Asymptotic formulas for Monster-related modular functions?

Question

Define the following,

$$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$

$$j_{2A}(\tau) =\Big(\big(\tfrac{\eta(\tau)}{\eta(2\tau)}\big)^{12}+2^6 \big(\tfrac{\eta(2\tau)}{\eta(\tau)}\big)^{12}\Big)^2 = \tfrac{1}{q} + 104 + \color{blue}{4372}q + 96256q^2 + 1240002q^3+\cdots \tag{2}$$

$$j_{3A}(\tau) =\Big(\big(\tfrac{\eta(\tau)}{\eta(3\tau)}\big)^{6}+3^3 \big(\tfrac{\eta(3\tau)}{\eta(\tau)}\big)^{6}\Big)^2 = \tfrac{1}{q} + 42 + \color{blue}{783}q + 8672q^2 +65367q^3+\dots \tag{3}$$

$$j_{4A}(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2 = \tfrac{1}{q} + 24+ \color{blue}{276}q + 2048q^2 +11202q^3+\dots\\ \tag{4}$$ $$j_{7A}(\tau)=\Big(\big(\tfrac{\eta(\tau)}{\eta(7\tau)}\big)^{2}+7 \big(\tfrac{\eta(7\tau)}{\eta(\tau)}\big)^{2}\Big)^2 = \tfrac{1}{q} + 10+ \color{blue}{51}q + 204q^2 +681q^3+\dots\\ \tag{5}$$

where $\eta(\tau)$ is the Dedekind eta function.

(Note that $196883, 4371, 782, 276, 51$ are degrees of the irreducible representations of the Monster, Baby Monster, Fischer Fi₂₃, Conway Co₁, and Held groups, respectively.)

The asymptotic formula (by Rademacher?) for the coefficient of $q^n$ of $(1)$ is given by,

$$a(n) \approx \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}\tag{for 1}$$

For example, let $a(n)$ be the coefficient, and $a'(n)$ given by the formula, then,

$$\begin{array}{cccc} n&100&200&300\\ a(n)&8.38\,\text{x}\,10^{52}&2.011\,\text{x}\,10^{75}&3.293\,\text{x}\,10^{92}\\ a'(n)&8.40\,\text{x}\,10^{52}&2.016\,\text{x}\,10^{75}&3.299\,\text{x}\,10^{92}\\ \end{array}$$

Question: What are the analogous coefficient formulas for $(2), (3), (4), (5)$?

$\color{blue}{\text{Edit}}$: After perusing the OEIS, it seems that the fourth has,

$$d(n) \approx \frac{e^{2\pi\sqrt{n}}}{2\,n^{3/4}}\tag{for 4}$$

$$\begin{array}{cccc} n&100&200&300\\ d(n)&3.04\,\text{x}\,10^{25}&3.64\,\text{x}\,10^{36}&1.26\,\text{x}\,10^{45}\\ d'(n)&3.06\,\text{x}\,10^{25}&3.66\,\text{x}\,10^{36}&1.27\,\text{x}\,10^{45}\\ \end{array}$$

though I have no proof that this is its correct asymptotic formula.

Answer 1

In a recent paper with Ken Ono and John Duncan, we give exact formulas for the coefficients of the Monsterous Moonshine modules. I will give a link to a preprint a few days when I can, but the relevant part relies on prior work of Bringmann and Ono (http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/115.pdf) which gives exact formulas for coefficients of modular forms assuming knowledge of their poles at cusps.

Each of your $j_g(\tau)=\sum a_g(n)q^n$ is a Hauptmodul for a group of the form $\Gamma_g=\left\langle \Gamma_0(N),\begin{pmatrix}0&-1\\N&0\end{pmatrix}\right\rangle,$ where $N=1,2,3,$ or $4$. In this case, the dominant term in the exact formulas for $a_g(n)$ is $$\frac{2\pi}{\sqrt{N n}}I_1\left(4\pi \sqrt{n/N}\right)\sim \frac{e^{4\pi \sqrt{n/N}}}{\sqrt{2}N^{1/4}n^{3/4}}.$$ Here $I_k$ is the usual $I$-Bessel function.

Answer 2

If we write $\psi(\tau) = \frac{\Delta(\tau)}{\Delta(2\tau)} = \sum_{n=-1}^\infty s(n) q^n$ and $\phi(\tau) = \frac{\Delta(2\tau)}{\Delta(\tau)} = \sum_{n=1}^\infty b(n) q^n$, then $j_{2A}(\tau) = \psi(\tau) + 128 + 2^{12}\phi(\tau)$. Asymptotics for $|s(n)|$ and $b(n)$ and explicit upper bounds are given in the paper http://arxiv.org/pdf/1408.1083.pdf, written with Kyle Pratt.

Specifically, we have $$|s(n)| \sim \frac{e^{2\pi\sqrt{n}}}{2n^{3/4}},$$ $$b(n) \sim \frac{2^{1/4}}{8192}\frac{e^{2\pi\sqrt{2n}}}{n^{3/4}}.$$

Answer 3

The coefficients of a modular form of non-positive weight can be given by an explicit formula that depends only on the poles at cusps (and constant terms when the weight is zero). The asymptotics are given by asymptotics of modified Bessel functions, which are well-known. Rademacher gave the formula in the level 1 case, and higher level cases are subsumed by inducing to a level 1 vector-valued form.

You can find an explicit formulation as Theorem 1.9 in Giménez's dissertation. While the formula in full generality is rather complicated, many of the terms will vanish for the case at hand, since you are working with Hauptmoduln of low level.

You can also extract the answer from earlier, slightly less explicit descriptions in section 2 of Dijkgraaf-Maldacena-Moore-Vafa, A Black Hole Farey Tail and Lemma 5.3 of Borcherds, Automorphic forms on $O_{s+2,2}(\mathbf{R})$ and infinite products