Elliptic curves, $j$-invariant and example of $j(\Lambda)=0$ First, consider $\Lambda=\mathbb{Z}\bigoplus\omega\mathbb{Z}$ with $\omega$ the third root of unity in the upper half plane. I know that the lattice is such that $g_2(\Lambda)=0$, where $g_2$ is the coefficient of the differential equation $$(\wp')^2=4\wp^3+g_2(\Lambda)\wp+g_3(\Lambda).$$
We know that for any $\tau\in\mathbb{C}$ $$g_3(\tau\Lambda)=\frac{1}{\tau^6}g_3(\Lambda).$$ We also know that $g_2(\tau\Lambda)=0$ for all $\tau$. Does this mean that given a curve of the kind $y^2=4x^3+b $ we can map $\mathbb{C}/\Lambda$ to it by just taking the $\wp$ function of the lattice $\tau\mathbb{Z}\bigoplus\tau\omega\mathbb{Z}$, $\tau=\sqrt[6]{\frac{g_3(\Lambda)}{b}}$?
More in general, by the lifting properties we know that the only way two complex tori are isomorphic is if their lattices satisfy $\Lambda'=\tau\Lambda$, in particular all the elliptic curves given before are isomorphic (obviously). We can also see that $$j(\Lambda)=1728\frac{g_2(\Lambda)^3}{g_2(\Lambda)^3-27g_3(\Lambda)^2}=j(\tau\Lambda),$$ so essentially it is invariant under isomorphisms of tori. Is this the motivation behind the definition of modular form(function), which is an equivalent condition for the above homogeneity condition?
Another question is:
Is it possible to show, in a not too sophisticated way, that the converse also holds, hence that the $j$-invariant is essentially injective up to isomorphism of lattices, without using similar results about the corresponding elliptic curves, and that $g_2$ and $g_3$ are surjective once we know the surjectivity of $j$ (which I have shown)?
 A: 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.
