5. Modular Forms of Congruence Subgroups

Question 1

Define the principal congruence subgroup. Give the special example.

Answer 1

A subgroup of Γ = SL2(Z) such that for (a,b,d,c) = (1,0,0,1) mod N

Γ(1) = SL2(Z)

Question 2

Define the Hecke congruence subgroup

Answer 2

Γ_0(N) = (a,b,d,c) = (,,*,0) mod N

I.e. c = 0 modN

Question 3

Define a congruence subgroup

Answer 3

A subgroup of SL2(Z) which contains the principle congruence subgroup.

Question 4

Give the relative orders of [SL2(Z): Γ_0(p)] and [SL2(Z): Γ_0(4)]

Answer 4

[SL2(Z): Γ_0(p)] = |SL2(Z)|/|Γ_0| = p+1

[SL2(Z): Γ_0(4)] = 6

Question 5

Give the conditions for a modular form of a congruence (the Hecke Γ_0) subgroup of SL2(Z)

Answer 5

1. f holomorphic on H
2. Modularity condition for generators of Γ_0
3. f is holomorphic at all cusps of Γ_0

Question 6

What limit is equivalent to ensuring holomorphicity at all cusps?

In what limit can we define a cusp form?

Answer 6

lim τ→i∞ j(a,τ) ^-k f(aτ)

this limit is 0

Question 7

How can we relate modular forms of Γ_0(n) for different n? Given f in M_k(Γ_0(n))

Answer 7

We can define some new modular form g(τ) := f(Nτ) where g(t) ∈ M_k(Γ_0(Nn))

Question 8

State the bound for the dimension of M_k(Γ_0(n))

Answer 8

≤ k/12[Sl2(Z): Γ_0(n)] + 1

Question 9

How can we relate Γ_0(n) to Eisenstein series

Answer 9

For every cusp of Γ_0(n) there is an Eisenstein series which gives an element of M_k(Γ_0(N)).

For any divisor d of N, we have E_k(dτ)