I have the following problem:
Let $h : Σ → Σ$ be a conformal self-map, different from the identity, of a compact Riemann surface $Σ$ of genus $p$. Show that $h$ has at most $2p+2$ fixed points.
The problem has a suggestion. Hint: Consider a meromorphic function $f : Σ → S^2$ with a single pole of order $≤ p + 1$ at some $z_0$ which is not a fixed point of $h$, and study $f(z) − f(h(z))$.
So let's go to what I thought ...
By theorem (see for example Jost p. 239), exists a non-constant meromorphic function $f : Σ → S^2$ with only one pole, of order $≤ p + 1$. Then for $z_0 \in Σ $ such that $h(z_0) \neq z_0$, we consider $f$ like mentioned with only one pole of order $≤ p + 1$ at $z_0$. Let $g(z):= f(z)-f(h(z))$.
Now i need study the set zeros of $g$ and conclude that $≤ 2p + 2$. My difficults is exactly here! After that i conclude that $h$ has at most $2p+2$ fixed points.
So that's it ... any help to solve my difficulty is welcome!Thanks!