moduli interpretations for modular curves

Question

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

Answer 1

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.

Answer 2

For your first question, I can give an application close to my own field. Perhaps a more Langlands-ish person can say something interesting about applications to modularity of Galois representations.

The moduli interpretation of level structures can appear naturally in orbifold 2D conformal field theory. In ordinary 2D CFT, one has a notion of correlation function, that takes as input a Riemann surface together with some data decorating closed submanifolds, and outputs a number. For elliptic curves with trivial decorations, one then obtains modular functions. In orbifold CFT, the input data is upgraded to a Riemann surface with a branched $G$-cover, for $G$ a finite group. In this case, for elliptic curves with trivial decorations, one obtains functions on moduli spaces of elliptic curves with $G$-torsors.

The simplest non-trivial case is for $G$ a cyclic group of order $N$. If we ignore the problem of fixing a basepoint, a $G$-torsor over an elliptic curve $E$ is a disjoint union of $N/k$ isomorphic elliptic curves, each with a cyclic $k$-isogeny to $E$ whose kernel has a distinguished generator, related by a cyclic group of isomorphisms, for some $k$ dividing $N$. Keeping this in mind, it is not hard to show that the (coarse) moduli space is a disjoint union of $Y_1(k)$ as $k$ ranges over divisors of $N$. In other words, correlation functions in this setting are given by a list of modular functions whose levels are divisors of $N$.

This particular family of examples has come up in my own work, because there is a natural notion of Hecke operator $T_n$ on these functions. The condition that $nT_nf$ is a monic polynomial in a function $f$ for lots of $n$ implies the restriction of $f$ to any component generates the function field for some genus zero quotient of $\mathcal{H}$. The monic polynomial condition then allows for the construction of holomorphic infinite products as Borcherds-Harvey-Moore lifts, and makes the analysis of some closely related infinite dimensional Lie algebras tractable.

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For your second question, I vaguely recall hearing about some work of Scholl to the effect that non-congruence forms have some connections to automorphic forms for higher-rank groups through their L-functions. Bold guesswork might suggest that non-congruence curves arise from natural moduli problems related to higher-rank motives, but presumably less straightforwardly than the way Shimura curves are moduli spaces of abelian surfaces with quaternionic multiplication.

Answer 3

Concerning q.2, this looks like about it: https://www.math.psu.edu/calendars/meeting.php?id=24844