Analytic Definition
The modular group SL(2, Z) acts on the upper half-plane by fractional linear transformations. The analytic definition of a modular curve involves a choice of a congruence subgroup Γ of SL(2, Z), i.e. a subgroup containing the principal congruence subgroup of level N Γ(N), for some positive integer N, where
The minimal such N is called the level of Γ. A complex structure can be put on the quotient Γ\H to obtain a noncompact Riemann surface commonly denoted Y(Γ).
Read more about this topic: Modular Curve
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