Elliptic curves, $j$-invariant and example of $j(\Lambda)=0$

Solution 1:

As you said you can reach all the $y^2=4x^3+b$ with the differential equation of the Weierstrass function of the lattice $\tau \Lambda,\tau=\sqrt[6]{\frac{g_3(\Lambda)}{b}}$.

$f$ is weight $k$ modular, ie. $f$ is analytic on the upper half-plane and $f(\frac{az+b}{cz+d})=(cz+d)^{-k} f(z)$ for all $a,b,c,d\in \Bbb{Z},ad-bc=1$ iff $F(a,b)= a^k f(\pm a/b)$ is analytic and $GL_2(\Bbb{Z})$ invariant on $\{ (a,b)\in \Bbb{C}^* \times\Bbb{C}^*,a/b\not \in \Bbb{R}\}$, ie. $F(a,b)=F(a\Bbb{Z}+b\Bbb{Z})$ is an analytic function on lattices which is homogeneous of degree $k$.

Not all analytic functions on lattices are modular forms, for example $F(\Lambda) = \sum_{\lambda\in \Lambda-0} \exp(1/\lambda^4)-1$, which is not homogeneous.

The cusp condition becomes $F(\Bbb{Z}+iy \Bbb{Z})$ is bounded as $y\to +\infty$, that we keep to make $M_k(SL_2(\Bbb{Z}))$ finite dimensional, and that we discard for the modular functions ($k=0$).

The modular forms for the congruence subgroups (things like $G_4(\tau)+G_4(N\tau)$) need a bit more work but they follow the same idea.