Homework Assignment 1 Math 252: Modular Abelian Varieties

William Stein

Sep. 17 (Due: Sep. 24)

There are 7 problems, each worth an equal number of points.

1. Verify that if g ∈ SL_{2}(R) and z ∈ C then

Im(g(z)) = \frac{Im(z)}{|cz + d|^{2}} .

2. Let D be the subset of points z ∈ C with |z| ≥ 1 and |Re(z)| ≤ 1/2. For each z in the following list, find an element g ∈ SL_{2}(Z) such that g(z) ∈ D:

−1 + \frac{i}{2}, −\frac{1}{5} + 3i, \frac{7}{8} + \frac{4}{5}i.

3. (a) Find an element ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) such that c = 7 and d = 9.

(b) Write the element you found in (a) in terms of S = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) and

T = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ).

4. (a) Find an element of g ∈ SL_{2}(Z) whose reduction modulo 5 is

(\begin{matrix}1 & 1 \\ 6 & 12 \end{matrix}) ∈ SL_{2}(Z/5Z).

(b) Let N be a positive integer. Prove that the reduction mod N map from SL_{2}(Z) to SL_{2}(Z/NZ) is surjective. [Hint: If you get stuck, you can find the proof somewhere in Shimura’s Introduction to the Arithmetic Theory of Automorphic Functions and reformulate it in your own way.]

5. Let N be a positive integer.Compute the index of each of the following

groups in SL_{2}(Z):

Γ(N), Γ_{1}(N), Γ_{0}(N)

6. Compute generators for Γ_{0}(3), Γ_{1}(3), and Γ(3).

7. Let p be a prime. Prove that Γ_{0}(p)\P^{1}(Q) has cardinality 2. What about Γ_{0}(p^{2})\P^{1}(Q)?