Stark's formula for the j-invariant In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the $j$-invariant (for some $\tau \in \mathcal{H}$ and $q=e^{2i\pi\tau}$)
$$ j(\tau) = \frac{(1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^n)^3}{q\prod_{n=1}^\infty (1  - q^n)^{24}} $$
with $\sigma_k(n) = \sum_{d|n} d^k$.
On the other hand the formula that I know for the $j$-invariant is :
$$
j(\tau) = 12^3 \frac{g_2^3(\tau)}{g_2^3(\tau) - 27g_3^2(\tau)}
$$
with
$$
g_2(\tau) = 60\sum_{(m,n) \neq (0,0)} \frac{1}{(m\tau +n)^4} = \frac{(2\pi)^4}{2^23}(1 + 240 \sum_{n=1}^\infty \sigma_3(n)q^n)
$$
and
$$
g_2(\tau) = 140\sum_{(m,n) \neq (0,0)} \frac{1}{(m\tau +n)^6} = \frac{(2\pi)^6}{2^23^3}(1 - 504 \sum_{n=1}^\infty \sigma_5(n)q^n)
$$
My question : How do you get Stark's formula from where "I" am ? (I obviously see how we get the "top" part of the fraction). i.e. why is it true (if it is) that
$$
g_2^3(\tau) - 27g_3^2(\tau) = q\prod_{n=1}^\infty (1  - q^n)^{24}
$$
modulo multiplication by some constant.
 A: Stark's formula is simply the $q$-expansion of the $j$-invariant. As you pointed out,
$$j(\tau)=1728\frac{g_2(\tau)^3}{\Delta(\tau)}=j(\tau)=1728\frac{g_2(\tau)^3}{g_2(\tau)^3-27g_3(\tau)^2}.$$
By taking the $q$-expansion, i.e. writing the Fourier expansion as a Laurent series, we get
$$ j(\tau) = \frac{(1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^n)^3}{q\prod_{n=1}^\infty (1  - q^n)^{24}}, $$
or better yet,
$$j(\tau)=1728\frac{(1 + 240 \sum_{n=1}^\infty \sigma_3(n)q^n)^2}{(1 + 240 \sum_{n=1}^\infty \sigma_3(n)q^n)^2-(1 - 504 \sum_{n=1}^\infty \sigma_5(n)q^n)^2}.$$
Let's take a look at the $q$-expansion of the modular invariants:
$$g_2=1+240q+2160q^2+6720q^3+⋯$$
$$g_3=1−504q−16632q^2−122976q^3-\cdots.$$
Now we can express $\Delta$ as
$$Δ=\sum_{n=1}^∞τ(n)q^n=q−24q^2+252q^3⋯.$$
Finally, we have the $j$-invariant which
 simplifies to something like $$j(\tau)=\frac{1}{q}+744+196884q...\quad\quad q=e^{2\pi i \tau}.$$
I don't remember the expansion beyond that but if you need any sources I recommend A First Course in Modular Forms by Diamond and Shurman or the first chapter of Silverman's Advanced Topics in the Arithmetic of Elliptic Curves
