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  "body": "\n## Definition\n\nLet $S$ be a scheme.\nAn *abelian scheme* over $S$ is a [[group scheme|TJ04VQP4NB]] $\\pi \\colon A \\to S$ such that:\n\n1. $\\pi$ is [[smooth|EXKGG9X6J0]];\n2. $\\pi$ is [[proper|9537RKBZPP]];\n3. every fibre of $\\pi$ is geometrically connected.\n\n## Remarks\n\nFor each point $s \\in S$, the fibre $A_s$ is an [[abelian variety|R7W03TXRS9]] over the residue field $\\kappa(s)$.\nAn abelian scheme over $\\operatorname{Spec} k$, for a field $k$, is the same as an abelian variety over $k$.\nThe group law of an abelian scheme is commutative.\n\nExample: let $S = \\operatorname{Spec} \\Z[1/2]$ and let $E \\subseteq \\mathbb{P}^2_S$ be the closure of $y^2 = x^3 - x$, with unit section the point at infinity.\nThe discriminant of this Weierstrass equation is $64 = 2^6$, a unit on $S$.\nSo every fibre is a smooth curve of genus $1$, and $E$ is an abelian scheme of relative dimension $1$.\n\nNon-example: the same equation over $\\operatorname{Spec} \\Z$ fails clause 1.\nModulo $2$, the curve $y^2 = x^3 + x$ is singular at the point $(1, 0)$, where both partial derivatives $2y$ and $3x^2 + 1$ vanish.\n",
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