Elliptic curve group

Definition
Let $$k$$ be a field and $$E = 0$$ be the equation of an elliptic curve (viz, a nonsingular cubic curve) over $$k$$. Let $$C$$ be the locus of $$E$$ in the projective plane over $$k$$. Then, we define the elliptic curve group over $$E$$ in the following equivalent ways:


 * It is the quotient group of the free abelian group on the set of all points of $$C$$, by the relations $$P + Q + R = 0$$ for every collinear triple of points $$P,Q,R$$ (here, each point is counted with multiplicity, so if a line is tangent at $$P$$ and also passes through $$Q$$, we get $$2P + Q = 0$$).
 * It is a group whose set of points is identified with the set of points in $$C$$, where the identity element is the inflection (at infinity) and the sum of any three collinear points is zero.

Equivalence of definitions
To prove the equivalence of the above definitions, we need to show that the addition defined by the collinearity relation is actually associative. This follows as a consequence of Bezout's theorem.

For a general cubic curve
If we remove the condition of nonsingularity, we still get a monoid analogous to the elliptic curve. The problem is at the singularity points.

Different fields for the same curve
If $$E$$ is the equation of an elliptic curve with coefficients over a field $$k$$, then we can consider $$E$$ as the equation of an elliptic curve over a field $$K$$ containing $$k$$.

The elliptic curve group over this bigger field will contain the elliptic curve group over a smaller field. A category-theoretic way of putting it is that any elliptic curve with coefficients in $$k$$ defines a functor from fields containing $$k$$, to groups containing the elliptic curve group.

Generalizations

 * Divisor class group
 * Hyperelliptic curve group