# What are the branch points of $X(n)\to X(1)$

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 books.google.fr/… –  Michalis Mar 14 '12 at 10:09