The closest sphere packings in 8 dimensions is the $E_8$ lattice $\Lambda_8$.
The closest sphere packings in 24 dimensions is the Leech lattice $\Lambda_{24}$.
Create the function, $f(\tau)$ defined by:
$$f(\tau) = \frac{\sum\limits_{x\in \Lambda_{24}} (-1)^{|x|^2\tau} } { \left(\sum\limits_{x\in \Lambda_{8}} (-1)^{|x|^2\tau}\right)^3 }$$
- We find $f(\tau)$ is a modular function of weight 0, i.e. it satisfies:
$$f(\tau) = f\left(\frac{ A\tau+B}{C\tau+D}\right)$$
for integers $A,B,C,D$ where $AD-BC=1$.
(*In particular we find $f(\tau)= 1-\frac{720}{j(\tau)}$ where $j$ is the j-invariant.)
Now is there any great meaning to this? Or is it entirely obvious that we would get a modular function of weight 0 from this fraction? And why not for other fractions involving other lattices in which the dimensions are multiples of each other?
Is there any great meaning behind the definition using the $E_8$ and Leech lattices, or does it more or less rely on the fact that $3\times 8=24$? I know that the Leech lattice in some sense can be though of as being made of 3 copies of $E_8$.
To me this is an amazing identity even more amazing than Euler's $e^{i\pi}+1=0$