16 1. THE CLASSICAL THEORY: PART I

Remark. Among the important subgroups of SL2(Z) are the congruence sub-

groups

Γ(N) =

a b

c d

=

1 0

0 1

(mod N) .

Then Γ(1) = SL2(Z). Geometrically the quotient spaces MΓ(N) := Γ(N)\H arise

as parameter spaces for complex tori Xτ plus additional “rigidifying” data. In this

case the additional data is “marking” the N-torsion points

Xτ (N) := (1/N)Λ/Λ

∼

=

(Z/NZ)2.

When we require that an ismorphism XΛ(N)

∼

=

XΛ(N) take marked points to

marked points the the equivalence classes of XΛ(N)’s are Γ(N)\H.

Later in these talks we will encounter arithmetic groups Γ which have compact

quotients.