A G ] 1 3 Se p 20 07 K 3 Surfaces of Finite Height over Finite Fields ∗ † (original) (raw)

Arithmetic of K3 surfaces defined over finite fields are investigated. In particular, we show that any K3 surface X of finite height over a finite field k of characteristic p ≥ 5 has a quasi-canonical lifting Z to characteristic 0, and that the endormorphism algebra of the transcendental cycles V (Z), as a Hodge module, is a CM field over Q. We also prove the Tate conjecture for any powers of such a K3 surface over k when the lifted Frobenius on V (Z) is irreducible. We illustrate by examples how to determine the associated formal Brauer group explicitly. Examples discussed here are all of hypergeometric type.