Automorphic and modular forms for subgroups of modular group and fuchsian groups 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 $SL_2(\mathbb{Z})$? 
 A: 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.
