Math 168: Homework Assignment 5

William Stein

Due: Wednesday, Nov 2, 2005

The problems have equal point value, and multi-part problems are of the same value.

1 Problems

1. Suppose γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) is a matrix with real entries and positive determinant. Prove that if z ∈ ℂ is a complex number with positive imaginary part, then the imaginary part of γ(z) = (az+b)/(cz+d) is also positive.

2. (a) Prove that a polynomial is an analytic function on ℂ.

(b) Prove that a rational function (quotient of two polynomials) is a meromorphic function on ℂ.

3. Suppose f and g are weakly modular functions with f ≠ 0.

(a) Prove that the product fg is a weakly modular function.

(b) Prove that 1/f is a weakly modular function.

(c) If f and g are modular functions, show that fg is a modular function.

(d) If f and g are modular forms, show that fg is a modular form.

4. Suppose f is a weakly modular function of odd weight k. Show that f = 0.

5. (a) Prove that Γ_{1}(N) is a group.

(b) Prove that Γ_{1}(N) has finite index in SL_{2}(ℤ) (Hint: it contains the kernel of the homomorphism SL_{2}(ℤ) → SL_{2}(ℤ/Nℤ).)