Over the past several lectures, we have focused on triangle groups and ways to tesselate both the plane and the sphere. In order to generalize this, we will focus on ways to tesselate compact, connected Riemann surfaces. Today we’ll begin our discussion on how to do this by focusing on complex functions with lots of symmetries: we give a historical approach using elliptic modular functions.
Elliptic Modular Functions
In the late 1800’s, several German mathematicians, including Robert Fricke (1861 – 1930), Felix Klein (1849 – 1925), and Karl Weierstrass (1815 – 1897), considered those functions having two very nice properties:
- Meromorphic: is the ratio of two analytic functions and . Recall that analytic functions satisfy the Cauchy-Riemann equations:
- Modular: for all . This implies that for .
It is easy to see that the constant functions are examples of elliptic modular function. Klein constructed a nontrivial example of such a function:
There are nonconstant elliptic modular functions. For example, one is the function which maps a complex number in the upper half plane to the extended complex number
where and is the divisor function.
This function is known today as Klein’s elliptic modular -invariant. It is nontrivial to show that this function is modular. It is this function which motivates the study of the subject of elliptic modular functions.
Consider the function defined by . Then and .
Note that the denominator of the -invariant involves .
Domains of Definition
Recall that we introduced Riemann surfaces in order to define the domain of definition of analytic function. Elliptic modular functions have many symmetries, so they are actually defined on a fundamental domain: Define in terms of
We have the following standard result.
Denote as the extended upper half plane.
- The complex numbers , , and are all elements of the boundary , and hence elements of . In fact, , , and .
- is surjective. In fact, the restriction is a bijection.
- For each complex number in , there exists a unique and in such that .
For this reason, we sometimes write . In particular, the elliptic modular function may be viewed as a well-defined function when composing with stereographic projection .
You may wish to compare the image with the points discussed in Lecture 9.
Upon coloring this region one shade, we can give colors to the “translated” regions for each . The “translates” give a “triangulation” of the upper-half of the extended complex plane, that is, for each .
We state a result which we will explain over the course of the next two lectures.
Let be a compact, connected Riemann surface. Assume that there exists an -fold covering map such that is a finite group.
Then, for each -fold covering map which is meromorphic on , that is, the composition is the ratio of two analytic functions on the subset for each index , there exists a rational function which makes the following diagram commute:
Sketch of Proof: First we show the existence of . Since is surjective, there exists a retraction , that is, an analytic map such that the composition is the identity. The map satisfies , so we can indeed express as a function of .
Next, we show that is a rational function of . For each , consider the inverse image
By assumption, there are exactly elements in the preimage and $m$ elements in the preimage — counting multiplicities — so also has exactly elements.
For each , there are only finitely many -conjugates of . Similarly, acts trivially on :
Hence we may view as a group which acts on the variables but leaves invariant. That is, we have a map which sends in to in .
In general, we construct the objects above as follows. Fix a finite group . Let be a finite set so that the disjoint union is a compact, connected Riemann surface endowed with the canonical surjection which sends . We seek a surjective, meromorphic function such that is a rational function of having automorphism group . In fact, for each we would have
(Our convention will be that acts on , whereas acts on .) Eventually, we will call a Belyĭ map. We will discuss this construction in more detail in the next lecture.