189-726A: L-functions and Modular Forms Assignment 2

Due: Monday, October 10

1. Let Γ be a congruence subgroup of SL_{2}(Z) and let f(z) be a modular form on Γ of weight two.

a) Choose a point z_{0} in the complex upper half-plane. Show that the function m_{f} : Γ ⟶ C defined by

m_{f}(γ) = ∫_{z_{0}}^{γz_{0}} f(z)dz

does not depend on the choice of z_{0}, and is a homomorphism from Γ to C. b) Show that the function f ↦ m_{f} from M_{2}(Γ) to hom(Γ, C) is injective.

c) Use a) and b) to show that M_{2}(Γ) = 0 if Γ has no infinite abelian quotients.

d) Show that the abelianisation of SL_{2}(Z) is a finite group of order 12, and conclude that M_{2}(SL_{2}(Z)) = 0.

2. Let f = ∑_{n} a_{n}q^{n} be a cusp form of weight k on Γ_{0}(N) ⊂ PSL_{2}(R).

a) Show that the matrix ( \begin{matrix}0 & −1 \\ N & 0 \end{matrix} ) belongs to the normaliser of Γ_{0}(N)

in PSL_{2}(R). Conclude that the transformation

f ↦ N^{−k/2}z^{−k}f(\frac{−1}{Nz})

is an involution on S_{k}(Γ_{0}(N)).

b) Assume that f is an eigenvector for this involution, with associated eigenvalue w_{f} ∈ {1, −1}. Show that the L-series L(f, s) extends to an analytic function of s ∈ C and satisfies a functional equation relating its values at s and k − s. (Hint: follow the approach explained in class in the case where N = 1, with the involution of part a) playing the role of the matrix

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

c) Explain how to evaluate the central critical value L(f, k/2) numerically.

d) How many fourier coefficients of f would you need to know in order to evaluate L(f, k/2) with a numerical error of at most ϵ > 0? (Give your answer as a function of N, k, and ϵ.)