## Homework Assignment 3

Recall that the upper half plane $\mathbb H^2$ consists of complex numbers $z = x + i \, y$ with $y > 0$. Let $\overline{\mathbb D} = \left \{ x + i \, y \in \mathbb C \, \bigl| \, x^2 + y^2 \leq 1 \right \} = \mathbb D \cup \partial \mathbb D$ denote the closed unit disk as the union of the unit disk $\mathbb D$ with its boundary $\partial \mathbb D = \left \{ x + i \, y \in \mathbb C \, \bigl| \, x^2 + y^2 = 1 \right \} \simeq S^1(\mathbb R)$. This assignment is meant to explain the pictures at the Wolfram Demonstrations site as well as on Wikipedia:

This assignment is due Friday, October 18, 2013 at the start of class.

Problem 1.

• Let $GL_2(\mathbb R)$ denote the collection of those $2 \times 2$ real matrices with nonzero determinant. Prove the Iwasawa Decomposition:
$\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] = \left[ \begin{matrix} y & x \\ 0 & 1 \end{matrix} \right] \left[ \begin{matrix} r \, \cos \theta & - r \, \sin \theta \\ r \, \sin \theta & \ \ r \, \cos \theta \end{matrix} \right]$
in terms of the real numbers $x = \dfrac {a \, c + b \, d}{c^2 + d^2}$, $y = \dfrac {a \, d - b \, c}{c^2 + d^2}$, $r = \sqrt{c^2 + d^2}$, and $\theta = \tan^{-1} \dfrac {c}{d}$.

• Let $GL_2(\mathbb R)^+$ denote those $2 \times 2$ real matrices with positive determinant. Consider the map $\phi: GL_2(\mathbb R)^+ \to \mathbb H^2$ defined by $\left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \mapsto \dfrac {a \, i + b}{c \, i + d}$. Show that $\phi$ is surjective, whereas
\begin{aligned} \phi^{-1}(i) & = \left \{ \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right] \in GL_2(\mathbb R)^+ \ \biggl| \ \dfrac {a \, i + b}{c \, i + d} = i \right \} = \left \{ A \in GL_2(\mathbb R)^+ \, \biggl| \, A^T \, A = (\det A) \, I_2 \right \} \\[5pt] & = \mathbb R^\times \cdot SO_2(\mathbb R). \end{aligned}.

Problem 2. Show that $\mathbb H^2 \simeq GL_2(\mathbb R)^+ / \bigl( \mathbb R^\times \cdot SO_2(\mathbb R) \bigr)$. Hint: Use the Iwasawa Decomposition.

Problem 3. Define the the modular curve $X(1) = Y(1) \cup \partial Y(1)$ 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 \} \\[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}
The normalized hyperbolic area differential on the upper half plane $\mathbb H^2$ is given by $d \mu(z) = \dfrac {3}{\pi} \, \dfrac {dx \, dy}{y^2}$.

• Show that $\text{Vol} \bigl( Y(1) \bigr) = 1$.

• Conclude that $X(1) = Y(1) \cup \partial Y(1)$ is a compact Riemann surface with finite volume.

Problem 4. Let $z_0 = x_0 + i \, y_0$ be a complex number with $z_0 \notin \overline{\mathbb D}$. In the Poincare Disk Model, a “line” is given by $L = \left \{ x + i \, y \in \mathbb A^1(\mathbb C) \, \biggl| \, x^2 + y^2 - 2 \, x_0 \, x - 2 \, y_0 \, y + 1 = 0 \right \}$.

• Identifying the affine complex line $\mathbb A^1(\mathbb C)$ with the affine real plane $\mathbb A^2(\mathbb R)$, show that $L$ is a circle of radius $r = \sqrt{|z_0|^2 - 1}$ centered at $(x_0, y_0)$ which intersects the boundary $\partial \mathbb D$ at the two distinct complex points $z_0 / (1 \pm i \, r)$. Hint: Solve the simultaneous equations $x^2 + y^2 = x_0 \, x + y_0 \, y = 1$.

• Show that $L$ is perpendicular to $\partial \mathbb D$ at the points where they intersect. Hint: Consider the inner product of the vectors $\left( \dfrac {x}{\sqrt{x^2 + y^2}}, \ \dfrac {y}{\sqrt{x^2 + y^2}} \right)$ and $\left( \dfrac {x - x_0}{\sqrt{(x - x_0)^2 + (y - y_0)^2}}, \ \dfrac {y - y_0}{\sqrt{(x - x_0)^2 + (y - y_0)^2}} \right)$ which are normal to $\partial \mathbb D$ and $L$, respectively.

Problem 5.Using the isomorphism
$\begin{matrix} \mathbb D & \simeq & X = \left \{ (u,v,w) \in \mathbb A^3(\mathbb R) \, \biggl| \, u^2 + v^2 - w^2 = -1, \ w > 0 \right \} \\[10pt] x + i \, y = \dfrac {u + i \, v}{1 + w} & \leftrightarrow & (u,v,w) = \left( \dfrac {2 \, x}{1 - x^2 - y^2}, \ \dfrac {2 \, y}{1 - x^2 - y^2}, \ \dfrac {1 + x^2 + y^2}{1 - x^2 - y^2} \right) \end{matrix}$
show that the intersection of the unit disk $\mathbb D$ and the “line” $L$ is in bijection with the intersection of the hyperboloid $X$ and the plane $x_0 \, u + y_0 \, v = w$. Hint: Show that points $x + i \, y$ in $L \cap \mathbb D$ map to $(u,v,w) = \left( \dfrac {x}{1 - x_0 \, x - y_0 \, y}, \ \dfrac {y}{1 - x_0 \, x - y_0 \, y}, \ \dfrac {x_0 \, x + y_0 \, y}{1 - x_0 \, x - y_0 \, y} \right)$.