Recall that the upper half plane

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

The normalized hyperbolic area differential on the upper half plane

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

**Problem 4.** Let

- 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 vectorsand which are normal to and , respectively.

**Problem 5.**Using the isomorphism

show that the intersection of the unit disk *Hint:* Show that points