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

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.Comment: 28 pages. Final version identical (modulo style) to the galley proofs. To appear in Doc. Mat

Families of p-divisible groups with constant Newton polygon

A p-divisible group over a base scheme in characteristic p in general does not admit a slope filtration. Let X be a p-divisible group with constant Newton polygon over a normal noetherian scheme S; we prove that there exists an isogeny from X to Y such that Y admits a slope filtration. In case S is regular this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We give an example of a p-divisible group over a non-normal base which does not admit an isogeny to a p-divisible group with a slope filtration.Comment: To be published in Documenta Mathematic

On the Drinfeld moduli problem of p-divisible groups

Let $O_D$ be the ring of integers in a division algebra of invariant $1/n$ over a p-adic local field. Drinfeld proved that the moduli problem of special formal $O_D$-modules is representable by Deligne's formal scheme version of the Drinfeld p-adic halfspace. In this paper we exhibit other moduli spaces of formal $p$-divisible groups which are represented by $p$-adic formal schemes whose generic fibers are isomorphic to the Drinfeld p-adic halfspace. We also prove an analogue concerning the Lubin-Tate moduli space.Comment: Expanded introductio

On the Arithmetic Fundamental Lemma in the minuscule case

The arithmetic fundamental lemma conjecture of the third author connects the derivative of an orbital integral on a symmetric space with an intersection number on a formal moduli space of $p$-divisible groups of Picard type. It arises in the relative trace formula approach to the arithmetic Gan-Gross-Prasad conjecture. We prove this conjecture in the minuscule case.Comment: Referee's comments incorporated; in particular, the existence of frames for using the theory of displays in the proofs of Theorems 9.4 and 9.5 is clarified. To appear in Compositio Mat

Program to enhance tae transfer of new technology to potential industrial, governmental, and academic users in the oklahoma area Quarterly status report, 1 Jul. - 30 Sep. 1967

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