Fixed points of an automorphism and a meromorphic map.

Question

I want to solve this problem

Let $X$ a compact Riemann surface and $f:X\rightarrow \mathbb{P}^1$ a non constant meromorphic function in $X$. Let $h\in\text{Aut }(X) $ and automorphism of finite order $\text{ord } (h)$. If the number of fixed points of $h$ is bigger than $2\,\text {deg }(f)$ then $f=f\circ h$ and $\text {deg } (f)$ is a multiple of $\text {ord }{h}$.

I've been playing around with the Riemann-Hurwitz equations without arriving to something useful.

Any hint?

Answer 1

$f$ is holomorphic $X\to\Bbb{P}^1$ and meromorphic $X\to\Bbb{C}$.

Composing $f$ with a Möbius transformation we can assume the given fixed points of $h$ are not poles of $f$.

How many zeros does $f-f\circ h$ have ? How many poles does $f$ thus $f\circ h$ have ?