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.
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