Recall that the upper half plane consists of complex numbers with . Let denote the closed unit disk as the union of the unit disk with its boundary . 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 denote the collection of those real matrices with nonzero determinant. Prove the Iwasawa Decomposition:

in terms of the real numbers , , , and . - Let denote those real matrices with positive determinant. Consider the map defined by . Show that is surjective, whereas

.

**Problem 2.** Show that . *Hint:* Use the Iwasawa Decomposition.

**Problem 3.** Define the the modular curve in terms of

The normalized hyperbolic area differential on the upper half plane is given by .

- Show that .
- Conclude that is a compact Riemann surface with finite volume.

**Problem 4.** Let be a complex number with . In the Poincare Disk Model, a “line” is given by .

- Identifying the affine complex line with the affine real plane , show that is a circle of radius centered at which intersects the boundary at the two distinct complex points .
*Hint:*Solve the simultaneous equations . - Show that is perpendicular to at the points where they intersect.
*Hint:*Consider the inner product of the vectors and which are normal to and , respectively.

**Problem 5.**Using the isomorphism

show that the intersection of the unit disk and the “line” is in bijection with the intersection of the hyperboloid and the plane . *Hint:* Show that points in map to .