Ordinary K3 surfaces over a finite field

TL;DR

This paper characterizes ordinary K3 surfaces over finite fields using linear algebra data, extending classical results for abelian varieties and providing new insights into their structure under certain conditions.

Contribution

It offers a linear algebraic description of ordinary K3 surfaces over finite fields, refining previous work and extending results to specific cases unconditionally.

Findings

Provides a categorical description of ordinary K3 surfaces over finite fields.

Extends Deligne's description from abelian varieties to K3 surfaces.

Offers unconditional results for K3 surfaces with large Picard rank or small degree.

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.