I want to show the following:

Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$.

In the above it holds that $F$ is an isomorphism.

I am pretty sure that I should use the formula of riemann-hurwitz which can be found here. But I wasnt able to proof the statement.

Hopefully someone can help me.

  • $\begingroup$ Hint: try to show that $N=1$ and there is no branched point. $\endgroup$ – user99914 Jul 17 '16 at 12:54
  • $\begingroup$ Well, if there would be no branced points, then we would have $N=1$ and this yields that $F$ is an isomorphism (I already proved this fact). But with so less information I don't know how to prove that there is no branched point. $\endgroup$ – Marc Jul 17 '16 at 13:13
  • $\begingroup$ Note that $2g-2$ is strictly positive (since $g\ge 2$) and $\sum (e_p - 1)$ is nonnegative. $\endgroup$ – user99914 Jul 17 '16 at 13:15
  • $\begingroup$ Oh okay, quite easy. ^.^. Thanks. $\endgroup$ – Marc Jul 17 '16 at 13:18
  • $\begingroup$ You may post an answer, @Marc. That finishes the question. $\endgroup$ – user99914 Jul 17 '16 at 14:14

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