K3 surfaces of finite height over finite fields

Abstract

Arithmetic of K3 surfaces defined over finite fields is investigated.In particular, we show that any K3 surface $X$ of finite height over a finite field $k$ of characteristic $p \geq 5$ has a quasi-canonical lifting $Z$ to characteristic $0$, and that for any such $Z$, the endormorphism algebra of the transcendental cycles $V(Z)$, as a Hodge module, is a CM field over $\mathbb{Q}$.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.