Some big picture questions -


1. What are some applications of the moduli interpretation for congruence curves? Specifically, the interpretations for congruence curves parametrizing elliptic curves with additional level structure.


(Edit: I know there must be tons, though I'm still having trouble finding good resources on this. I'd appreciate some references or maybe a quick summary)
2. Are there any known moduli interpretations for noncongruence modular curves? (Ie, either $X/\Gamma$ or $\mathcal{H}/\Gamma$ for some finite index non-congruence subgroup $\Gamma\subseteq\text{SL}\_2(\mathbb{Z})$).


Would people be interested if there was one?


I understand that this would depend on the interpretation itself - for example, for any normal subgroup $\Gamma\subset\text{SL}\_2(\mathbb{Z})$, you could always trivially say that $\mathcal{H}/\Gamma$ parametrizes isomorphism classes of elliptic curves together with the additional structure of a choice of an element in $\text{SL}\_2(\mathbb{Z})/\Gamma$.


This is why I asked my first question. In particular, what kinds of moduli interpretations for non-congruence curves would be interesting, if it existed?


thanks



moduli interpretations for modular curves
Over $\mathbb{C}$, elliptic curves with, say, a point of order $N$ can be identified with the quotient of the upper half plane $\mathbb{H}$ by $\Gamma\_1(N)$ just by associating to the class of $\tau\in \mathbb{H}$ the isomorphism class of the air $(\mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau), 1/N)$. Under this identification, modular forms of level $N$ can be realized as sections of a certain line bundle on "the space" (using the term loosely) of such isomorphism classes that is natural in a sense because is has only to do with this moduli interpretation (roughly speaking, the fiber at each point is a tensor power of the space of differentials on the associated elliptic curve).


One can take this observation a lot further to get a good notion of modular forms over base rings other than $\mathbb{C}$ by studying sections of these natural invertible sheaves on modular curves over these more general bases. In particular, one gets a good notion of $p$-adic analytic modular forms by looking at rigid-analytic moduli spaces of elliptic curves.


ADDED IN EDIT:


Regarding your second question, here is perhaps one reason related to my answer above to "believe" that such a moduli interpretation shouldn't exist. I'm not sure how convincing it is...


If there were such an interpretation, then one should be able to mimic the stuff in my answer above to get a geometric notion of modular forms for non-congruence subgroups over a more general class of rings, including, say, $\mathbb{Z}$. Then basic facts from algebraic geometry would kick in and tell you that the Fourier coefficients of such forms should have bounded denominators, as they do for congruence subgroups. This, however, is false for modular forms on non-congruence subgroups. I'm not sure of the history behind these results, but I know that Winnie Li and her collaborators have proven theorems in this area.


