2 1. BASIC FACTS

Remark 1.2. If N = 1, then

SL2(Z) = Γ0(1) = Γ1(1) = Γ(1).

If Γ is a congruence subgroup and τ ∈ H, then let Γτ denote the isotropy

subgroup of Γ for τ .

Definition 1.3. Suppose that τ ∈ H, and that Γ is a congruence subgroup.

(1) If

−1 0

0 −1

∈ Γ and

Γτ = ±

1 0

0 1

,

then τ is called an elliptic fixed point of order

1

2

|Γτ|.

(2) If

−1 0

0 −1

∈ Γ and

Γτ =

1 0

0 1

,

then τ is called an elliptic fixed point of order |Γτ|.

Example 1.4. Let Γ = SL2(Z), and let S and T be the matrices

S =

0 −1

1 0

,

T =

1 1

0 1

.

It is not diﬃcult to verify that SL2(Z) is generated by S and T . If τ ∈ F, then we

have

Γτ =

±{I,

S} if τ = i,

±{I, ST, (ST )2} if τ = ω,

±{I} otherwise.

Therefore, τ = i (resp. τ = ω) is an elliptic fixed point of order 2 (resp. 3).

Definition 1.5. Suppose that Γ is a congruence subgroup of SL2(Z). A cusp

of Γ is an equivalence class in

P1(Q)

= Q ∪ {∞} under the action of Γ.

Example 1.6. There is just one cusp when Γ = SL2(Z), and it is customary

to select the point at ∞ as its canonical representative.

The following formulas are often useful.

Proposition 1.7. If N is a positive integer, then

[Γ0(1) : Γ0(N)] = N

p|N

1 +

1

p

,

[Γ0(1) : Γ1(N)] = N

2

p|N

1 −

1

p2

,

where the products are over the prime divisors p of N.