Past exam of the geometry course of the University of Oxford / 2008 / B / 8 / iv / Solution

If a nonidentity holomorphic automorphism had more than $2g+2$ fixed points, choose that many distinct ones and apply the function constructed above. The zero-and-pole count would give $f=f\circ \phi$, and the local reciprocal coordinate would force $\phi$ to be the identity, a contradiction. Therefore