Good reduction of an abelian variety over a Dedekind scheme

Let $S$ be a Dedekind scheme (Dedekind domains) with function field $K$, and let $A_K$ be an abelian variety over $K$ (Abelian varieties over a field). One says that $A_K$ has good reduction over $S$ if there exists an abelian scheme $A\to S$ (Abelian schemes over a base) together with an isomorphism $A{\times }_S\mathrm{Spec}K\cong A_K$ of $K$-schemes; such an $A$ is an abelian scheme model of $A_K$ over $S$. For a discrete valuation ring $R$ with fraction field $K$ (Discrete valuation rings) this is the local notion at its closed point, and $A_K$ has potential good reduction if there is a finite extension $K^{\mathrm{\prime }}\mathrm{/}K$ such that $A_K{\otimes }_K{K}^{\mathrm{\prime }}$ has good reduction over the normalization of $R$ in $K^{\mathrm{\prime }}$.

For a closed point $s\in S$, the reduction of $A$ at $s$ is $A_s=A{\times }_S\mathrm{Spec}\kappa (s)$, which is an abelian variety over the residue field $\kappa (s)$ of dimension $\dim A_K$. Good reduction is a property of the pair $(A_K,S)$; the definition asserts no existence statement, and in particular no claim is made that every abelian variety has good reduction.