Manin problems for Shimura varieties of Hodge type

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 k̄. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. We solve them if either p ≥ 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.