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

Let p be a prime. Let (R, 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 \ {m} extends to an abelian scheme over Spec R. We show that such extensions always exist if e ≤ p -1, exist in most cases if p ≤ e ≤ 2p -3, and do not exist in general if e ≥ 2p - 2. The case e ≤ 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.