A few things confused me:
$H$ is the upper half plane and $G$ the modular group which acts on it in the usual way.
Let $G_2$ be the 2nd Eisenstein Series ($k=2$, weight 4) and $G_3$ the 3nd Eisenstein Series ($k=3$, weight 6)
$g_2$ = $60G_2$
$g_3$ = $140G_3$
Define $\Delta$ to be the cusp form of weight $12$ ($k = 6$) $g_2^3 - 27g_3^2$
Let $j$ = $\frac{1728g_2^3}{\Delta}$
Serre proves $j$ is a modular function of weight $0$, that it's holomorphic in $H$ and has a simple pole at infinity, and that
"It defines by passage to quotient" a bijection of $\frac{H}{G}$ onto $\mathbb{C}$.
To exhibit the bijection he says consider $\lambda$ in $H$. Construct the modular form $f_\lambda = 1728g_2^3 - \lambda\Delta$, argues that it has a unique zero modulo $G$, and this unique zero serves as the image of $\lambda$.
(Q1)
It's among the first times I see such an "indirect" bijection (one not arising as sending a variable to some algebraic expression involving it). Can we really exhibit bijections in this manner?
Serre proceeds to prove if $f$ is a modular function of weight $0$, then it is a rational function of $j$.
(Q2)
He starts off by saying we can assume $f$ is holomorphic in $H$ because we are free to multiply it by a suitable polynomial in $j$ (and we know $j$ is holomorphic in $H$). I am not sure why this is.
My thinking is that if $f$ is simply meromorphic and thus has an isolated singularity $z_0$ in $H$, then it's Laurent expansion would contain some terms with negative powers of $(z-z_0)$. While since $j$ is holomorphic it's Laurent expansion at $z_0$ contains only nonnegative powers of $(z-z_0)$, and so we can raise the Laurent expansion of $j$ to the power equalling $-n$ where $n$ is the smallest power of $z-z_0$ in the Laurent series for $f$ at $z_0$, and possibly subtract by a constant to avoid preserving the negative powers, and then multiply the result by the Laurent series for $f$ at $z_0$.
I'm not sure about the details of what I wrote, like are my thoughts correct about possibly having to subtract by a constant after raising $j$ to the $-n$ power?
Another step of the argument I am unsure of is where Serre says since $\Delta$ is zero at infinity, $g = \Delta^nf$ is holomorphic at infinity.
Again my intuition tells me reason can be seen by looking at Laurent series expansions for each:
if $f$ and $\Delta$ are functions on the $z-plane$ we can identify their Laurent expansions about infinity with their Laurent expansions about $0$ in the $w = \frac{1}{z}$ plane. So we do this change of variables. Now if $f$ was meromorphic at infinity in $z-plane$ its expansion about $0$ in $w-plane$ contains terms with negative powers of $w$. Since $\Delta$ is holomorphic (and $0$) at infinity in z-plane, it's expansion about $0$ in $w-plane$ contains only positive powers of $w$. So multiplying $\Delta$'s expansion by the power of the smallest negative power in $f$'s expansion, and multiplying the result with $f$'s expansion, kills off all the negative powers of $w$, and so there indeed exists an $n$ such that $g=\Delta^nf$ is holomorphic at $w=0$, and thus at $z = infinity$.
Is my thinking above correct?
(Q3)
Now in the final part of the proof, Serre reduces the problem to showing that $\frac{G_2^3}{\Delta}$ and $\frac{G_3^2}{\Delta}$ are rational functions of $j$.
I believe I got the answer but I'm not sure.
Should $\frac{G_2^3}{\Delta}$ $\approx$ $j$
and $\frac{G_3^2}{\Delta}$ $\approx$ $\frac{-1}{j^2} + \frac{1}{j}$
where $\approx$ means ignoring constants.
Is this correct?
Then Serre says $\hat{\frac{H}{G}}$ denote the compactification of $\frac{H}{G}$ where $H$ is the upper half plane and $G$ the modular group which acts on it in the usual way.
He says there is a complex analytic structure on $\hat{\frac{H}{G}}$.
He says that by the bijection exhibited above (in my Q1) $j$ defines an isomorphism of $\hat{\frac{H}{G}}$ onto the Riemann Sphere $S_2$ - which I'm OK with accepting, although I don't quite understand compactifications yet.
Then he says that the fact that the following 3 statements are equivalent amounts to the well-known fact that the only meromorphic functions on $S_2$ are the rational functions:
for a meromorphic function $f$ on $H$:
1) $f$ is a modular function of weight $0$ 2) $f$ is a quotient of 2 modular forms of the same weight 3) $f$ is a rational function of $j$.
Q4)
How does the equivalence of these 3 statements amount to the well-known fact that the only meromorphic functions on the Riemann Sphere are the rational functions?