2018
DOI: 10.1515/crelle-2018-0023
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Ordinary K3 surfaces over a finite field

Abstract: We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu.Our main result is conditional on a conjecture on potential semi-stable reduction of K3 surfaces over p-adic fields. We give unconditional versions for K3 surfaces of large Picard rank and for K3 surfaces of … Show more

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Cited by 7 publications
(21 citation statements)
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“…Remark 2.7. The proof of the above statement shows how, over finite fields, one can count the number of Enriques quotients of an ordinary K3 surface from the linear algebra data provided by Taelman in [12]. More precisely, to any ordinary K3 surface over a finite field k, we can associate a triplet (M, F, K), where M := H 2 (X ι can , Z), F : M → M is a lift of Frobenius and K is the ample cone in NS(X ι can ).…”
Section: Ordinary Enriques Surfaces and Canonical Liftsmentioning
confidence: 93%
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“…Remark 2.7. The proof of the above statement shows how, over finite fields, one can count the number of Enriques quotients of an ordinary K3 surface from the linear algebra data provided by Taelman in [12]. More precisely, to any ordinary K3 surface over a finite field k, we can associate a triplet (M, F, K), where M := H 2 (X ι can , Z), F : M → M is a lift of Frobenius and K is the ample cone in NS(X ι can ).…”
Section: Ordinary Enriques Surfaces and Canonical Liftsmentioning
confidence: 93%
“…In [5], the author makes restrictions on the ground field (which is assumed to be algebraically closed and of odd characteristic) and the proof makes strong use of derived categories and Taelman's result [12,Theorem C]. Here, we would like to propose a different approach which uses the canonicity of the canonical lift and deformation theory.…”
Section: Ordinary Enriques Surfaces and Canonical Liftsmentioning
confidence: 99%
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“…Proof. It is enough to show that the cardinality of Y (F 2 ) is even (see [Tae20]). Y is the projective F 2 -variety defined by the equation…”
Section: Construction Of the Quaternion Algebra Amentioning
confidence: 99%