Homework Assignment 2

Posted on September 17, 2013 by Edray Herber Goins, Ph.D.

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.

Homework Assignment 2 Download

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}.

About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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