Take any elliptic curve over a finite field $\mathbb{F}_q$ . If you fix a collection of affine rational points on the curve $P_1,...P_n$, there is a differential $\omega$ such that $\mathcal{v}_{P_i}(\omega)=-1$ and $\omega_{P_i}(1)=1$. My goal is to "find" such a differential for specific cases.
There is a one to one correspondence between functions on the curve and Weil differentials. Is there a natural way of constructing a function $f$ such that $df$ has the properties above.
There is no important reason for specifying an elliptic curve but I thought it may be the easiest place to start.