Abelian variety

Summary

An abelian variety is a complete algebraic variety that is also an algebraic group, with group operations given by morphisms. Its group law is automatically commutative, and the variety is automatically projective. Abelian varieties of dimension one are the elliptic curves, and the Jacobian variety of a curve is a basic example in higher dimension. They are central in arithmetic geometry, for instance in the Mordell–Weil theorem.