= Fixed-point bound for automorphisms of a compact Riemann surface
{title2=$\#\operatorname{Fix}(\varphi)\leq2g+2\quad(\varphi\ne1)$}
A nonidentity holomorphic <automorphism> of a compact connected <Riemann surface> of <genus> $g$ has at most $2g+2$ fixed points. If there were more, choose $g+1$ distinct fixed points and let $D$ be their sum. The <Riemann-Roch theorem> gives $\ell(D)\geq2$, so there is a nonconstant <meromorphic function> with at most simple poles at these points and no other poles. The function $h=f-f\circ\varphi$ has at most $g+1$ poles and at least $n-g-1$ zeros at the remaining fixed points. Since a nonzero <principal divisor> has degree zero, $n>2g+2$ forces $h=0$. At an actual pole fixed by $\varphi$, the reciprocal $1/f$ is a local coordinate because its zero is simple. The identity $f=f\circ\varphi$ therefore forces $\varphi$ to be the identity near that pole, and the <identity theorem on a Riemann surface> makes it the identity everywhere.
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