189-726A: L-functions and Modular Forms Assignment 4

Due: Monday, November 7

1. Let f_{i} = ∑_{n} a_{n}^{(i)}q^{n} (for i = 1, …, r) be r Hecke eigenforms on SL_{2}(Z) (of possibly different weights.) Show that the Dirichlet series

D(f_{1}, …, f_{r}, s) := ∑_{n=1}^{∞} a_{n}^{(1)} ⋯ a_{n}^{(r)}n^{−s}

has an Euler product factorisation of the form ∏_{p} R_{p}(p^{−s}), where R_{p} is a rational function. What can you say about the degree of R_{p}?

2. (This exercise is taken from p. 41 of Zagier’s notes on modular forms). For any non-negative integer n and k > 2, define the n-th Poincaré series

of weight k by

P_{n}(z) = ∑_{(c,d)∈(Z^{2})′}(cz + d)^{−k}e^{2πin \frac{az+b}{cz+d}} ,

where (Z^{2})′ denotes the set of pairs (c, d) ∈ Z^{2} with gcd(c, d) = 1, and for any such (c, d) the pair (a, b) is chosen (arbitrarily) in such a way that ad−bc = 1.

a) Show that this series converges absolutely and defines a modular form of weight k on SL_{2}(Z).

b) Show that, if f = ∑ a_{n}q^{n} is any cusp form of weight k on SL_{2}(Z), then

⟨P_{n}, f⟩_{k} = (k − 2)!(4πn)^{1−k}a_{n},

where ⟨ , ⟩_{k} denotes the Peteersson scalar product on forms of weight k, as defined in class.