Suppose $E$ is an elliptic curve over $k$, and $(E,+)$ is an abelian group(suppose we fix some closed point as identity). Let $[p]$ denote the Weil divisor corresponding to the closed point $p \in E$.
It is claimed that for any closed point $p_0, p, a \in E$, the Weil divisor $$[p_0 + a]+[p]-[p_0]-[a+p]$$ is trivial.
I know how to show this when $k = \mathbb{C}$: the proof uses the residue of periodic function and basically it says that for a Weil divisor to be trivial iff (1) the number of zeros equal to the number of poles and (2) the sum of zeros equal to the sum of poles. However, I don't know how to show this for arbitrary field. I guess one might use Riemann-Roth in some way, but did not figure out what to do.