# Questions on the $j$-invariant

The j-invariant as a modular function is typically defined $$j(\tau) = \frac{E_4(\tau)^3}{\Delta(\tau)}$$ since $$E_4$$ is a modular form of weight 4 and $$\Delta$$ has weight 12, it follows that $$j$$ is a modular function.

My first question: Is there a different derivation of the $$j$$-invariant? Perhaps one using the geometry of the modular curve $$X(1) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}^*$$ or one more algebraic in nature? I.e., if I wanted to construct a function that is $$\mathrm{SL}_2(\mathbb{Z})$$-invariant, how should I start?

For my second question, There is the well known result that the field of all modular functions is equal to $$\mathbb{C}(j)$$. This is typically using the q-expansions of $$j$$ and a residue like theorem for modular forms.

However, $$X(1) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}^*$$ has a model so that $$X(1)(F)$$ parametrizes $$\bar{F}$$-isomorphism classes of elliptic curves defined over $$F$$. So, one might expect that there is an algebraic proof that the $$j$$-invariant generates all modular functions that works over any field. Does such a proof exist?

The first question comes naturally under Schwarz's theory of uniformization of hyperbolic triangles: let $$\tau_0=(1+\sqrt{-3})/2$$, $$\tau_1=i$$, and $$\tau_\infty=i\infty$$, and denote by $$\Delta$$ the hyperbolic triangle with vertices $$\tau_z$$, so one half of the usual fundamental domain for $$X(1)$$. If $$H^*$$ denotes the closure of the upper-half plane, Schwarz theory tells you that there exists a unique analytic map from $$H^*$$ to $$\Delta$$ such that $$D(z)=\tau_z$$ for $$z=0$$, $$1$$, $$\infty$$. Denote by $$J$$ the inverse map (so $$J$$ is from $$\Delta$$ to $$H^*$$ and satisfies $$J(\tau_z)=z$$). By Schwarz's reflection principle, coming from the tesselation by $$\Delta$$, one can extend $$J$$ into a meromorphic function from $$H^*$$ to $$P^1(\mathbb C)$$ which will be invariant under the subgroup of orientation preserving maps of the group generated by reflections along the sides of $$\Delta$$, here $$SL_2(\mathbb Z)$$, and of course $$j(\tau)=1728 J(\tau)$$. This can of course be done for any hyperbolic triangle.