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Let me first define the algebra $M$ I am talking about: let us fix a prime $p$, an integer $N$ not divisible by $p$. For $k$ an integer, let me call $N_k$ the $\mathbb{Z}$-module of modular forms of level $\Gamma_0(N)$ (or $\Gamma_1(N)$, or $\Gamma(N)$, whichever you prefer) of some weight $\leq k$, and whose $q$-expansion (at infinity) is in $\mathbb{Z}[[q]]$. Let me call $M_k \subset \mathbb{F}_p[[q]]$ the image of $N_k$ by the map $f \mapsto \tilde f$: "reduction mod $p$ of the $q$-expansion of $f$". It is clear that $M_k \subset M_{k+1}$ and let $M = \cup_{k \geq 0} M_k \subset \mathbb{F}_p[[q]]$. It is clear that $M$ is a sub-algebra of $\mathbb{F}_p[[q]]$, that one can call the algebra of mod $p$ modular forms of level $\Gamma_0(N)$ (or $\Gamma_1(N)$, or $\Gamma(N)$).

When $N=1$, the structure of $M$ has been determined by Swinnerton-Dyer in his famous paper in Antwerpen III (and there is an also famous Bourbaki seminar by Serre on this subject): when $p=2,3$, $M=\mathbb{F}_p[\tilde \Delta]$ and when $p\geq5$, $M=\mathbb{F}_p[\tilde E_4,\tilde E_6]/(A_p(\tilde E_4,\tilde E_6)-1)$, where $A_p$ is a polynomial in two variables, square-free, and homogeneous of degree $p-1$ if we consider $\tilde E_4$ of degree $4$ and $\tilde E_6$ of degree $6$. In particular, $M$ is in any case a noetherian domain of dimension $1$.

My question is:

In which cases $N>1$ has the structure of $M$ been determined? What is known about its structure?

I have not been able to find any reference discussing this question, but such references certainly exist. (I have found many references discussing the actions of the Hecke operators on the $M_k$ and $M$, and the algebras they generate, but this is not the same question: there is no Hecke operators in the one I am asking.)

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Dear Joel, For $X_1(N)$ you might look in Gross's Tameness criterion paper. The algebra $M$ is the same as the algebra of regular functions on the ordinary locus of the Igusa curve covering $X_1(N)$ in char. p. (I think that Serre obliquely remarks on this in article ; I vaguely remember that there is a comment that Spec $M$ has genus zero for $p < 13$, but has genus one when $p = 13$, and that this is related to the Igusa curve. I guess the point is that the genus of the Igusa curve (say for $N =1$, which is Serre's case) is equal to $1/2$ genus$(X_1(p) - X_0(p))$, which vanishes ... –  Emerton Apr 4 '12 at 2:02
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... for $p < 13$, but equals $1$ for $p = 13$.) Best wishes, Matt –  Emerton Apr 4 '12 at 2:03
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P.S. It will be the same for $X_0(N)$. –  Emerton Apr 4 '12 at 2:04
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P.P.S. And so $M$ is the coordinate ring of a smooth affine curve over $\mathbb F_p$, whose genus and number of punctures are easily computable. –  Emerton Apr 4 '12 at 2:06
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Dear Joel, Regarding your question about Dedekind domains, I guess the answer is yes. (I'd not thought of this explicitly before.) Sorry not to have written an answer; I'm following the path once taken by BCnrd, and restricting myself to comments. Best wishes, Matt –  Emerton Apr 5 '12 at 2:41
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