## Homework Assignment 2

Recall that the collection of Möbius transformations is the group
$\text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr) = \left \{ \gamma: \mathbb P^1(\mathbb C) \to \mathbb P^1(\mathbb C) \, \biggl| \, \gamma(z) = \dfrac {a \, z + b}{c \, z + d} \ \text{for some} \ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in GL_2(\mathbb C) \right \},$
while the (ordinary) triangle group is the set
$D(m,n,k) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \bigl| \, {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.$
This homework set is meant to discuss the relationship with these two sets. This assignment is due Friday, September 20, 2013 at the start of class.

Problem 1. Fix an integer $n \geq 3$. Upon denoting the positive real number $\lambda_n = 2 \, \cos \dfrac {\pi}{n} = \zeta_{2n} + {\zeta_{2n}}^{-1}$ in terms of a primitive $2n$-th root of unity $\zeta_{2n} = \cos \dfrac {\pi}{n} + i \, \sin \dfrac {\pi}{n}$, consider the $2 \times 2$ matrices $S = \left[ \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]$ and $T = \left[ \begin{matrix} 1 & \lambda_n \\ 0 & 1 \end{matrix} \right]$.

• Show that $S^2 = (S \, T)^n = -I_2$. Hint: Verify that
$S \, T = \left[ \begin{matrix} 0 & -1 \\ 1 & \lambda_n \end{matrix} \right] = \left[ \begin{matrix} 1 & -\zeta_{2n} \\ -\zeta_{2n} & 1 \end{matrix} \right] \left[ \begin{matrix} \zeta_{2n} & 0 \\ 0 \zeta_{2n}^{-1} \end{matrix} \right] \left[ \begin{matrix} 1 & -\zeta_{2n} \\ -\zeta_{2n} & 1 \end{matrix} \right]^{-1}.$

• Show that $T^N = \left[ \begin{matrix} 1 & N \, \lambda_n \\ 0 & 1 \end{matrix} \right]$ for any $N \in \mathbb Z$. Conclude that $T$ has infinite order.

Problem 2. Define the modular group as the set
$\Gamma(1) = D(2,3,\infty) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \bigl| \, {\gamma_0}^2 = {\gamma_1}^3 = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.$
More generally, fix an integer $n \geq 3$, and define the Hecke group as the set
$H_n = D(2,n,\infty) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \bigl| \, {\gamma_0}^2 = {\gamma_1}^n = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.$
Observe that $H_3 = \Gamma(1)$. Consider the map $\phi: H_n \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$ which sends $\phi \bigl(\gamma_0 \bigr)(z) = - \dfrac {1}{z}$, $\phi \bigl( \gamma_1 \bigr)(z) = - \dfrac {1}{z+ \lambda_n}$, and $\phi \bigl( \gamma_\infty \bigr)(z) = z - \lambda_n.$

• Show $\phi$ is a well-defined group homomorphism which is injective but not surjective.

• Denote the extended complex numbers $P = \zeta_4$, $Q = \zeta_{2n}^{n-1}$, and $R = i \, \infty$. Show that $\phi \bigl(\gamma_0 \bigr)(P) = P$, $\phi \bigl(\gamma_1 \bigr)(Q) = Q$, and $\phi \bigl(\gamma_\infty \bigr)(R) = R.$ Conclude that the Hecke group $H_n$ tiles the extended upper half plane $\mathbb H^\ast = \mathbb H^2 \cup \mathbb P^1 \bigl( \mathbb Q(\lambda_n) \bigr)$ with $0^\circ - (180/n)^\circ - 90^\circ$ triangles $V = \{ P, \, Q, \, R \}$. (Observe that the sum of the angles is less than $180^\circ$.)

Problem 3. When $n = 3$, the special linear group
$SL_2(\mathbb Z) = \left \{ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in \text{Mat}_{2 \times 2}(\mathbb Z) \, \biggl| \, a \, d - b \, c = 1 \right \}$
is generated by $S$ and $T$ since $\lambda_3 = 1$.

• Consider the map $\varphi: SL_2(\mathbb Z) \to \text{Aut} \bigl( \mathbb P^1(\mathbb C) \bigr)$ defined by $\varphi \left( \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \right)(z) = \dfrac {a \, z + b}{c \, z + d}$. Show that $\varphi \bigl( S \bigr)(z) = - \dfrac {1}{z}$, $\varphi \bigl( S \, T \bigr)(z) = - \dfrac {1}{z+1}$, and $\varphi \bigl( T^{-1} \bigr)(z) = z-1$.

