Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 23 / 3 / a

The map $X(\Gamma )\to X(\Gamma (1))\cong {\mathbb{P}}^1$ has degree $d$, the effective projective index. Ramification occurs only over the two elliptic points and the modular cusp. Over the order-two elliptic point, $r_2$ points have index one and the other $(d-r_2)/2$ points have index two, contributing $(d-r_2)/2$ to the ramification divisor. The analogous order-three contribution is $2(d-r_3)/3$.

The local degree at a modular cusp is its effective cusp width. The cusp widths sum to $d$, so the modular cusp contribution is $d-r_{\infty }$. The Riemann-Hurwitz formula, $2g(X)-2=d(2g(Y)-2)+{\sum }_P(e_P-1)$, now yieldsTherefore the genus formula for a modular curve isUsing the index in ${\mathrm{SL}}_2$ without accounting for its center would give the wrong degree when $-I\notin \Gamma$.