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 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.
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