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It is well known that a Riemann surface $C$ such that there exists a meromorphic function with just one simple pole on it, then $C $ is the Riemann sphere.

What can be said if there exists a meromorphic function with exactly two simple poles?

And if it is one double pole instead of two simple ones?

Carlo
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    I think you mean a compact Riemann surface? – Steven Gubkin Jun 29 '15 at 17:49
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    In the language of algebraic geometry - A curve (complete Riemann surface) is called hyperelliptic if its genus $\ge2$, and admits a finite morphism to $P^1$ (Riemann sphere) of degree 2 - https://en.wikipedia.org/wiki/Hyperelliptic_curve might be a good place to start looking. – peter a g Jun 29 '15 at 18:00

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