16 1. THE CLASSICAL THEORY: PART I
Remark. Among the important subgroups of SL2(Z) are the congruence sub-
(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
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