Néron model

Summary

Let $R$ be a discrete valuation ring with fraction field $K$. The Néron model of an abelian variety over $K$ is its best smooth model over $R$. It is a smooth group scheme of finite type over $R$, characterized by a universal property for maps from smooth $R$-schemes. André Néron proved that it always exists. Its special fibre describes the reduction of the abelian variety, even when the reduction is bad.