2. Modular Forms

Question 1

Define a weakly modular function of weight k

Answer 1

f(γτ) = (cτ + d)^k f(τ) = j(γ, τ)^k f(τ)

Question 2

Define a meromorphic function on domain D

Answer 2

A function which is holomorphic (complex differentiable) on D - S, where S is a subgroup of D containing all isolated poles of f.

Question 3

Define a meromorphic modular form of weight k

Answer 3

A weakly modular, meromorphic function which is meromorphic at infinity.

Question 4

Briefly describe how we can understand a function to be meromorphic.

Answer 4

Consider f(τ) = f(τ+1). We can map τ to q = e^2iπτ since this is periodic by τ→ τ+1. This maps a width 1 vertical strip of H to a punctured unit disc.
There exists a unique meromorphic (holomorphic) function f(τ)=f ̃(q)
As q → 0, τ→ ∞ and we can write a Laurent series for q.

Question 5

Define the Laurent series for q

Answer 5

f(τ) = f ̃(q) = SUM_n»-∞ a_n q^n

Question 6

What are the conditions for a Laurent expnasion to be holomorphic?

Answer 6

a_n = 0 for n < 0

Hence st. f(∞) = f ̃(0) = a_0

Question 7

Give the three conditions for a holomorphic modular form of weight k

Answer 7

1. f holomorphic on H
2. f weakly modular function
3. Laurent expansion

Since f holomorphic on H, the Laurent expansion converges on all H.

Question 8

Define a cusp form

Answer 8

A modular form which vanishes at the cusp ∞:
a_0 = 0

Question 9

For which values of k are there no modular (or cusp) forms? Why?

Answer 9

For k odd. Consider γ = -I under the modular function condition. If k odd we get f(τ) = -f(τ)

Question 10

Where is a modular form typically not bounded?

Answer 10

For Im(τ) → 0 and τ → -d/c

Question 11

What is sufficient to show a function is weakly modular of weight k?

Answer 11

Check the transformation properties of S and T and show holomorphicity at the cusp.

Question 12

Give the modularity conditions for S and T on a function f

Answer 12

f(Sτ) = f(-1/τ) = τ^k f(τ)
f(Tτ) = f(τ+1) = f(τ)

Question 13

What quantity is Γ-invariant?

Answer 13

Im(τ)^k/2 |f(τ)|

Question 14

Give the two bound facts for a cusp form (Hecke bound)

Answer 14

1. Im(τ)^k/2 |f(τ)| is bounded on H
2. There exists some constant C>0 such that |a_n| <= Cn^k/2 for all n

Question 15

Define the Ramanajuan-Petersson conjecture

Answer 15

a_n = O(n^(k-1)/2+ε)

I.e. there exists ε>0 and some constant C=C(ε) such that a_n is bounded by Cn^(k-1)/2+ε