What are all the subgroups of SL_2(Z)?

Question

I'm learning about modular forms and I dislike the congruence subgroups. They feel inadequately motivated to me, and they beg the question of what other subgroups of SL_2(Z) are there that may be just as significant? So are there other subgroups that have cool properties?

Answer 1

For each $p$, we have a map $\text{SL}_2(\mathbb{Z}) \to \text{SL}_2(\mathbb{F}_p)$.

This map is surjective, although it is not obvious that it is surjective (you can lift any matrix, but how do you know you can you find a lift with determinant one?). The surjectivity is known, in fancy language, as "strong approximation for $\text{SL}_2$", although there are elementary (but not easy) proofs.

Now, there's a ton of "theory" that applies to $\text{SL}_2(\mathbb{F}_p)$ because these are groups of rational points of an algebraic group (google "finite group of Lie type").

In particular you have the abstract theory of Borel subgroups, unipotent subgroups, tori, etc., and for any such subgroup of $\text{SL}_2(\mathbb{F}_p)$ you can consider its pre-image in $\text{SL}_2(\mathbb{Z})$.

That's where all the congruence subgroups are coming from, and why they are important and we can prove things about them.

But there are non-congruence subgroups of finite index, so congruence groups aren't the whole world, either.