Lecture 11: Friday, September 13, 2013

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 $f : \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ having two very nice properties:

• Meromorphic: $f(\tau) = p(\tau)/q(\tau)$ is the ratio of two analytic functions $p(\tau)$ and $q(\tau)$. Recall that analytic functions $\omega: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ satisfy the Cauchy-Riemann equations:
\dfrac {\partial u}{\partial x} = \dfrac {\partial v}{\partial y} \qquad \text{and} \qquad \dfrac {\partial u}{\partial y} = - \dfrac {\partial v}{\partial x} \qquad \text{where} \qquad \left \{ \begin{aligned} \tau & = x + i \, y, \\[5pt] \omega & = u + i \, v. \end{aligned} \right.

• Modular: $f(\tau+1) = f(-1/\tau) = f(\tau)$ for all $\tau \in \mathbb P^1(\mathbb C)$. This implies that $f \left( \dfrac {a \, \tau + b}{c \, \tau + d} \right) = f(\tau)$ for $\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in SL_2(\mathbb Z) = \left \{ \gamma \in \text{Mat}_{2\times 2}(\mathbb Z) \ \biggl| \ \det \, \gamma = 1 \right \}$.

It is easy to see that the constant functions are examples of elliptic modular function. Klein constructed a nontrivial example of such a function:

Theorem.
There are nonconstant elliptic modular functions. For example, one is the function $J: \mathbb H^2 \to \mathbb P^1(\mathbb C)$ which maps a complex number $\tau$ in the upper half plane $\mathbb H^2$ to the extended complex number

$J(\tau) = \dfrac {\displaystyle \left[ 1 + 240 \sum_{n=1}^{\infty} \sigma_3(n) \, q_1^n \right]^3}{\displaystyle 1728 \, q_1 \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)^{24}} = 1 + \dfrac {\displaystyle \left[ 1 - 504 \sum_{n=1}^{\infty} \sigma_5(n) \, q_1^n \right]^2}{\displaystyle 1728 \, q_1 \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)^{24}},$

where $q_h = e^{2 \pi i \tau/h}$ and $\sigma_h(n) = \sum_{d | n} d^h$ is the divisor function.

This function is known today as Klein’s elliptic modular $J$-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.

Following the German mathematician Richard Dedekind (1831 – 1916), we will use the Dedekind eta Function to construct more examples of elliptic modular functions.

Theorem.
Consider the function $\eta: \mathbb H^2 \to \mathbb P^1(\mathbb C)$ defined by $\eta(\tau) = \displaystyle q_1^{1/24} \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr)$. Then $\eta(\tau + 1) = e^{\pi i/12} \cdot \eta(\tau)$ and $\eta(-1/\tau) = \sqrt{\tau / i} \cdot \eta(\tau)$.

Note that the denominator of the $J$-invariant involves $1728 \, \eta(\tau)^{24}$.

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 $X(1) = Y(1) \cup \partial Y(1)$ in terms of
\begin{aligned} Y(1) & = \left \{ x + i \, y \in \mathbb C \ \biggl| \ 1 < x^2 + y^2, \ - \dfrac 12 < x < \dfrac 12, \ 0 < y \right \}, \\[5pt] \partial Y(1) & = \left \{ x + i \, y \in \mathbb C \ \biggl| \ x = - \dfrac 12, \ \dfrac {\sqrt{3}}{2} \leq y \right \} \cup \left \{ x + i \, y \in \mathbb C \ \biggl| \ 1 = x^2 + y^2, \ - \dfrac 12 \leq x \leq 0, \ 0 < y \right \} \cup \{ i \, \infty \}. \end{aligned}

We have the following standard result.

Theorem.
Denote $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb Q)$ as the extended upper half plane.

• The complex numbers $P = \zeta_4 = i$, $Q = {\zeta_6}^2 = - \dfrac {1}{2} + i \, \dfrac {\sqrt{3}}{2}$, and $R = i \, \infty$ are all elements of the boundary $\partial Y(1)$, and hence elements of $\mathbb H^\ast$. In fact, $J(P) = 1$, $J(Q) = 0$, and $J(R) = \infty$.

