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78 LECTURE 8. THE MODULAR CURVE Example 8.1 . Let Γ = Γ(1). Let us show that there exists a holomorphic isomorphism H / SL(2 , Z ) = C . This shows that the set of isomorphism classes of elliptic curves has a natural structure of a complex manifold of dimension 1 isomorphic to the complex plane C . Since g 3 2 and Δ are of the same weight, the map H → P 1 ( C ) , τ ( g 2 ( τ ) 3 , Δ( τ )) is a well defined holomorphic map. Obviously it is constant on any orbit of Γ(1), hence factors through a holomorphic map f : H / Γ(1) P 1 ( C ) . Since Δ does not vanish on H , its image is contained in P 1 ( C ) \ {∞} = C . I claim that f is one-to-one onto C . In fact, for any complex number c the modular form f = g 3 2 - c Δ is of weight 6. It follows from Lemma 7.1 that f has either one simple zero, or one zero of multiplicity 2 at the elliptic point of order 2, or a triple zero at the elliptic point of index 3. This shows that each c Z occurs in the image of j on H / Γ and only once. We leave to the reader the simple check that a bijective map between two complex manifolds of dimension 1 is an isomorphism. Notice that the explicit isomorphism H / SL(2 , Z ) C is given by the holomorphic function
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