Hi,


Using the isomorphism between an elliptic curve $E$ and its $Pic\_1(E)$ group, one can
easily give $E$ the structure of a group variety after choosing a point $O\in E$. The
operation that one gets is:
$$P+Q+R = 0\text{ in $E$ iff }P+Q+R-3O\text{ (as divisors) is a principal divisor}.$$


* Question:
Why is the condition $P+Q+R-3O$ being a principal divisor equivalent to $P+Q+R$ being the
intersection divisor of $E$ with a line?


Thanks!



definition of group operation in elliptic curves
This is true if $O$ is an inflexion point. In this case $3O$ is cut out by a line, so $3O$ is linearly equivalent to the intersection of a line with a curve. Thus $P + Q + R$ is linearly equivalent to $3O$ if and only if it is cut out by a line.


