## Lecture 13: Wednesday, September 18, 2013

In Lecture 11 and Lecture 12, we discussed examples of Riemann Surfaces $X$ and their relation with the Riemann Sphere $X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R)$. 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 $\mathbb H^\ast \to \mathbb P^1(\mathbb C)$ which are defined on the extended upper half plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C)$. For example,
\begin{aligned} 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}} \\[5pt] J_{2,0}(\tau) & = \left( \dfrac {\eta(\tau)}{\eta(2 \, \tau)} \right)^{24} = \displaystyle \left[ q_1 \, \prod_{n=1}^{\infty} \bigl( 1 + q_1^n \bigr)^{24} \right]^{-1} \\[5pt] \lambda(\tau) & = 16 \, \dfrac {\eta(\tau/2)^8 \, \eta(2 \, \tau)^{16}}{\eta(\tau)^{24}} = \displaystyle 16 \, q_2 \prod_{n=1}^{\infty} \dfrac { \bigl(1+q_2^{2n} \bigr)^{16}}{\bigl( 1 + q_2^n \bigr)^8} \\[5pt] \eta(\tau) & = \displaystyle q_1^{1/24} \prod_{n=1}^{\infty} \bigl( 1 - q_1^n \bigr) \end{aligned}
where $q_h = e^{2 \pi i \tau/h}$ and $\sigma_h(n) = \sum_{d | n} d^h$ is the divisor function. These functions were constructed by considering various subsets $I$ in $SL_2(\mathbb Z) = \left \{ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in \text{Mat}_{2 \times 2}(\mathbb Z) \, \biggl| \, a \, d - b \, c = 1 \right \}$.

In this week’s homework assignment, we see that $SL_2(\mathbb Z)$ is generated by the $2 \times 2$ matrices $S = \left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]$ and $T = \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right]$. Using the map $SL_2(\mathbb Z) \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$, we see that $S$, $S \, T$, and $T^{-1}$ map to the Möbius transformations $\gamma_0(z) = - \dfrac {1}{z}$, $\gamma_1(z) = - \dfrac {1}{z+1}$, and $\gamma_{\infty}(z) = z - 1$, respectively. In fact, the image of $SL_2(\mathbb Z)$ is the triangle group
$\Gamma(1) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^2 = {\gamma_1}^3 = \gamma_0 \, \gamma_1 \, \gamma_\infty \right \rangle = D(2,3,\infty) \simeq PSL_2(\mathbb Z).$
This is called the modular group.

The points $P = \zeta_4 = \sqrt{-1}$, $Q = {\zeta_6}^2 = - \dfrac 12 + i \, \dfrac {\sqrt{3}}{2}$, and $R = i \, \infty$ are the unique fixed points of $\gamma_0$, $\gamma_1$, and $\gamma_\infty$, respectively, so that we find a $0^\circ - 60^\circ - 90^\circ$ hyperbolic triangle $V = \{ P, \, Q, \, R \}$. The images $\gamma \, V = \{ \gamma \, P, \ \gamma \, Q, \ \gamma \, R \}$ of the triangles for $\gamma \in \Gamma(1)$ yield a tessellation of the extended upper half plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C)$.

### Hecke’s Modular Group

We can completely generalize this. Fix an integer $n \geq 3$, and consider the root of unity $\zeta_{2n} = \cos \dfrac {\pi}{n} + i \, \sin \dfrac {\pi}{n}$. Upon defining the positive real number $\lambda = \zeta_{2n} + {\zeta_{2n}}^{-1} = 2 \, \cos \dfrac {\pi}{n}$, we consider the $2 \times 2$ matrices $S = \left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]$ and $T = \left[ \begin{matrix} 1 & \lambda_n \\ 0 & 1 \end{matrix} \right]$. The group $\left \langle S, \, T \, \biggl| \, S^2 = T^n = -I_2 \right \rangle \subseteq SL_2(\mathbb R)$, so we consider the same map $SL_2(\mathbb R) \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$ as before. We see that $S$, $S \, T$, and $T^{-1}$ map to the Möbius transformations $\gamma_0(z) = - \dfrac {1}{z}$, $\gamma_1(z) = - \dfrac {1}{z+ \lambda_n}$, and $\gamma_{\infty}(z) = z - \lambda_n$, respectively. In fact, the image of the group generated by $S$ and $T$ is the triangle group
$H_n = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^2 = {\gamma_1}^n = \gamma_0 \, \gamma_1 \, \gamma_\infty \right \rangle = D(2,n,\infty) \hookrightarrow PSL_2(\mathbb R).$
This is called the Hecke modular group. Since this is a discrete subgroup of $PSL_2(\mathbb R)$, we call $H_n$ a Fuchsian group.

