# Elliptic curve group

This article defines a natural context where a group occurs, or is associated, with another algebraic, topological or analytic structure

View other occurrences of groups

This term is related to: algebraic number theory

View other terms related to algebraic number theory | View facts related to algebraic number theory

## Contents

## Definition

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

- It is the quotient group of the free abelian group on the set of all points of , by the relations for every collinear triple of points (here, each point is counted with multiplicity, so if a line is tangent at and also passes through , we get ).
- It is a group whose set of points is identified with the set of points in , 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 is the equation of an elliptic curve with coefficients over a field , then we can consider as the equation of an elliptic curve over a field containing .

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 defines a functor from fields containing , to groups containing the elliptic curve group.