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.

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