Group Law Elliptic Curve with Divisor Class Group

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 $$E(k)$$.

• $(O)$ isn't a principal divisor. Principal divisors have degree zero. – Lord Shark the Unknown Jan 11 at 21:27
• @LordSharktheUnknown: ah sure sure, $(O)-(O)$ should be the prinipal one and is obviously one. Thank you. But the point is why in $Cl(E)$ the group law is already determined. – KarlPeter Jan 11 at 21:31
• This is Proposition III.3.4 (p. 61) in Silverman. Have you looked there? – André 3000 Jan 11 at 21:33
• Every degree zero divisor is congruent, modulo principal divisors, to exactly one of the form $(P)-(O)$. – Lord Shark the Unknown Jan 11 at 21:33
• It suffices to show (or define) that $P+Q-R-O$ is a principal divisor iff $P + Q= R$ in the group law. From this then $\sum_j P_j-Q_j = R-O+div(f)$ where $R = \sum_j P_j-Q_j$ in the group law and $div(f) = O-R+\sum_j P_j-Q_j$ where $f$ is meromorphic – reuns Jan 11 at 22:05