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  "body": "\n## Definition\n\nLet $k$ be a field.\nAn *abelian variety* over $k$ is an [[algebraic group|GH81SYAPPN]] $A$ over $k$ such that:\n\n1. $A$ is a geometrically integral [[variety|HV6YBS0YDH]] over $k$;\n2. $A$ is complete, that is, the structure morphism $A \\to \\operatorname{Spec} k$ is [[proper|9537RKBZPP]].\n\nHere the group structure is given by a multiplication morphism $m \\colon A \\times_k A \\to A$, an inverse morphism $i \\colon A \\to A$ and a unit point $e \\in A(k)$ that satisfy the group axioms.\n\n## Remarks\n\nExample: an elliptic curve $E$ over $k$, with its chord-and-tangent group law and the point at infinity as unit, is an abelian variety of dimension $1$.\nIt is a smooth projective curve, so it is geometrically integral and complete, and the group law is given by morphisms.\n\nNon-example: the multiplicative group $\\mathbb{G}_m = \\operatorname{Spec} k[t, t^{-1}]$ is a geometrically integral algebraic group, but it is not complete.\nIt is an open subset of the projective line, and it is affine of positive dimension, so it is not proper over $k$.\n\nThe following three facts follow from the definition and are not part of it.\nThe group law of an abelian variety is commutative, so it is often written additively.\nEvery abelian variety is projective.\nAn abelian variety is smooth over $k$, since it is a geometrically reduced group variety.\n",
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