Purity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic

Abstract

Let $p$ be a prime. Let $(R,\ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $\Spec R\setminus\{\ideal{m}\}$ extends to an abelian scheme over $\Spec R$. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of N\'eron models over $O$. If $p>2$ and index $p-1$, the examples are new.