Suppose that with . Then consider the lattice . We can form a torus as . We call meromorphic functions on elliptic functions with respect to .
Now suppose that is an elliptic function with the additional property that we can almost scale the arguments: for . Then in particular we can scale to 1: , where now is defined on . If we fix and consider as a function on , then we have for any . (If , then we define .)
We call a function satisfying for all a modular function.
The point of all that is the elliptic functions can be naturally turned into modular functions.