s \rightarrow 1^+\). Then, \(j_{N, m}(z)\) is the constant term of the Fourier expansion of \(F_{N, m}(z, 1)\) when \(t \in \mathcal {C}_N\).

Next we define the differential operator \(\xi _0\) that plays an important role in the study of Harmonic Maass forms [12], defined in the next section.

### Proposition 2.3

## 3 Proof of main result

Given this setup, the exponents of the infinite product expansion of a modular form can be obtained by applying the Möbius inversion formula to the result obtained in [4].

### Proposition 3.1

*f*(

*z*) of weight

*k*on \(\Gamma _0(N)\) for \(N > 1\) with infinite product expansion as in (1.1). Then, we have that

A normalized modular form with integral coefficients \(f(z) = q^h + \sum _{n = h+1}^{\infty } a(n)q^n\) has a product expansion \(f(z) = q^h \prod _{m = 1}^\infty (1-q^m)^{c_f(m)}\) where the \(c_f(m)\) are integers. This can be seen by expanding the infinite product and solving for the coefficients of the Fourier expansion. Now we consider \(c_f(m)\) as \(m \rightarrow \infty \) by computing the growth of terms in the above expression.

### 3.1 Growth of \(j_{N, m}(z)\)

We note that \(F_{N,m}(z,s)\) is a harmonic Maaßform (defined below) and cite a Lemma giving an explicit computation for the values for the analytic continuation \(j_{N,m}(z)\).

### Definition 3.2

*Definition 7.1*[12]) A smooth function\(f: \mathcal {H}\rightarrow \mathbb {C}\) is a weak harmonic Maaß form of weight

*k*for \(\Gamma _0(N)\) if the following conditions are satisfied:

- (1)
*f*transforms like a modular form under the action of \(\Gamma _0(N)\), - (2)
*f*is in the kernel of the weight*k*hyperbolic Laplacian$$\begin{aligned} \Delta _k = -y^2 \left( \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} \right) + iky\left( \frac{\partial }{\partial x} +i\frac{\partial }{\partial y} \right) , \end{aligned}$$ - (3)
*f*has at most linear exponential growth at the cusps of*f*.

### Lemma 3.3

*t*: If \(t \sim \infty \),

### Lemma 3.4

Let \(z = x + iy \in \mathcal {F}_N \cup \mathcal {C}_N^*\). As \(m \rightarrow \infty \), we have that \(j_{N, m}(z) \asymp \left( \frac{e^{2\pi m y}}{\sqrt{m}} \right) .\)

### Proof

*m*th coefficient of the Maaß-Eisenstein series for \(\Gamma _0(N)\) and \(K_{\nu }(y), I_{\nu }(y)\) are the modified Bessel functions of the first and second kind with special values given above. Then, apply the relations on these modified Bessel functions, noting that for \(y \in \mathbb {R}\setminus \{0\}\), (see 10.2.13 and 10.2.17 in [1]),

### 3.2 Regularized integration

### Lemma 3.5

- (1)
If the genus of \(X_0(N)\) is 0, the regularized integral as defined above vanishes.

- (2)
If genus of \(\Gamma _0(N)\) is 1, the regularized integral vanishes for infinitely many positive integers

*m*.

### Proof

Recall that \(\xi _0(j_{N,d}(z))\) is a weight two cusp form. Suppose that the genus of \(X_0(N)\) is zero. Then the space of cusp forms of weight two is trivial, so \(\xi _0(j_{N, d}(z))\) must be 0. It follows that the regularized integral vanishes.

*N*whose Hasse-Weil

*L*-series coincides with

*L*-function for

*g*(

*z*). In other words, for all values of

*p*that do not divide

*N*,

*p*, the number of points of \(E_g/ \mathbb {F}(p)\) is exactly \(p+1\) and \(a(p) = 0\). So we have that for an odd

*m*, \(j_{N, m}(z)\) identically vanishes if and only if \(E_g\) is supersingular at some

*p*|

*m*. Since there exist infinitely many supersingular primes for every elliptic curve over \(\mathbb {Q}\) (see [5]), there are infinitely many positive integers

*m*such that \(j_{N, m}(z)\) is holomorphic on \(\mathcal {H}\) (i.e. \(\xi _0 j \equiv 0\)). \(\square \)

This regularized integral *R*(*m*) may not vanish if the genus of \(X_0(N)\) is not 0. Below we address the growth of *R*(*m*). First, we characterize the growth of the Fourier cofficient \(b_m(1,0)\) of the weight 0 Poincaré series of index *m*.

### Lemma 3.6

*m*Poincaré series given in Eq. 2.1 with Fourier expansion

### Proof

*d*modulo

*m*. Weil’s bound gives

*m*(see 10.41 in [11]). Then as \(m \rightarrow \infty \), we obtain

### Lemma 3.7

*f*(

*z*) in a chosen fundamental domain is \(\{z_j = x_j + iy_j\}_{j = 1, \ldots , r}\) with \(y_1 \le \cdots \le y_r\), \(r \ge 1\). As \(m \rightarrow \infty \) we can bound

*R*(

*m*) as:

### Proof

*R*(

*m*).

*t*defined in [4]. Lemma 3.6 shows that term \(b_m(1,0)a(0) \ll m^{1/4}e^{4\pi \sqrt{m}}\). Moreover, the term

*a*(

*m*) is negligible, since the coefficients of the modular form

*f*(

*z*) has polynomial growth in

*m*. Now, the cusps in \(\mathcal {C}_N^*\) are rational points. From the proof of Lemma 3.4, we have

*f*(

*z*). Since the

*m*-dependence of this term also comes from \(j_{N,m}(t)\), the growth of this term is dominated by the zeros of

*f*(

*z*). Thus we have

*R*(

*m*) is dictated by the above. Thus, given the integral expansion in (3.3),

### 3.3 Proof of results

We begin by proving the second theorem since many of the arguments used to prove the first theorem parallel the ones employed here:

### Proof of Theorem 1.2

*m*, we have

*N*|

*m*or not is bounded with respect to

*m*. Then, since

*m*dominates the other divisors of

*m*as \(m \rightarrow \infty \), we evaluate the expression \(\sum _{z \in \mathcal {F}_N \cup \mathcal {C}_N^{*}} \nu _{z}^{(N)} (f(z)) j_{N, d}(z)\) only at \(d = m\). Further \( \nu _{z}^{(N)} \) is a constant that does not depend on

*m*. Thus we arrive at

Now, suppose we relax the conditions on the genus of \(X_0(N)\). We now prove the upper bound on \(c_f(m)\) in general given in Theorem 1.1.

### Proof of Theorem 1.1

*N*|

*m*or not is bounded as \(m \rightarrow \infty \). Evaluating the remaining expression at \(d = m\) we have

Finally, we see that the bounds yielded by our calculation are consistent with earlier work in [7].

### Proof of Corollary 1.3

*f*(

*z*) has no roots or poles on the upper half plane, the first term in the above summation,

*d*|

*m*to compute the desired bound for \(c_f(m)\):

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

### Acknowledgements

The authors would like to thank the NSF and the Emory REU (especially Dr. Mertens and Professor Ono) for their support of our research.

## Authors’ Affiliations

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