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.

Homework Assignment 3 Download

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}.

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).

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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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