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Let $X$ be a compact Riemann surface, and $p\neq q\in X$ be two distinct points. Given any positive integer $n$, can one find a meromorphic function on $X$ which exactly has a zero of order $n$ at $p$ and a pole of order $n$ at $q$?

The statement reminds me of Mittag-Leffler theorem, but I'm not sure whether it holds for compact Riemann surfaces. Any help would be appreciated.

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  • $\begingroup$ Do you mean to require that the function has no zeroes and poles besides $p$ and $q$? $\endgroup$ Commented Jul 5, 2018 at 5:32
  • $\begingroup$ @EricWofsey : Yes, that's what I require. $\endgroup$ Commented Jul 5, 2018 at 5:37

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This is not possible in general. For instance, for $n=1$, the meromorphic function would have to have degree $1$ and thus be an isomorphism between $X$ and $\mathbb{P}^1$, so no such function can exist if $X$ has nonzero genus. More generally, if $X$ has genus $g$, then for any $q$ there are $g$ different positive integers $n$ such that $X$ has no meromorphic functions with a pole of order $n$ at $q$ and no other poles, called "Weierstrass gaps". This follows from Riemann-Roch; you can find a sketch of the argument on Wikipedia.

(I don't know how constraining the additional condition of having $p$ as the only zero is. It seems plausible to me that for a generic curve $X$ of sufficiently high genus, there does not exist such a meromorphic function for any $p,q,$ and $n$.)

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