189-726B: Modular Forms II Assignment 10

Due: Wednesday, April 26

1. If f and g are modular forms of weight k and ℓ respectively, on some congruence group Γ_{0}(N), and having fourier coefficients in a field K, show that the expression

[f, g] := \frac{1}{2πi}(ℓf′g − kfg′)

is a modular form of weight k + ℓ + 2 with fourier coefficients in K. (Here, f′ is just the usual derivative of f with respect to the variable τ.)

2. The operation (f, g) ↦ [f, g] is called the Rankin-Cohen bracket. Show that this operation endows the space M := ⊕_{k≥0}M_{k} of classical modular forms (with coefficients in a field K) with the structure of a (graded) Lie algebra over K.

3. Compute the Rankin-Cohen bracket [Q, R] where Q and R are the usual normalised Eisenstein series of weights 4 and 6 respectively.