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Manin problems for Shimura varieties of Hodge type
, 2007
"... Let k be a perfect field of characteristic p> 0. We prove the existence of ascending and descending slope filtrations for Shimura pdivisible objects over k. We use them to classify rationally these objects over ¯ k. Among geometric applications, we mention two. First we formulate Manin problems ..."
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Cited by 10 (6 self)
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Let k be a perfect field of characteristic p> 0. We prove the existence of ascending and descending slope filtrations for Shimura pdivisible objects over k. We use them to classify rationally these objects over ¯ k. Among geometric applications, we mention two. First we formulate Manin problems for Shimura varieties of Hodge type. Under two mild conditions (checked for p ≥3 in [45]) we solve them. Second we formulate integral Manin problems. We solve them for some Shimura varieties of PEL type.
CM lifts for Isogeny Classes of Shimura Fcrystals over Finite Fields
, 2007
"... We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of CM lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with endomorphisms over finite fields. These contexts are general enough in ..."
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Cited by 6 (1 self)
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We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of CM lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with endomorphisms over finite fields. These contexts are general enough in order to apply to the Langlands–Rapoport conjecture for all special fibres of characteristic at least 5 of integral canonical models of Shimura varieties of Hodge type.
THE RELATIVE BREUILKISIN CLASSIFICATION OF pDIVISIBLE GROUPS AND FINITE FLAT GROUP SCHEMES
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Good Reductions of Shimura Varieties of Preabelian Type in Arbitrary Unramified Mixed Characteristic, I
, 2003
"... ABSTRACT. We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of ..."
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ABSTRACT. We prove the existence of weak integral canonical models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence of integral canonical models of Shimura varieties of preabelian (resp. of abelian) type in mixed characteristic (0, p) with p ≥3 (resp. with p = 2) and with respect to hyperspecial subgroups; if p = 3 (resp. if p = 2) we restrict in this part I either to the An, Cn, DH n (resp. An and Cn) types or to the Bn and DR n (resp. Bn, DH n and DR n) types which have compact factors over R (resp. which have compact factors over R in some pcompact sense). Though the second application is new just for p ≤ 3, a great part of its proof is new even for p ≥5 and corrects [Va1, 6.4.11] in most of the cases. The second application forms progress towards the proof of a conjecture of Milne. It also provides in arbitrary mixed characteristic the very first examples of general nature of projective varieties over number fields which are not embeddable into abelian varieties and which have Néron models over certain local rings of rings of integers of number fields.