# Explicit Differential on a Curve

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.

• 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. – Lord Shark the Unknown Oct 26 '17 at 5:40
• Im a dirty liar, not unique..... Ill update it @LordSharktheUnknown – Tom Lewia Oct 26 '17 at 5:41