In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces and their relation with the Riemann Sphere . Today, we put things into perspective by considering the larger picture: we discuss tesselations of the upper half plane, covering spaces, and deck transformations.
The Modular Group
We have seen several functions which are defined on the extended upper half plane . For example,
where and is the divisor function. These functions were constructed by considering various subsets in .
In this week’s homework assignment, we see that is generated by the matrices and . Using the map , we see that , , and map to the Möbius transformations , , and , respectively. In fact, the image of is the triangle group
This is called the modular group.
The points , , and are the unique fixed points of , , and , respectively, so that we find a hyperbolic triangle . The images of the triangles for yield a tessellation of the extended upper half plane .
Hecke’s Modular Group
We can completely generalize this. Fix an integer , and consider the root of unity . Upon defining the positive real number , we consider the matrices and . The group , so we consider the same map as before. We see that , , and map to the Möbius transformations , , and , respectively. In fact, the image of the group generated by and is the triangle group
This is called the Hecke modular group. Since this is a discrete subgroup of , we call a Fuchsian group.
The points , , and are the unique fixed points of , , and , respectively, so that we find a hyperbolic triangle . The images of the triangles for yield a triangulation of the extended upper half plane .
When , we see that is the usual modular group. In fact, the Fundamental Domain for this group is the set
When , the description of the Fundamental Domain gets a bit more complicated. A very nice paper by Ronald Evans entitled A fundamental region for Hecke’s modular group sheds some light on this problem.
Meromorphic functions in general and elliptic modular functions in particular can be cumbersome and unintuitive, but we introduce a rather technical idea to simplify the discussion.
Let and be compact, connected Riemann surfaces for finite indexing sets and , respectively. A map is called an -fold covering map or an -fold covering projection if, for each we can find an open set containing (that is, is open for each ) such that the following holds:
- Surjectivity: The inverse image is a union of nonempty open sets , that is, is open for each and .
- Disjoint Sheets: for with .
- Local Homeomorphisms: in for .
- Analytic: For every and , each is a connected, open subset of . Moreover, the transition maps are analytic functions.
An automorphism is an invertible map such that the composition is an analytic function on the subset for each index . Given a covering map , we denote the deck transformation group as the set
Elements are called deck transformations.
Constructing Covering Spaces
The examples discussed in the previous lectures considered covering maps of the Riemann Surface
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 , there exists a rational function which makes the following diagram commute:
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 .
For instance, recall that in Lecture 12 we considered the sets
Then we have the covering maps
explicitly defined by
The deck transformations can be associated with those Möbius transformations which fix . In the specific case above, the rational function is invariant under and . Since , we see that is invariant under the symmetric group , while is invariant under the cyclic group . This means we have the following diagram of subfields and corresponding Galois groups:
Let denote the Riemann Sphere. We have seen that there are -fold covering maps for certain values of . Is there a universal cover such that each cover corresponds to a cover such that ? We will return to this question.