Let $C$ be an elliptic curve in $\mathbb{P}^3$. By Riemann-Roch we have $\ell(D)=\deg D$ for a divisor $D$, since the genus $g$ equals 1. So if $a$ is a point on the curve, then the complete linear system of $D=1\cdot a$ will only contain $D$ itself. Apart from Riemann-Roch, one can decude this by translating rational functions on $C$ to elliptic functions on the corresponding torus. And one sees that such elliptic functions can't have poles of order one, and so again the complete linear system of $D=1\cdot a$ just contains itself. I've learned most of this watching Borcherds' youtube series.
My question is, is there a more direct or elementary way of understanding this fact about the complete linear system of $1\cdot a$? Let's consider a rational function $f/g$ with $\deg f=\deg g$. Why is it not possible to choose $g$ to have a zero at $a$, $f$ to be non-zero at $a$ and that $g$ and $f$ otherwise have the same zeros on $C$?