K3 surfaces of finite height over finite fields

TL;DR

This paper studies K3 surfaces over finite fields, showing they can be lifted to characteristic zero with specific properties, and proves the Tate conjecture for certain products, providing explicit examples of their formal Brauer groups.

Contribution

It demonstrates the quasi-canonical lifting of finite height K3 surfaces and establishes the CM field structure of their transcendental cycles, advancing understanding of their arithmetic properties.

Findings

Finite height K3 surfaces over finite fields have quasi-canonical liftings.

The endomorphism algebra of transcendental cycles is a CM field.

The Tate conjecture is proved for products of certain K3 surfaces.

Abstract

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for any such lifting, the endormorphism algebra of the transcendental cycles, as a Hodge module, is a CM field. The Tate conjecture for the product of certain two K3 surfaces is also proved. We illustrate by examples how to determine explicitly the formal Brauer group associated to a K3 surface over k. Examples discussed here are all of hypergeometric type.