I'm trying to "decode" Corollary 2.6 of Katz's text in the case $m = 1$. A shortened background:
Fix a prime $p > 5$ and $N \in \mathbb{N}$ prime to $p$. Let $F := \mathbb{F}_{p}(\zeta_{N})$ with $\zeta_{N}$ a primitive $N$-th root of unity. For $m \in \mathbb{N}\cup\{\infty\}$ let $W_{m} := W_{m}(F)$ be the Witt vectors of length $m$, with $W := W_{\infty}(F)$. The unique primitive $N$-th root of unity in $W$ which lifts $\zeta_{N}$, the "Teichmiiller representative", will also be denoted $\zeta_{N}$.
Let $M^{0}$ be the moduli scheme over $W$ W which classifies isomorphism classes of elliptic curves over $W$-algebras together with a level-$N$ structure of determinant $\zeta_{N}$, and let $M$ be its canonical compactification. For each $m \in \mathbb{N}$ we put $M^{0}_{m} := M^{0} \otimes_{W} W_{m}$ and $M_{m} := M \otimes_{W} W_{m}$.Let $S_{m}$ denote the open subscheme of $M_{m}$ ) where the Hasse invariant mod $p$ (or equivalently is invertible. Then $S_{m}$ is an affine smooth curve over $W_{m}$.
Let $T_{m, n} \rightarrow S_{m}$ be the etale covering of $S_{m}$ defined by the kernel of the character $\chi_{n}:\pi_{1}(S_{m}) \rightarrow (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$.
Let $R_{m}$ be the graded ring of holomorphic modular forms defined over $W_{m}$, of level $N$, and of type $\zeta_{N}$. That is, $$R_{m} = \bigoplus_{k\geq 1} H^{0}(M_{m}, \underline{\omega}^{\otimes k}),$$
where $\underline{\omega}$ is an invertible sheaf coming from the invertible section $\omega_{can}$ on $T_{m, m}$ (see p. 335).
Corollary 1.6. Let $\sum f_{i} \in R_{m}$ and $m_{1} \leq m$. If at one cusp of $M_{m}$ we have $\sum f_{i}(q) \equiv 0 \pmod{p^{m_{1}}}$ in $W_{m}[[q]]$, then it is true at every cusp.
My question: Letting $m = 1$ I have figured out that $W_{1}= F$. Is it true in this case then that $R_{1} = \bigoplus_{k\geq 1}M_{k}(\Gamma(N))$ and $M_{1} = X(N)$, where $M_{k}(\Gamma(N))$ is the weight-$k$ space of holomorphic modular forms and $X(N)$ is the corresponding modular curve? Or maybe I need to adjust $\Gamma(N)$ to $\Gamma_{1}(N)$ or similar.
The term "moduli scheme" has a very packed definition that I have tried to unpack. But it requires so much background that I don't have -- seemingly years worth of studying hardcore algebraic geometry.