Elliptic Modular Forms

In connection with the question which complex numbers can be written as the absolute invariant of a lattice, we were led to analytic functions with a new type of symmetries. These functions are analytic functions on the upper half-plane with a specific transformation law with respect to the action of the full elliptic modular group (or of certain subgroups) on H, namely

$$f\left(\frac{az+b}{cz+d}\right)=(cz + d)^k f(z).$$

Functions with such a transformation behavior are called modular forms.

We will see that the elliptic modular group is generated by the substitutions

$$z \mapsto z + 1\ {\rm{and}}\ z \mapsto -\frac{1}{z}.$$

It is thus enough to check the transformation behavior only for these substitutions. There is an analogy to the transformation behavior of elliptic functions under translations in a lattice L, where it was also sufficient to check the invariance under the two generating translations ω₁, ω₂. But in contrast to the translation lattice L, the elliptic modular group is not commutative. Hence the theory of modular forms is more complicated than the theory of elliptic functions. This could be already observed in the construction of a fundamental domain for the action of the modular group Γ on the upper half-plane H, V.8.7.