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

TL;DR

This paper establishes criteria for extending abelian schemes over regular local rings of mixed characteristic, with implications for the existence and uniqueness of integral models of Shimura varieties and Néron models.

Contribution

It provides new conditions determining when abelian schemes extend over regular bases of mixed characteristic, including cases with ramification index up to 2p-3.

Findings

Extensions exist if ramification index e ≤ p-1.

Extensions mostly exist if p ≤ e ≤ 2p-3.

Extensions generally do not exist if e ≥ 2p-2.

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.