Math 168: Homework Assignment 6

William Stein

Due: Wednesday, Nov 9, 2005

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

1 Problems

1. Let k be an integer, and for any function f : 𝔥^{∗} → ℂ and γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(ℚ), set f|[γ]_{k}(z) = (cz + d)^{−k}f(γ(z)). Prove that if γ_{1}, γ_{2} ∈ SL_{2}(ℤ), then for all z ∈ 𝔥 we have

f|[γ_{1}γ_{2}]_{k}(z) = ((f|[γ_{1}]_{k})|[γ_{2}]_{k})(z).

(I mostly did this on the blackboard in class.)

2. Prove that for any α, β ∈ ℙ^{1}(ℚ), there exists γ ∈ SL_{2}(ℤ) such that γ(α) = β.

3. By hand write down the coefficients of 1, q, q^{2}, and q^{3} of the Eisenstein series E_{8}.

4. Explicitly compute the Victor Miller basis for M_{28}(SL_{2}(ℤ)).

5. Let f = E_{4}E_{6}Δ.

(a) What is the weight of f?

(b) Determine the location of all the zeros of f.