TITLE: Riemann Surface, existence of meromorphic function.
QUESTION [7 upvotes]: There is a question which has been perplexing me. Given $S$ a compact Riemann surface, $p$ and $q$ two distinct points. Is it always possible to find a meromorphic function on $S$ which is zero on $p$ but not on $q$?
Thanks!

REPLY [7 votes]: Suppose the genus of $S$ is $g$ then consider the divisor $m\cdot p - q$. By Riemann-Roch theorem, we have 
$$ l(m\cdot p - q) - l(K - m\cdot p +q) = m - g $$
where $K$ is a canonical divisor. Let $m$ be large enough such that $deg(K - m\cdot p + q) = 2g - 1 - m < 0$ i.e. $l(K - m\cdot p + q) = 0$, then the dimension of $L(m\cdot p - q)$ grows linearly. Thus, there exists a meoromorphic function $f \in L((m+1)\cdot p - q)$ but $f \notin L(m\cdot p - q)$. That is, there exists a function $f$ that has a pole of order $m+1$ at $p$ and a zero at $q$ as you want.