Last time, we introduced the Fundamental Domain in terms of the extended complex plane . Felix Klein showed the existence of a function , invariant under , giving these isomorphisms. Today, we use Richard Dedekind‘s function to create similar functions .

### Covering Spaces

Recall the main result from the previous lecture:

Proposition.

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:

Eventually, we will call a Belyĭ map. This is a very strange result which seems to have little importance. We explain via a series of examples how this motivated the entire study of modular functions.

### Modular Curve

Consider the set for the finite set . We can draw this finite union using Helena Verrill‘s Fundamental Domain drawer, or even Andrzej Kozlowski‘s Wolfram Demonstration. Either way, we see that is a compact, connected, Riemann Surface.

For each subset , we see that Klein’s is invariant under the action , so that is a trivial function: indeed, it factors through the projection which sends . However, a nontrivial function is given by

(Recall that .) More information about this function can be found on the Wikipedia page on Classical Modular Curves as well as the On-Line Encyclopedia of Integer Sequences.

Proposition.

Define in terms of .

- is a bijection.
- We are able to express the -invariant as a rational function of this function:

Observe that the projection map sending is 3-to-1, and the rational function such that in terms of has . This is not a coincidence!

### Modular Curve

Now consider the compact, connected, Riemann Surface for the finite set

As before, we can draw this finite union using Helena Verrill‘s Fundamental Domain drawer, or even Andrzej Kozlowski‘s Wolfram Demonstration.

We have two projections so that both and are functions from . A nontrivial function which has as its domain of definition is

(Recall that .) The parameter is often called the nome, while the function is often called the elliptic modular lambda function. More information about this function can be found on the On-Line Encyclopedia of Integer Sequences.

Proposition.

Define in terms of .

- is a bijection.
- and .

The following diagram may be useful to keep track of everything:

where we map

### Relation with Galois Groups

In these examples, we have seen how the projection maps yield rational functions , but we have not yet introduced the group . Observe that for each we have

In fact, we can say more:

in terms of the rational functions and . Since , we see that the following subgroup of acts on :

We call this realization of the symmetric group the Anharmonic Group.

Let’s focus on the subgroups of for the moment. It is easy to check that is invariant under while is invariant under both and . In other words, we may think of as the Galois group of the extension over ; and we may think of the subgroup as the Galois group of the extension over . Similarly, we may think of and as the Galois groups of over and , respectively.

### Open Question

What about the subgroup ? This corresponds to the Galois group over . (Recall that is a root of unity. It appears that the function

is a nontrivial function which has the Riemann surface as its domain of definition. Is this function a classical one in the literature? The coefficients do not appear in the On-Line Encyclopedia of Integer Sequences.

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