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Let $x_{1}, . . . , x_{n}$ be distinct points in a compact Riemann surface $\Sigma$ and let $w_{1}, . . . w_{n}$ be distinct points in $\mathbb{C}$. Show that there is a meromorphic function on which maps $x_{i}$ to $w_{i}$ for $i = 1, . . . , n$.

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This îs not a trivial fact, as even the existence of a non-constant meromoprhic function is not that easy to show. We'll use some big guns here, some consequence of Riemann-Roch theorem that we'll state:

Given $n$ points on a Riemann surface $x_1$, $\ldots$,$x_n$, there exists a meromorphic function on it with zeroes at all $x_i$, $i\ne 1$ and a pole at $x_1$. Compose this meromorphic function on the left with $z\mapsto \frac{z}{z+1}$ we get a meromorphic function $\phi_1$ which takes value $1$ at $x_1$ and $0$ at all the other points. Do this similarly for all the $x_i$ and get $\phi_i$, and now take $\phi\colon = \sum w_i \phi_i$.

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  • $\begingroup$ How can we ensure that there exists a meromorphic function on it with zeroes at all $x_i$, $i\ne 1$ and a pole at $x_1$? $\endgroup$ – Manoel Jan 6 '17 at 23:44
  • $\begingroup$ This question (math.stackexchange.com/questions/634141/…) is also about that. $\endgroup$ – Manoel Jan 6 '17 at 23:48

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