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

### Summary

Summary: 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 $ele p-1$, exist in most cases if $ple ele 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $ele p-1$ 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 Néron models over $O$. If $p>2$ and index $p-1$, the examples are new.

### Mathematics Subject Classification

11G10, 11G18, 14F30, 14G35, 14G40, 14K10, 14K15, 14L05, 14L15, 14J20

### Keywords/Phrases

rings, group schemes, $p$-divisible groups, breuil windows and modules, abelian schemes, Shimura varieties, and Néron models