The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes

Messing, William

doi:10.1007/bfb0058301

Cited by 265 publications

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“…The crystalline cohomology class The preceding lemma allows us to control the deformation theory of the pair (A 1 , ψ 1 ) as in the sketch of proof above. Indeed, standard deformation theory arguments and, for instance, Grothendieck-Messing theory as in [24] -see also [11,Theorem 2.4] for a summary -shows that the obstruction to deforming ψ 1 with A 1 is controlled by the group H 2 (X, O X ). As this k-vector space is one-dimensional, the versal deformation space of (A 1 , ψ 1 ) is a divisor Σ in the formal neighborhood…”

Section: 3mentioning

confidence: 99%

The Tate conjecture for K3 surfaces over finite fields

Charles

2012

Invent. math.

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Abstract. Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p ≥ 5. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p ≥ 5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p ≥ 5.

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Section: 3mentioning

confidence: 99%

The Tate conjecture for K3 surfaces over finite fields

Charles

2012

Invent. math.

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“…For given x ∈ G n (A), since G n is finitely presented there is a finitely generated ideal I ′ ⊆ I such that x lifts to an element x ′ ∈ G n (B/I ′ ). Now we can use that G is formally smooth by [Me1,Th. 3.3.13].…”

Section: From P-divisible Groups To Dieudonné Displaysmentioning

confidence: 99%

Relations between Dieudonné displays and crystalline Dieudonné theory

Lau

2014

Algebra Number Theory

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“…We provide a short overview of the theory of one-dimensional Barsotti-Tate O Kmodules , for O K the ring of integers of a p-adic local field K, following [7], [2]. We exclusively focus on those aspects of the theory which are relevant for this paper.…”

Section: Preliminariesmentioning

confidence: 99%

A compactification of Igusa varieties

Mantovan

2007

Math. Ann.

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Abstract. We investigate the notion of Igusa level structure for a one-dimensional Barsotti-Tate group over a scheme X of positive characteristic and compare it to Drinfeld's notion of level structure. In particular, we show how the geometry of the Igusa covers of X is useful for studying the geometry of its Drinfeld covers (e.g. connected and smooth components, singularities).Our results apply in particular to the study of the Shimura varieties considered in [3]. In this context, they are higher dimensional analogues of the classical work of Igusa for modular curves and of the work of Carayol for Shimura curves. In the case when the Barsotti-Tate group has constant p-rank, this approach was carried-out by Harris and Taylor in [3].

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The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes

Cited by 265 publications

References 0 publications

The Tate conjecture for K3 surfaces over finite fields

The Tate conjecture for K3 surfaces over finite fields

Relations between Dieudonné displays and crystalline Dieudonné theory

A compactification of Igusa varieties

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