I'm reading Jürgen Jost's book; Compact Riemann Surface. In the chapter of elliptical curves I find the following theorem:
"Theorem 5.10.3 In the group structure on the elliptic curve $\Sigma$ given by $y^{2}=4x^{3}-g_{2}x -g_{3}$, we will have $p_{1}+p_{2}+p_{3}=0$ if and only if $p_{1},p_{2},p_{3}$ lie on line in $\mathbb{P}^{2}$."
I read the demonstration, but I do not understand ... I wonder if I can find it in another reference, another book about this result.
thank you
There are tons of books about elliptic curves. One cheap one is Milne's book "Elliptic curves".
I'll try to explain the group structure in more than one sentence:
So a group structure on $\Sigma$ means that we have a rule for adding two points and getting a third point. Furthermore, there is an identity element, the group operation is commutative and associative.
The rule is this: Let $P,Q \in \Sigma$. Then we define $P+Q$ to be the following point: let $L_{PQ}$ be the line between $P$ and $Q$. Since $\Sigma$ is a curve a degree $3$, this line intersect $\Sigma$ in a third point $R'$. Then we define $P+Q$ to be the reflection of $R'$ about the $x$ axis.
Note that this definition makes commutativity clear: $L_{PQ}=L_{QP}$. Associativity is much harder to prove.
The identity element is the "point at infinity", it has homogeneous coordinates $(0:0:1)$. Think of it as a point "infinitely high above the finite plane". Then $L_{P0}$ (letting 0 denote the point at infinity) is just a vertical line. Then (if you draw an example), you'll see that $P+0=P$.