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Sources say that if you have a curve defined over $\mathbb{F}_q$ and a collection of $\mathbb{F}_q$-rational points $P_1 \cdots P_n$. There exists a differential, $\omega$ on the curve with simple poles at the $P_i$'s and Res$_{P_i}(\omega)=1$.

Can this be explicitly written down for any cases? I don't even know how to start? What about for $\mathbb{P}^1$? You can assume all the points are away from the place at infinity as well.

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  • $\begingroup$ Sounds unlikely to me. Are you allowed poles other than $P_1,\ldots,P_n$? If not I can't see existence, if so I can't see uniqueness. $\endgroup$ – Lord Shark the Unknown Oct 26 '17 at 5:40
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    $\begingroup$ Im a dirty liar, not unique..... Ill update it @LordSharktheUnknown $\endgroup$ – Tom Lewia Oct 26 '17 at 5:41

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