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Unformatted text preview: 80 LECTURE 8. THE MODULAR CURVE . The natural inclusion U H * factors through the map U/ x H * / 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 * / . Let h be the index of the cusp. Then x consists of matrices 1 mh 1 and hence the map e 2 i/h sends U/ 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 / = U \{ c } . Also it is easy to see that the map * (1) extends to the composition of holomorphic maps. It remains to prove the last assertion, the compactness of H * / . First of all, we replace by a subgroup of finite index which is normal in (1). Then H * / = ( H * / ) / ( / ) , H * / (1) = ( H * / ) / ( / (1)) It remains to use the following simple fact from topology: Lemma 8.2.Lemma 8....
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 Fall '09
 ONTONKONG
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