189-726B: Modular Forms II Assignment 4

Due: Wednesday, February 6

1. Show that the action of the Hecke operator T_{n} and of the complex conjugation c on the lattice Λ = H_{1}(X_{0}(N)(C), Z) ⊂ S_{2}(Γ_{0}(N))^{∨} that were defined in class commute with each other.

2. Prove the assertion made in class that the set of orbits for the action of Γ_{0}(N) on the set of ordered pairs of adjacent cusps in P_{1}(Q) is in natural bijection with P_{1}(Z/NZ).

3. Compute the space M^{Γ_{0}(11)} of modular symbols relative to the group Γ_{0}(11). Show that there is exactly one cusp form f on this group, and compute a_{2}(f) and a_{3}(f) using the modular symbol algorithm. Check that your answer agrees with

f(q) = q ∏_{n}(1 − q^{n})^{2}(1 − q^{11n})^{2}.