• Show that this map induces a short exact sequence $\{ 1 \} \longrightarrow \{ \pm I_2 \} \longrightarrow SL_2(\mathbb Z) \longrightarrow \Gamma(1) \longrightarrow \{ 1 \}$. That is, $\Gamma(1) \simeq PSL_2(\mathbb Z)$.

Problem 4. Let $N \geq 2$ be an integer. The canonical projection $\pi_N: \mathbb Z \twoheadrightarrow \mathbb Z / N \, \mathbb Z$ extends to a surjection $\pi_N: SL_2(\mathbb Z) \twoheadrightarrow SL_2(\mathbb Z / N \, \mathbb Z)$. Define the principal congruence subgroup of level $N$ as that subgroup $\Gamma(N)$ of $\Gamma(1)$ which makes the following diagram commute:

$\begin{matrix} & & \{ 1 \} & & \{ 1 \} & & \{ 1 \} & & \\ & & \downarrow & & \downarrow & & \downarrow & & \\ \{ 1 \} & \to & \{ \pm I_2 \} \cap \text{ker}(\pi_N) & \to & \text{ker}(\pi_N) & \to & \Gamma(N) & \to & \{ 1 \} \\ & & \downarrow & & \downarrow & & \downarrow & & \\ \{ 1 \} & \to & \{ \pm I_2 \} & \to & SL_2(\mathbb Z) & \to & \Gamma(1) & \to & \{ 1 \} \\ & & \downarrow & & \downarrow & & \downarrow & & \\ \{ 1 \} & \to & \{ \pm I_2 \ \text{mod} \ N \} & \to & SL_2(\mathbb Z / N \, \mathbb Z) & \to & \dfrac {SL_2(\mathbb Z / N \, \mathbb Z)}{\{ \pm I_2 \ \text{mod} \ N \}} & \to & \{ 1 \} \\ & & \downarrow & & \downarrow & & \downarrow & & \\ & & \{ 1 \} & & \{ 1 \} & & \{ 1 \} & & \\ \end{matrix}$

(Contrary to Wikipedia, $SL_2(\mathbb Z/N \, \mathbb Z) / \{ \pm I_2 \ \text{mod} \ N \}$ is not the same as $PSL_2(\mathbb Z/N \, \mathbb Z)$ in general.)

• Show that $T^N \in \text{ker}(\pi_N)$. Conclude that $\Gamma(N) = \left \langle {\gamma_0}^2, \, {\gamma_1}^3, \, {\gamma_\infty}^N \right \rangle \subseteq D(2,3,\infty)$.

• Show that $D(2,3,N) \simeq \Gamma(1) / \Gamma(N) \simeq SL_2(\mathbb Z/N \, \mathbb Z) / \{ \pm I_2 \ \text{mod} \ N \}$. Conclude that $\Gamma(1)/\Gamma(2) \simeq S_3$, $\Gamma(1)/\Gamma(3) \simeq A_4$, $\Gamma(1)/\Gamma(4) \simeq S_4$, and $\Gamma(1)/\Gamma(5) \simeq A_5$.

Problem 5. Assume that $p$ is an odd prime. The set $X(p)$ is that compact, connected Riemann surface formed by gluing the triangles associated with the triangle group $D(2,3,p)$.

• Show that the Euler characteristic is the integer
$\chi \bigl( X(p) \bigr) := \bigl| D(2,3,p) \bigr| \, \left( \dfrac {1}{2} + \dfrac {1}{3} + \dfrac {1}{p} - 1 \right) = - \dfrac {(p+1) \, (p-1) \, (p-6)}{12}.$
Hint: Using $\bigl| GL_2(\mathbb F_p) \bigr| = (p^2 - 1) \, (p^2 - p)$, show $\bigl| PSL_2(\mathbb F_p) \bigr| = (p^3 - p)/2$.

• The genus is that nonnegative integer $g = g \bigl( X(p) \bigr)$ such that $\chi \bigl( X(p) \bigr) = 2 - 2 \, g \bigl( X(p) \bigr)$. Show that $g \bigl( X(p) \bigr) = \dfrac {(p+2) \, (p-3) \, (p-5)}{24}$.