Abstract
Abstract We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over 𝐙 {{\mathbf{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.
Highlights
A K3 surface X over a perfect field k of characteristic p is called ordinary if it satisfies the following equivalent conditions: (i) the Hodge and Newton polygons of H2crys(X/W (k)) coincide, (ii) the Frobenius endomorphism of H2(X, OX ) is a bijection, (iii) the formal Brauer group of X has height 1. These are equivalent with |X(k)| ≡ 1 mod p
Using the strong version of the main theorem of CM for K3 surfaces of [19] we show that we can find a model of X over K := Frac W (Fq) ⊂ C such that (i) the GalK -module He2t(XK, Zp) = M ⊗ Zp decomposes as in Theorem C, (ii) for l = p, the GalK -module H2et(XK, Zl) = M ⊗ Zl is unramified, and Frobenius acts as F
Using Theorem C we show that XL is the canonical lift of its reduction, and deduce from this that X has already a smooth projective model X over OK
Summary
Citation for published version (APA): Taelman, L. (2020). Ordinary K3 surfaces over a finite field. Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date: Nov 2021
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