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Reconstructing pdivisible groups from their truncations of small level
 COMMENT. MATH. HELV. 85 (2010), 165–202
, 2010
"... Let k be an algebraically closed field of characteristic p>0. Let D be a pdivisible group over k. Let nD be the smallest nonnegative integer for which the following statement holds: if C is a pdivisible group over k of the same codimension and dimension as D and such that CŒp nD � is isomorph ..."
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Cited by 7 (4 self)
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Let k be an algebraically closed field of characteristic p>0. Let D be a pdivisible group over k. Let nD be the smallest nonnegative integer for which the following statement holds: if C is a pdivisible group over k of the same codimension and dimension as D and such that CŒp nD � is isomorphic to DŒp nD �, then C is isomorphic to D. To the Dieudonné module of D we associate a nonnegative integer `D which is a computable upper bound of nD. IfD is a product Q i2I Di of isoclinic pdivisible groups, we show that nD D `D; if the set I has at least two elements we also show that nD maxf1; nDi;nDi C nDj
A motivic conjecture of Milne
, 2007
"... Let k be an algebraically closed field of characteristic p> 0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates the étale cohomology with Zp coefficients to the crystalline cohomology with integral coefficients, in the wider context ..."
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Cited by 6 (5 self)
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Let k be an algebraically closed field of characteristic p> 0. Let W(k) be the ring of Witt vectors with coefficients in k. We prove a motivic conjecture of Milne that relates the étale cohomology with Zp coefficients to the crystalline cohomology with integral coefficients, in the wider context of pdivisible groups endowed with families of crystalline tensors over a finite, discrete valuation ring extension of W(k). The result implicitly extends work of Faltings. As a main new tool we construct global deformations of pdivisible groups endowed with crystalline tensors over certain regular, formally smooth schemes over W(k).
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.
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.