An abelian variety, named after Niels Abel, variety is a higher dimensional version of an elliptic curve.

Over a field $k$, an abelian variety consists of a smooth projective variety $X$ over $k$, a $k$-rational point 0 and morphisms $+$ : $X \times X \rightarrow X$ and $–$ : $X \rightarrow X$ which make it into a group.