# Elliptic curve group

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological or analytic structure
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This term is related to: algebraic number theory
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## 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.

## Facts

### 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.