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Let $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$ be a finite index subgroup. Let $X_\Gamma \to X(1)$ be the corresponding morphism of compact connected Riemann surfaces (obtained by adding the cusps).

Example. Take $\Gamma = \Gamma(n)$. Then $X_\Gamma = X(n)$.

What are the branch points of $X_\Gamma \to X(1)$.

Are they just the three points given by the elliptic points $0$ and $1728$ and the cusp $\infty$?

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    Yes, you're right. Check proposition 1.37 in Shimura http://books.google.fr/books?id=-PFtGa9fZooC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false2012-03-14

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