Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$? In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic geometry. It seems as if there is some relation between hyperbolic geometry and modular forms, for example, why is it precisely the set $\mathbb{H}$ (from which modular forms map into $\mathbb{C}$) that is also a model for a "weird" geometry in which the sum over the angles in a triangle is not $\pi$ or in which some axiom about parallel lines does not hold? It seems at first sight, as if these two mathematical areas are quite distant from each other.
If there is such a relation, can someone solve the following equation:
$$ \frac{\text{modular forms}}{\text{hyperbolic geometry}} = \frac{???}{\text{euclidean geometry}}$$
Of course, one can reinterpret modular forms as certain sections of line bundles over ... blah blah blah, but this is not the way you would ever describe what a modular form is to someone who has never heard about them.
cheers,
FW
 A: I would say that modular forms come from looking at the automorphism group of $\mathbb{H}$, which is $PSL_2(\mathbb{R})$. In particular, we look at nice discrete co-compact subgroups (i.e. subgroups for which the quotient $\mathbb{H}/\Gamma$ is compact), such as $PSL_2\mathbb{Z}$. In this sense, modular forms are a specific example of automorphic forms.
For $\mathbb{R}^2$, Euclidean space, the automorphism group is (I think?) $\mathbb{R}^2 \rtimes O(2)$. I'm not sure exactly what the discrete co-compact subgroups of this are, but I would suspect that they are a lot less interesting than those that arise from looking at the hyperbolic plane. Most likely, all that you get is the study of elliptic functions (i.e. functions that are defined on an elliptic curve, which is the quotient of $\mathbb{C} = \mathbb{R}^2$ modulo a lattice).
Now, you can combine these together to look at Jacobi forms...
A: If you were "working a lot with modular forms" and their relation with the Poincare model of hyperbolic geometry is a surprise for you, I recommend that you read some history to understand where modular forms came from and what are some of their uses. An excellent book to begin with is F. Klein, Lectures on development of mathematics in XIX century.
Or some old book on Elliptic functions.
