Why only congruence subgroups for modular forms?

Question

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of simplifying life or it has a more theoretic advantage?

Answer 1

One can consider other finite index subgroups, and people do. It may be that Shimura's book begins in this general context (I don't remember now).

But it is only in the context of congruence subgroups that one has the theory of Hecke operators and the resulting theory of Hecke eigenforms, and this is a big part of the theory for people interested in the connections between modular forms and arithmetic.

Also, the congruence case admits a treatment in terms of adeles, whereas the general case doesn't. Even for topics that don't necessarily require the subgroup to be congruence, this can makes the formulation simpler, and so provides another reason to restrict to the congruence case.

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If you would like to see some recent literature on the non-congruence case, you could google the "Atkin--Swinnerton-Dyer congruence relations''.

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Also, it seems worth remarking that when one passes from $GL_2$ (the case of classical modular forms) to the case of $GL_n$ for $n\geq 3$, then all finite index subgroups of $SL_n(\mathbb Z)$ are automatically congruence subgroups.