The modular group and the fundamental domain Seminar on Modular Forms Spring 2019
Johannes Hruza and Manuel Trachsler March 13, 2019
1 The Group SL2(Z) and the fundamental do- main
Definition 1. For a commutative Ring R we define GL2(R) as the following set: a b GL (R) := A = for which det(A) = ad − bc ∈ R∗ . (1) 2 c d
We define SL2(R) to be the set of all B ∈ GL2(R) for which det(B) = 1.
Lemma 1. SL2(R) is a subgroup of GL2(R). Proof. Recall that the kernel of a group homomorphism is a subgroup. Observe ∗ that det is a group homomorphism det : GL2(R) → R and thus SL2(R) is its kernel by definition. ¯ Let R = R. Then we can define an action of SL2(R) on C ( = C ∪ {∞} ) by az + b a a b A.z := and A.∞ := ,A = ∈ SL (R), z ∈ . (2) cz + d c c d 2 C
Definition 2. The upper half-plane of C is given by H := {z ∈ C | Im(z) > 0}.
Restricting this action to H gives us another well defined action ”.” : SL2(R)× H 7→ H called the fractional linear transformation. Indeed, for any z ∈ H the imaginary part of A.z is positive:
az + b (az + b)(cz¯ + d) Im(z) Im(A.z) = Im = Im = > 0. (3) cz + d |cz + d|2 |cz + d|2