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  "body": "\n## Definition\n\nLet $R$ be a [[discrete valuation ring|MGM3CNJW9V]] with fraction field $K$.\nLet $A_K$ be an [[abelian variety|R7W03TXRS9]] over $K$.\nA *Néron model* of $A_K$ is an $R$-scheme $N$ with an isomorphism $N \\times_R K \\cong A_K$ such that:\n\n1. $N$ is [[smooth|EXKGG9X6J0]], [[separated|X30G4V7VAJ]] and [[of finite type|AHTE5DVX61]] over $R$;\n2. for every smooth $R$-scheme $Y$ and every $K$-morphism $u_K \\colon Y \\times_R K \\to A_K$, there is a unique $R$-morphism $u \\colon Y \\to N$ that extends $u_K$.\n\nClause 2 is called the Néron mapping property.\n\n## Remarks\n\nA Néron model exists for every abelian variety $A_K$, and it is unique up to unique isomorphism by clause 2.\nClause 2 extends the group law of $A_K$, so $N$ is a group scheme over $R$.\nTaking $Y = \\operatorname{Spec} R$ in clause 2 shows that every point in $A_K(K)$ extends to a point in $N(R)$, so $N(R) = A_K(K)$.\n\nExample: if $A_K$ extends to an [[abelian scheme|JDDXZ89N33]] $A$ over $R$, then $A$ is the Néron model of $A_K$.\nThis is the case of good reduction.\n\nNon-example: let $E$ be an elliptic curve over $K$ with bad reduction, and let $W$ be the closure in $\\mathbb{P}^2_R$ of a Weierstrass equation for $E$ with coefficients in $R$.\nThe special fibre of $W$ is a singular cubic, so $W$ is not smooth over $R$ and fails clause 1.\n",
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