Growth of the coefficients of the inversion of the $j$-invariant function

Question

We have the $j$-invariant defined as

I have that $$ j(\tau)=\frac{1}{q}+\sum_{k\geq 0}c_kq^k, $$ where $q=e^{-2\pi t}$ ($\tau=it$) and $c_k\sim e^{4\pi\sqrt{k}}/(k^{3/4}\sqrt{2})$.

The inversion formula for the $j$-invariant is $$ q=j^{-1}+\sum_{k\geq 2}d_kj^{-k}. $$

Thus, I would like to know some upper bound or asymptotic formula for $d_k$.

Any hint or reference?

Answer 1

It's in the OEIS: https://oeis.org/A066396

There's a formula there that gives an approximation of the form (in your notation) $$ d_k \sim A \cdot (-1)^{k+1}\cdot B^k / k^{3/2} $$ where $A\approx1943.54943\dots$ and $B\approx2311.3945621\dots\,$.

Answer 2

Extended comment: Some analysis I've done recently in free probability theory gives the compositional inverse of a Laurent series of the form

$$f(z) = \frac{1}{z} + a_1 + a_2 z + a_3z^2 + \cdots$$

as

$$f^{(-1)}(z) =\frac{1}{z} + \frac{a_1}{z^2} + \frac{a_1^2+a_2}{z^3} + \frac{a_3+3a_1a_2+a_1^3}{z^4}+\cdots,$$

where the numerator polynomials are given in OEIS A134264 and are related to noncrossing partitions, Dyck lattice paths, and other combinatorial constructs.

This supports Somos' claim that OEIS A091406 gives the inverse of the j-invariant.

Answer 3

$j(\tau)=J(e^{2i\pi \tau})$, since $J(q)\in q^{-1}+\Bbb{Z}[[q]]$ then in formal series $$f(s)=J^{-1}(1/s)=\sum_{k\ge 1} d_k s^k \text{ is in }\Bbb{Z}[[s]] \tag{1}$$

$f^{-1}(q)=1/J(q)$ is surjective from $|q|< 1$ to $\Bbb{C}$ and locally biholomorphic away from $J(q)\in 0,1728,i\infty$. Whence in the branch where $f(0)=0$ then $f$ is analytic for $|q|<1/1728$, and it will be given by the series $(1)$.

The radius of convergence is $1/1728$ and $f'(1/1728)=\sum_{k\ge 1} \frac{k\, d_k}{1728^k}$ doesn't converge, this follows from that $J'(i)=0\implies \lim_{s\to 1/1728^-}f'(s)=\infty$.

To get an upper bound on the $d_k$, let $m(y)=\sup_{\Im(\tau)>y} |1/j(\tau)|$. For $|s|<m(y)$ we'll have $J^{-1}(1/s)<e^{-2\pi y}$ so that $|d_k| \le m(y)^{-k-1} e^{-2\pi y}$ whenever $m(y)< 1/1728$. A numerical check shows that $m(2)< 10^{-5}$.

The monodromy group of $j^{-1}(s)$ and $f(s)$ is interesting:

• in the branch where $j^{-1}(\infty)= i\infty$ when rotating around $\infty$ it transforms $j^{-1}(s)$ to $j^{-1}(s)-1=\pmatrix{1&-1\\0&1}j^{-1}(s)$.

• In the branch where $j^{-1}(1728)=i$ when rotating around $1728$ it transforms $j^{-1}(s)$ to $\pmatrix{0&1\\-1&0} j^{-1}(s)$.

• In the branch where $j^{-1}(0))=e^{2i\pi/3}$ when rotating around $0$ it transforms $j^{-1}(s)$ to $\pmatrix{0&-1\\1&1} j^{-1}(s)$.

• In the branch where $j^{-1}(\infty)=\pmatrix{a&b\\c&d}i\infty=a/c$ it gets different, rotating around $\infty$ transforms $j^{-1}(s)$ to $\pmatrix{a&b\\c&d} \pmatrix{1&-1\\0&1} \pmatrix{a&b\\c&d}^{-1} j^{-1}(s)$. Whence in the branch such that $f(0)=\exp(-2i\pi a/d)$ then $f(s) = \exp(-2i\pi j^{-1}(s))$ isn't analytic at $0$ anymore.