Group structure on the elliptic curve. 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
 A: 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 $P+0=P$. 
A: Take the book: "Algebraic Curves" by Fulton, pag 63 and exercise 5.41 (case m = 1).
