Modular Forms: Problem Sheet 7
29 March 2016
1. (a) Let Γ be a congruence subgroup, k ∈ Z, f ∈ Mk(Γ), α ∈ GL+2(Q), and
denote Γ0= Γ ∩ α−1Γα. Prove that f |kα ∈ Mk(Γ0) (provide all details).
(b) Prove Proposition 4.1 from the notes.
Note: you may use exercises from previous chapters.
2. Read the proof of Lemma 4.2 from the notes and check that 1 0 b 1 b ∈ F p ∪ 0 cN 1 1 mod p ,
where c is any integer with cN ≡ −1 (mod p), indeed forms a system of coset representatives for the quotient given at the bottom of page 51 in the proof . 3. Let N be a positive integer, let p be a prime number, and let
α = 1 0 0 p , Γ = Γ0(N ) (instead of Γ1(N )), Γ0= Γ ∩ α−1Γα.
Determine a system of coset representatives for the quotient Γ0\Γ. 4. Prove that for any even integer k ≥ 4 and prime p we have
TpGk = σk−1(p)Gk
for the Eisenstein series Gk and the Hecke operator Tp on Mk(SL2(Z)).
