189-726B: Modular Forms II Assignment 6

Due: Wednesday, February 20

1. Let P = E_{2} = 1 − 24 ∑_{n=1}^{∞} σ_{1}(n)q^{n} be the weight two Eisenstein series. Prove the claim made in class that 12θP − P^{2} belongs to the space M_{4}.

2. Let Δ = (Q^{3} − R^{2})/1728 be Ramanujan’s Δ-function. Compute ∂_{12}(Δ) and conclude that 12P is the logarithmic derivative of Δ. Deduce from this the infinite product formula for Δ.

3. Show that

θR = −Δ (mod 5), θQ = 2Δ (mod 7),

and conclude the well-known congruences for the Ramanujan τ function:

τ(n) ≡ nσ_{5}(n) (mod 5), τ(n) ≡ nσ_{3}(n) (mod 7).

At this point, let me correct a mistake I made in class: the coefficient appearing in the formula for the weight 6 Eisenstein series R = E_{6} is 504, not 540 (I miscopied...) So in fact, the only primes that require special treatment are 2 and 3, everything works for p = 5, as Shahab said.

4. Show that the mod 5 modular form θ(E_{10}) has filtration (weight) 12 and relate it to Δ. How does this compare to what you calculated in 3?

5. Prove the assertion made in class that ∂B = −QA, where A and B are the homogeneous polynomials of degrees p − 1 and p + 1 respectively satisfying A(Q, R) = E_{p−1} and B(Q, R) = E_{p+1}.