Are some congruence subgroups better than others?

Question

When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins to focus on the three types which are usually denoted by $\Gamma(N),$ $ \Gamma_0(N)$ and $\Gamma_1(N)$, where $N\in \mathbb{Z}_{\geq 1}$ and pretty much forgets about the rest. Back then I thought it was because these were the only ones that had a nice moduli space description. But, now that I know that $(\operatorname{GL}_2,\mathcal{H}^{\pm})$ is a PEL-type Shimura datum, I am not entirely convinced by my old reasoning either.

I have two possible guesses as to why this might be happening:

1. Since the Shimura variety attached to $(\operatorname{GL}_2,\mathcal{H}^{\pm})$ is the inverse system $\{\operatorname{Sh}_K(\operatorname{GL}_2,\mathcal{H}^{\pm})\}_K$ for all small enough $K$, it is enough that we care about the 'really small $K$s', because we can study the cohomologies, Galois action, Hecke operators just using these ones.

2. Another theory I have is related to the models over the reflex field. The complex points of the varieties of different levels sure have a nice moduli description using elliptic curves, but I believe there does not exist a moduli description over the reflex field (which I am guessing is $\mathbb{Q}$ in this case). What lead me to believe this, is the issues caused by Weil pairing at the level $\Gamma(N)$. So maybe in combination with the first point, it is better to study those really small $K$s, where we have a moduli description over some number field and Galois action and Hecke operators can possibly be understood using the moduli description?!

I am not entirely sure about either of these arguments though. Thanks in advance for your comments and answers.

Edit: Please note that although I have marked the answer by Joe Silverman as the correct one, this rather open question could have multiple possible answers. Hence, I strongly recommend that you read all the other answers as well!

Answer 1

Let me add two other guesses:

1. Especially in a class and/or for students first encountering this material in a book, it is best to start with concrete representative cases. $\Gamma(N)$, $\Gamma_0(N)$, and $\Gamma_1(N)$ certainly qualify as concrete, while also being sufficiently general to illustrate many of the fundamental ideas.

2. $\Gamma(N)$, $\Gamma_0(N)$, and $\Gamma_1(N)$ correspond to three extremely natural answers to the question: "What sort of level structure should I add to the moduli space of elliptic curves?" Very early in the study of elliptic curves, one sees the importance of the torsion subgroup. So it makes sense to classify an elliptic curve with a cyclic subgroup of order $N$, or a point of order $N$, or all of its points of order $N$. (Yes, I know $\Gamma(N)$ doesn't quite do the latter, but close enough.)

Answer 2

From the point of view of modular forms, each of the three congruence subgroups $\Gamma_0(N), \Gamma_1(N), \Gamma(N)$ has a useful feature:

Forms on $\Gamma_1(N)$ are exactly forms arising from representations of conductor dividing $N$.

Forms on $\Gamma_0(N)$ are exactly forms arising from representations of conductor dividing $N$ with trivial central character.

Forms on $\Gamma(N)$ have an action of $SL_2( \mathbb Z/N)$, which you can use to make the connection of modular forms to representation theory.

For these reasons, all of these subgroups come up often in research. The first two properties have analogues for Shimura varieties arising from forms of $GL_n$ (at any place where the group splits to $GL_n$) using the mirabolic subgroup and the last property has analogues for every Shimura variety.

Answer 3

This question already has multiple nice answers, but I am going to add one more thing which isn't quite covered by the existing posts.

One distinctive advantage of the $\Gamma_0(N)$ and $\Gamma_1(N)$ subgroups comes from new-vector theory: for these levels, we can split up the space of modular forms into an "old part" (built up in a rather explicit fashion from $\Gamma_1(M)$ with $M < N$) and a "new part" (the stuff that genuinely belongs at level $N$). Then, for newforms, we have very powerful uniqueness results like the strong multiplicity one theorem (which says that if two Hecke eigenforms in the new subspace have the same Galois representation, they are equal).

These properties of $\Gamma_1(N)$ would break down if you substituted some other random family of congruence subgroups in place of $\Gamma_1(N)$. Moreover, finding families of subgroups that give you similar multiplicity-one results for bigger reductive groups in place of $GL_2$ is a highly nontrivial question and an active topic of research right now.