Elliptic curves carry the structure of an abelian group, which in terms of curves defined over the complex numbers is the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.
Elliptic curves carry the structure of an abelian group with the distinguished point as the identity of the group law. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions.