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Is there a well-understood correspondence between subgroups G of $SL_2(\mathbb{Z})$ (not necessarily of finite index) and graded algebras of modular forms invariant under G?

Given an algebra of modular forms (on the upper half-plane) known to be an algebra of forms M(G) invariant under some modular subgroup G can one reconstruct the subgroup G? (Possibly, the group G should arise as a deck transformation group for the covering Proj M(G) -> Proj M($SL_2(\mathbb{Z}))=(?)\mathbb{P}^1(\mathbb{C})$)) arising from the natural embedding M($SL_2(\mathbb{Z})$)) -> M(G) .)

(By the way, what is a good reference for general properties of Proj M(G)?)

If this is not enough can we reconstruct the (modular) subgroup from the knowledge of its invariant algebras of p-adic modular forms?

What can we say about the situation in case of discrete subgroups of $PSL_2(\mathbb{R})$?

Do you know any papers considering the situation of fuchsian groups (structure of algebra of modular forms etc.)?

Do you know any papers considering algebras of classical and p-adic modular forms for the case of subgroups G of infinite index in $SL_2(\mathbb{Z})$?

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What is your definition of a $p$-adic modular form? –  Marc Palm Aug 11 '12 at 10:53
    
You're asking, when I understand the question correctly, whether knowing the algebra abstractly is equivalent to knowing the Fuchsian group abstractly? Note that the fundamental group of $X= \Gamma \backslash H$ is abstractly isomorphic to the Fuchsian group, but you need the representation of the Fuchsian group inside $SL_2(R)$ to reconstruct the Riemann surface $X$ up to iso. The algebras of non-isomorphic Riemann surfaces are (necessarily?) different. –  Marc Palm Aug 11 '12 at 11:10
    
Okay, I see now the representation is fixed because you regard them as subgroups of $SL_2(\mathbb{Z})$. Nevertheless you might want to mod out conjugation. –  Marc Palm Aug 11 '12 at 11:14

1 Answer 1

If you know them as algebras over $M(SL_2(\mathbb Z))$, almost exactly the procedure you describe works. Since the modular forms with weight a multiple of $12$ are sections of powers of an ample line bundle on the modular curve associatied to that subgroup, $\operatorname{Proj} $ of the ring is just that modular curve. The $SL_2(\mathbb Z)$-invariant subalgebra defines a map to $X(1)$ with certain ramification data (index $1$ or $2$ over one elliptic point and $1$ or $3$ over the other, any ramification over the cusp, unramified everyhere else), which implies that the monodromy action on the fiber over an unramified point factors through $PSL_2(\mathbb Z)$, and the stabilizer of a point (or its lift to $SL_2(\mathbb Z)$, rather) is the subgroup you want.

Fully abstractly, you can't. Consider two modular curves isomorphic to $\mathbb P^1$ with four cusps and no elliptic points, say $X(3)$ and $X_1(5)$. Their associated algebras $M(\Gamma(3))$ and $M(\Gamma_1(5))$ are both the algebra of even-degree functions on $\mathbb P^1$, so they are isomorphic.

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@Mrc Plm: to tell the truth, I don't know exactly which definition to use. Yes, probably the reconstruction will work only up to a congugation. –  N B Aug 12 '12 at 4:22
    
@Will Sawin: well, now I want to formulate it in the following terms: Is it possible to define (in a reasonable way) a scheme over $\mathbb{Z}$ such that it's fiber over $\mathbb{C}$ is given by $Proj M(\Gamma)$ and this scheme determines uniquely the group $\Gamma$ in $SL_2(\mathbb{Z})$? Your comment shows that one really needs to understand what the fibers over finite fields are. I don't know if it is possible to define such a scheme in order to get the fibers over $\mathbb{Q}_p$ as spectrum of what I call "p-adic modular forms". –  N B Aug 12 '12 at 4:22
    
Or maybe, one needs an analogue of modular forms over finite fields to obtain fibers of our scheme over finite fields as spectrums of corresponding algebras over $\mathbb{F}_p$. –  N B Aug 12 '12 at 4:22
    
@Will Sawin: When you say that "modular forms with weight multiple of 12 are sections of powers of an ample line bundle on the modular curve", how are we thinking of that modular curve? For example, for X(1), which as a scheme is isomorphic to $\mathbb{P}^1/\mathbb{C}$, the only line bundles should be the twisted sheaves $\mathcal{O}(n)$. Are you saying that in the case of forms on curves isomorphic to $\mathbb{P}^1$, this ample line bundle is just one of these $\mathcal{O}(n)$? In that case, which one is it? –  oxeimon Aug 12 '12 at 5:21

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