I'm studying Serre's "Formes modulaires et fonctions zêta p-adiques". There is a point which is not at all clear to me. He says that the algebra $\widetilde{M}^0$ of modular forms mod $p$ of weight congruent to $0$ mod $p-1$ is "l'algebre affine d'une courbe lisse". I have not found the definition of affine algebra of a curve, but I think - meaning I'm not sure - he means the coordinate ring, over $\mathbb{F}_p$, of such curve.
Even so, it is not clear to me which curve he is talking about, although I recall you that Swinnerton-Dyer showed that $\widetilde{M}\cong \mathbb{F}_p[X,Y]/(\tilde{A}-1)$, with $A$ the unique polynomial such that $A(E_4,E_6)=E_{p-1}$.
In "Congruences et formes modulaires" he makes a more precise description: he states that $\widetilde{M}^0$ is isomorphic to l'algebre affine over $\mathbb{F}_p$ of the curve $X$ obtained from the projective line by removing the values of the modular function $j$ mod $p$ corresponding to curves with zero Hasse invariant, i.e., supersingular curves.
But then, how would this isomorphism be defined? Why would this curve $X$ be smooth?
Consider that I am not at all familiar with Katz's approach, and scheme theory in general, so keep it elementary please!
A partial answer.
By this $E_{p-1}\equiv 1\bmod p$.
Let $M_{k(p-1)}(p)$ be the weight $k(p-1)$ modular forms for $SL_2(\Bbb{Z})$ with coefficients in $\Bbb{Z}_{(p)}$
and $R$ the subring of $\Bbb{Q}(j)$ of modular functions with $q$-expansion in $\Bbb{Z}_{(p)}$ and no poles outside of the zeros of $E_{p-1}$.
If $f\in M_{k(p-1)}(p)$ then $f\equiv f/E_{p-1}^k\bmod p$ where $f/E_{p-1}^k$ is a modular function whose poles are located at the zeros of $E_{p-1}$. Also its coefficients in $\Bbb{Z}_{(p)}$ give that $f/E_{p-1}^k\in R$.
Conversely, for any modular function $g\in R$ we'll have that $g E_{p-1}^k \in M_{k(p-1)}(p)$ for $k$ large enough.
Whence the $\Bbb{F}_p$ algebra generated by the reduction $\bmod p$ of all the $M_{k(p-1)}(p) $ will be $R/(p)$.
Let $S=\{j(a)\in \Bbb{C},E_{p-1}(a)=0\}$. Remove from $S$ the $j(a)$ that are not algebraic over $\Bbb{Q}$ and not integral over $\Bbb{Z}_{(p-1)}$. Let $h$ be the product of their $\Bbb{Z}_{(p)}[x]_{monic}$ minimal polynomials. Then $R/(p)$ will be the coordinate ring of $\Bbb{P}^1_{\Bbb{F}_p}$ minus the (Galois orbit of) $[b:1]$ for each root $b\in \overline{\Bbb{F}}_p$ of $h\bmod p$.
This is a smooth affine curve.