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Unformatted text preview: 80 LECTURE 8. THE MODULAR CURVE Γ. The natural inclusion U → H * factors through the map U/ Γ x → H * / Γ x . Taking c small enough and using (8.8) we see that this map is injective. Its image is an open neighborhood ¯ U of the cusp c ∈ H * / Γ. Let h be the index of the cusp. Then Γ x consists of matrices ± „ 1 mh 1 « and hence the map τ → e 2 πiτ/h sends U/ Γ x into C with the image isomorphic to an open disk. This defines a natural complex structure on the neighborhood ¯ U . Notice that it is consistent with the complex structure on ¯ U ∩H / Γ = ¯ U \{ c } . Also it is easy to see that the map π * Γ(1) extends to the composition of holomorphic maps. It remains to prove the last assertion, the compactness of H * / Γ. First of all, we replace Γ by a subgroup of finite index Γ which is normal in Γ(1). Then H * / Γ = ( H * / Γ ) / (Γ / Γ) , H * / Γ(1) = ( H * / Γ ) / (Γ / Γ(1)) It remains to use the following simple fact from topology: Lemma 8.2.Lemma 8....
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