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 plus additional “rigidifying” data. In this
case the additional data is “marking” the N-torsion points
(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.
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