189-726B: Modular Forms II Assignment 2

Due: Wednesday, January 23

1. Prove the recurrence formula

T̃_{p^{r−1}} T̃_{p}([Λ]) = T̃_{p}T̃_{p^{r−1}}([Λ]) = T̃_{p^{r}}([Λ]) + pT̃_{p^{r−2}}([pΛ])

for the abstract Hecke operators on lattices that was used in class on Monday.

2. Show that the Hecke operators T_{n} acting on the space S_{k}(Γ_{0}(N)) are selfadjoint with respect to the Petersson scalar product, when gcd(n, N) = 1. You may eventually have to extend some definitions that were only given for SL_{2}(Z). Try to use only your notes, and avoid peeking into a textbook!

3. Show that the modular forms f(τ) and f(d′τ) belong to S_{k}(Γ_{0}(N)) when dd′ divides N and f belongs to S_{k}(Γ_{0}((d)). Show that if f is an eigenvector for T_{n} with gcd(n, N) = 1, then the same holds for f(d′τ), and that the eigenvalues are equal.

4. Let T be the Q-algebra of Hecke operators T_{n} acting on S_{24}(SL_{2}(Z)). Express T as a product of specific number fields, and write down a basis of eigenforms for S_{24}(SL_{2}(Z)). (This fun calculation was done by Hecke.)