• $J: \mathbb H^\ast \to \mathbb P^1(\mathbb C)$ is surjective. In fact, the restriction $J: X(1) \to \mathbb P^1(\mathbb C)$ is a bijection.

• For each complex number $\omega = u + i \, v$ in $\mathbb H^\ast$, there exists a unique $\gamma \in SL_2(\mathbb Z)$ and $\tau = x + i \, y$ in $X(1)$ such that $\omega = \gamma \, \tau$.

For this reason, we sometimes write $X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast$. In particular, the elliptic modular function $J: \mathbb H^2 \to \mathbb P^1(\mathbb C)$ may be viewed as a well-defined function $\sigma \circ J: X(1) \to S^2(\mathbb R)$ when composing with stereographic projection $\sigma$.

$\begin{matrix} X(1) & \longrightarrow & \mathbb P^1(\mathbb C) & \longrightarrow & S^2(\mathbb R) \\[10pt] \tau & & J(\tau) = (z_1 : z_0) & & \sigma(z) = \left( \dfrac {2 \, \text{Re}(z_1 \, \overline{z_0})}{|z_1|^2 + |z_0|^2}, \ \dfrac {2 \, \text{Im}(z_1 \, \overline{z_0})}{|z_1|^2 + |z_0|^2}, \ \dfrac {|z_1|^2 - |z_0|^2}{|z_1|^2 + |z_0|^2} \right) \\[10pt] P & & (1:1) & & (+1,0,0) \\ Q & & (0:1) & & (0,0,-1) \\ R & & (1:0) & & (0,0,+1) \end{matrix}$
You may wish to compare the image $\sigma \circ J: V = \{ P, \, Q, \, R \} \hookrightarrow S^2(\mathbb R)$ with the points discussed in Lecture 9.

Upon coloring this region $X(1)$ one shade, we can give colors to the “translated” regions $\gamma \, X(1) = \left \{ \gamma \, \tau = \dfrac {a \, \tau + b}{c \, \tau + d} \in \mathbb C \ \biggl| \ \tau \in X(1) \right \}$ for each $\gamma = \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in SL_2(\mathbb Z)$. The “translates” give a “triangulation” of the upper-half of the extended complex plane, that is, $J \circ \gamma: V = \{ P, \, Q, \, R \} \hookrightarrow \mathbb H^\ast$ for each $\gamma \in SL_2(\mathbb Z)$.

Covering Spaces

We state a result which we will explain over the course of the next two lectures.

Proposition.
Let $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ be a compact, connected Riemann surface. Assume that there exists an $n$-fold covering map $\Pi: X \twoheadrightarrow S^2(\mathbb R)$ such that $\text{Aut}(\Pi)$ is a finite group.

Then, for each $m$-fold covering map $\phi: X \to \mathbb P^1(\mathbb C)$ which is meromorphic on $X$, that is, the composition $\phi \circ {\mu_\alpha}^{-1}$ is the ratio of two analytic functions on the subset $\mu_\alpha(U_\alpha) \subseteq \mathbb P^1(\mathbb C)$ for each index $\alpha \in I$, there exists a rational function $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ which makes the following diagram commute:

$\begin{matrix} X & \overset{\gamma}{\longrightarrow} & X & \overset{\phi}{\longrightarrow} & \mathbb P^1(\mathbb C) & & P & \mapsto & z = \phi(P) \\[10pt] \downarrow^\Pi & & \downarrow^\Pi & & \downarrow^f & & \downarrow & & \downarrow \\[10pt] S^2(\mathbb R) & = & S^2(\mathbb R) & & & & \Pi(P) & & \\[10pt] \downarrow^\simeq & & \downarrow^\simeq & & \downarrow & & \downarrow & & \downarrow \\[10pt] X(1) & = & X(1) & \overset{J}{\longrightarrow} & \mathbb P^1(\mathbb C) & & \tau = \bigl( J^{-1} \circ \sigma^{-1} \circ \Pi \bigr)(P) & \mapsto & J(\tau) = f(z) \end{matrix}$