The points $P = \zeta_4$, $Q = {\zeta_{2n}}^{n-1}$, and $R = i \, \infty$ are the unique fixed points of $\gamma_0$, $\gamma_1$, and $\gamma_\infty$, respectively, so that we find a $0^\circ - (180/n)^\circ - 90^\circ$ hyperbolic triangle $V = \{ P, \, Q, \, R \}$. The images $\gamma \, V = \{ \gamma \, P, \ \gamma \, Q, \ \gamma \, R \}$ of the triangles for $\gamma \in H_n$ yield a triangulation of the extended upper half plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1(\mathbb C)$.

When $n = 3$, we see that $H_3 = \Gamma(1)$ is the usual modular group. In fact, the Fundamental Domain for this group is the set
\begin{aligned} X(1) & = \left \{ x + i \, y \in \mathbb C \ \biggl| \ 1 < x^2 + y^2, \ - \dfrac 12 < x < \dfrac 12, \ 0 < y \right \} \\[5pt] & \cup \left \{ x + i \, y \in \mathbb C \ \biggl| \ x = - \dfrac 12, \ \dfrac {\sqrt{3}}{2} \leq y \right \} \\[5pt] & \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}
When $n \geq 4$, 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.

### Covering Spaces

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 $\left( X, \, \{ U_\alpha \}, \, \{ \mu_\alpha \} \right)$ and $\left( Y, \, \{ V_\beta \}, \, \{ \nu_\beta \} \right)$ be compact, connected Riemann surfaces for finite indexing sets $I$ and $J$, respectively. A map $\Pi: X \to Y$ is called an $n$-fold covering map or an $n$-fold covering projection if, for each $Q \in Y$ we can find an open set $V \subseteq Y$ containing $Q$ (that is, $\nu_\beta(V) \subseteq \mathbb C$ is open for each $\beta \in J$) such that the following holds:

• Surjectivity: The inverse image $\Pi^{-1}(V) = U_1 \cup U_2 \cup \cdots \cup U_n$ is a union of nonempty open sets $U_i \subseteq X$, that is, $\mu_\alpha(U_i) \subseteq \mathbb C$ is open for each $\alpha \in I$ and $1 \leq i \leq n$.

• Disjoint Sheets: $U_i \cap U_j = \emptyset$ for $1 \leq i, j \leq n$ with $i \neq j$.

• Local Homeomorphisms: $\Pi(U_i) \simeq V$ in $Y$ for $1 \leq i \leq n$.

• Analytic: For every $\alpha \in I$ and $\beta \in J$, each $\bigl( \nu_\beta \circ \Pi \bigr)(U_\alpha)$ is a connected, open subset of $\mathbb C$. Moreover, the transition maps $\omega_{\beta \alpha} = \nu_\beta \circ \Pi \circ {\mu_\alpha}^{-1}$ are analytic functions.

$\begin{matrix} \mathbb C & & X & & Y & & \mathbb C \\[10pt] \uparrow & & \uparrow & & \uparrow & & \uparrow \\[10pt] \mu_\alpha \bigl( U_\alpha \cap \Pi^{-1}(V_\beta) \bigr) & \overset{{\mu_\alpha}^{-1}}{\longrightarrow} & U_\alpha \cap \Pi^{-1}(V_\beta) & \overset{\Pi}{\longrightarrow} & \Pi(U_\alpha) \cap V_\beta & \overset{\nu_\beta}{\longrightarrow} & \nu_\beta \bigl( \Pi(U_\alpha) \cap V_\beta \bigr) \end{matrix}$

An automorphism $\gamma \in \text{Aut}(X)$ is an invertible map $\gamma: X \to X$ such that the composition $\gamma \circ {\mu_\alpha}^{-1}$ is an analytic function on the subset $\mu_\alpha(U_\alpha) \subseteq \mathbb C$ for each index $\alpha \in I$. Given a covering map $\Pi: X \to Y$, we denote the deck transformation group as the set
$\text{Aut}(\Pi) = \left \{ \gamma \in \text{Aut}(X) \ \biggl| \ \bigl( \Pi \circ \gamma \bigr)(P) = \Pi(P) \ \text{for all} \ P \in X \right \}.$
Elements $\gamma \in \text{Aut}(\Pi)$ are called deck transformations.

