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 small degree.

On ordinary Enriques surfaces in positive characteristic

“…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 ).…”

“…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.…”

“…We now want to apply the results of the previous paragraph to give a description of Enriques surfaces over finite fields by using linear algebra data in a similar way to what Taelman does in [12] for K3 surfaces. Let Y be an Enriques surface over a finite field F q , and choose, like before, an embedding i : W → C. Let again Y ι can be the complex Enriques surfaces obtained from Y can via extension of scalars through ι.…”

“…Finally in Section 5, we construct a groupoid of ordinary Enriques surfaces over finite fields and, as in Taelman's work [12], we construct a fully faithful functor to a suitable category of linear algebra data. This functor turns out to be an equivalence of categories if one assumes a strong form of potential semi-stable reduction (this is condition (⋆) in [12]).…”

An example of a Brauer--Manin obstruction to weak approximation at a prime with good reduction

“…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…”