Manin problems for Shimura varieties of Hodge type

Abstract

Let k be a perfect field of characteristic p>0. We prove the existence of ascending and descending slope filtrations for Shimura p-divisible objects over k. We use them to classify rationally these objects over \bar k. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. We solve them if either p\Ge 3 or p=2 and two mild conditions hold. Second we formulate integral Manin problems. We solve them for certain Shimura varieties of PEL type.

43 pages. Final version as close to the galley proof version as the styles allow. To appear in J. of the Ramanujan Math. Soc

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