### Constructing Covering Spaces

The examples discussed in the previous lectures considered covering maps of the Riemann Surface $Y = X(1) = SL_2(\mathbb Z) \backslash \mathbb H^\ast \simeq \mathbb P^1(\mathbb C) \simeq S^2(\mathbb R).$

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 $\phi: X \to \mathbb P^1(\mathbb C)$, there exists a rational function $f: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C)$ which makes the following diagram commute:

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

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 $G$.

For instance, recall that in Lecture 12 we considered the sets
\begin{aligned} I & = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & -1 \\ 1 & \ \ 0 \end{matrix} \right], \ \left[ \begin{matrix} 0 & -1 \\ 1 & \ \ 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} \ \ 0 & 1 \\ -1 & 0 \end{matrix} \right], \ \left[ \begin{matrix} -1 & \ \ 0 \\ \ \ 1 & -1 \end{matrix} \right] \right \} \\[10pt] J & = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right], \ \left[ \begin{matrix} 1 & -1 \\ 1 & \ \ 0 \end{matrix} \right], \ \left[ \begin{matrix} 0 & -1 \\ 1 & \ \ 1 \end{matrix} \right] \right \} \\[5pt] K & = \left \{ \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \right \} \end{aligned}

Then we have the covering maps

$\begin{matrix} X(2) = \bigcup_{\gamma \in I} \gamma \, X(1) & \longrightarrow & X_0(2) = \bigcup_{\gamma \in J} \gamma \, X(1) & \longrightarrow & X(1) = \bigcup_{\gamma \in K} \gamma \, X(1) \\[10pt] \downarrow & & \downarrow & & \downarrow \\[10pt] \mathbb P^1(\mathbb C) & \longrightarrow & \mathbb P^1(\mathbb C) & \longrightarrow & \mathbb P^1(\mathbb C) \end{matrix}$

explicitly defined by
$\begin{matrix} \tau & \mapsto & \tau & \mapsto & \tau \\[10pt] \downarrow & & \downarrow & & \downarrow \\[10pt] z= \lambda(\tau) & \mapsto & w = J_{2,0}(\tau) = 256 \, \dfrac {1 - z}{z^2} & \mapsto & J(\tau) = \dfrac {(w + 256)^3}{1728 \, w^2} = \dfrac {4}{27} \, \dfrac {( z^2 - z+ 1 \bigr)^3}{z^2 \, (z- 1)^2} \end{matrix}$

The deck transformations can be associated with those Möbius transformations which fix $J = f(z)$. In the specific case above, the rational function $J = f(z)$ is invariant under $r(z) = \dfrac {z-1}{z}$ and $s(z) = \dfrac {z}{z-1}$. Since $s^2 = r^3 = (s \circ r)^2 = 1$, we see that $J = \dfrac {4}{27} \, \dfrac {( z^2 - z+ 1 \bigr)^3}{z^2 \, (z- 1)^2}$ is invariant under the symmetric group $S_3 = \left \langle r, \, s \, \bigl| \, s^2 = r^3 = (s \circ r)^2 = 1 \right \rangle$, while $J_{2,0}(\tau) = 256 \, \dfrac {1 - z}{z^2}$ is invariant under the cyclic group $Z_2 = \left \langle 1, \, s \right \rangle$. This means we have the following diagram of subfields and corresponding Galois groups:

$\begin{matrix} X(2) & \qquad & \mathbb C(\lambda) & \qquad & \left \langle 1 \right \rangle \\[10pt] \downarrow & & \uparrow & & \downarrow \\[10pt] X_0(2) & & \mathbb C(J_{2,0}) & & Z_2 \\[10pt] \downarrow & & \uparrow & & \downarrow \\[10pt] X(1) & & \mathbb C(J) & & S_3 \end{matrix}$

### Motivating Questions

Let $Y = S^2(\mathbb R)$ denote the Riemann Sphere. We have seen that there are $n$-fold covering maps $\Pi: X \to Y$ for certain values of $n$. Is there a universal cover $\phi: \widetilde{Y} \to Y$ such that each cover $\Pi: X \to Y$ corresponds to a cover $\widetilde{\Pi}: \widetilde{Y} \to X$ such that $\phi = \Pi \circ \widetilde{\Pi}$? We will return to this question.