Let $E$ be an elliptic curve and $k$ a field. It is well know that $E(k)$ has an (additive) group structure and indeed there are a lot of sources describing what geometrically there is going on.
The point of my interest is to find a derivation of this group law in language of divisors - especially using properties of the divisor class group.
Indeed the divisors $Div(E)$ are formal sums $\sum_{P \in E(k)} n_P (P)$ with $n_P \in \mathbb{Z}$ and the principal divisors $div(f) = \sum_P ord_P(f) (P)$ form a subgroup of $Div(E)$; denote it by $PrDiv(E)$.
The divisor class group is the quotient $Cl(E)= Div(E)/PrDiv(E)$.
Obviosly we can define canonically a map $E(k) \to Cl(E), p \to (P)-(O)$ where $O$ is the special point (=neutral element).
How to show that this map determine the group law on $E(k)$.