Sketch of Proof: First we show the existence of $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$. Since $\phi: X \to \mathbb P^1(\mathbb C)$ is surjective, there exists a retraction $r: \mathbb P^1(\mathbb C) \to X$, that is, an analytic map such that the composition $r \circ \phi: X \to X$ is the identity. The map $f = \sigma^{-1} \circ \Pi \circ r$ satisfies $J \circ \bigl( J^{-1} \circ \sigma^{-1} \circ \Pi \bigr) = f \circ \phi$, so we can indeed express $J = f(z)$ as a function of $z$.

Next, we show that $J = f(z)$ is a rational function of $z$. For each $w \in \mathbb P^1(\mathbb C)$, consider the inverse image
$f^{-1}(w) = \left \{ z \in \mathbb P^1(\mathbb C) \, \biggl| \, f(z) = w \right \} \overset{r}{\longrightarrow} \left \{ P \in X \, \biggl| \, \Pi(P) = \sigma(w) \right \} = \Pi^{-1} \bigl( \sigma(w) \bigr).$
By assumption, there are exactly $n$ elements in the preimage $\Pi^{-1}(Q)$ and $m$ elements in the preimage $\phi^{-1}(w)$ — counting multiplicities — so $f^{-1}(w)$ also has exactly $m \, n$ elements.

For each $P \in X$, there are only finitely many $\text{Aut}(\Pi)$-conjugates $\gamma \, z = \bigl( \phi \circ \gamma \bigr)(P)$ of $z = \phi(P)$. Similarly, $\text{Aut}(\Pi)$ acts trivially on $f$:
$f ( \gamma \, z) = \bigl( \sigma^{-1} \circ \Pi \circ r \bigr)(\gamma \, z) = \bigl( \sigma^{-1} \circ \Pi \circ \gamma \bigr)(P) = \bigl( \sigma^{-1} \circ \Pi \bigr)(P) = f(z).$
Hence we may view $\text{Aut}(\Pi)$ as a group which acts on the variables $z$ but leaves $J = f(z)$ invariant. That is, we have a map $\phi^\ast: \text{Aut}(\Pi) \to \text{Aut}(f)$ which sends $\gamma: P \mapsto \gamma(P)$ in $\text{Aut}(\Pi)$ to $\phi^\ast(\gamma): z \mapsto \bigl( \phi \circ \gamma \circ r \bigr)(z)$ in $\text{Aut}(f)$. $\square$

Remarks

In general, we construct the objects above as follows. Fix a finite group $G$. Let $I \subseteq SL_2(\mathbb Z)$ be a finite set so that the disjoint union $X = \bigcup_{\gamma \in I} \gamma \, X(1)$ is a compact, connected Riemann surface endowed with the canonical surjection $\Pi: X \twoheadrightarrow X(1)$ which sends $\gamma \, \tau \mapsto \tau$. We seek a surjective, meromorphic function $\phi: X \to X(1)$ such that $J = f(z)$ is a rational function of $z = \phi(\tau)$ having automorphism group $G$. In fact, for each $\gamma \in I$ we would have
$\phi(\gamma \, \tau) = \dfrac {a \, \phi(\tau) + b}{c \, \phi(\tau) + b} \quad \text{for some} \quad \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in GL_2(\mathbb C) = \left \{ \gamma \in \text{Mat}_{2\times 2}(\mathbb C) \ \biggl| \ \det \, \gamma \neq 0 \right \}.$
(Our convention will be that $SL_2(\mathbb Z)$ acts on $\tau$, whereas $GL_2(\mathbb C)$ acts on $z$.) Eventually, we will call $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ a Belyĭ map. We will discuss this construction in more detail in the next lecture.