4. Valence Formula, Structure Theory and Dimension Formula

Question 1

For meromorphic function f and point p ∈ H ⋃ ∞, define ord_p(f)

Answer 1

If P = 0: order of vanishing of f a P>0

If P, a pole: - order of pole at P<0

Else: 0

Question 2

For meromorphic function f and point p ∈ H ⋃ ∞, define ord_p(∞)

Answer 2

Index of the first non-zero coefficient in the q-expansion of f.

Question 3

State the valence (or k/12) formula

Answer 3

ord_∞(f) + 1/2 ord_i(f) + 1/3 ord_ω(f) + SUM_ P∈Γ ord_p(f) = k/12

Question 4

What does it mean for two modular forms to coincide?

Answer 4

Two modular forms of the same weight k coincide if their first [k/12] +1 Fourier coefficients coincide

Question 5

Give the definition of the discriminant function in terms of Eisenstein series

Answer 5

Δ(τ) = 1/1728 (E_4^3(τ) E_6^2(τ))

Question 6

Define the relation between the set of Modular forms, Cusp forms and Eisenstein series of weight k.

Answer 6

M_k(Γ) = S_k(Γ) ⊕ CE_k

Question 7

Define the relation ship between cusp forms and modular forms of different weights

Answer 7

S_k(Γ) = ΔM_k-12(Γ)

Question 8

Give the dimension criteria for modular forms

Answer 8

dim M_k(Γ) = { [k/12]+1, if k!= 2 (mod12)
{ [k/12], if k=2 (mod12)

Question 9

Relate the dimensions of modular and cusp forms of weight k

Answer 9

dim M_k(Γ) = dim S_k(Γ) + 1

Question 10

How can we relate all Modular forms to E_4 and E_6?

Answer 10

Every modular form can be expressed uniquely as a polynomial in E_4 and E_6. That is, {E_4^a E_6^b; 4a+6b=k} defines a basis for M_k

Question 11

Define the j-function

Answer 11

j(τ) = E_4^3/Δ

Question 12

Give the properties of the j-function

Answer 12

j(τ) = j(γτ)

J is a meromorphic modular function of weight 0, holomorphic on H.