Modular forms are usually considered as complex-valued functions on the upper half-plane quite invariant by a discrete subgroup $\Gamma$ of isometries and satisfying smoothness and growth condition.
My question is about $\Gamma$, which is generally an arithmetic subgroup, or even assumed to be a congruence subgroup. We can think of other kinds of interesting isometry subgroups:
- cocompact subgroups (e.g. orders of division quaternion algebras)
- thin subgroups (with polygonal fundamental domain but with a whole edge on the boundary)
I wonder whether or not the theory adapt to those cases (for instance the Fourier expansion and the cusp notion do not carry on the cocompact setting... but we do study Maass forms, and they are as well interesting but we know that the proper setting is the arithmetical subgroups only, otherwise there is no more Weyl's law) and if they are interesting.
Any new insight coming from elliptic curves, Riemannian geometry, arithmetic or elsewhere is welcome.
