Tate’s conjecture for $K3$ surfaces of finite height

Abstract

This paper extends the proof [16] of the Tate conjecture for ordinary $K3$ surfaces over a finite field to the more general case of all $k3$’s of finite height. As in [16], our method is to find a lifting of the $K3$ to characteristic zero with sufficiently many Hodge cycles. In the ordinary case, the so-called “canonical lifting” of Deligne and Illusie [7] did the job, and a study of the Galois action on $p$-adic étale cohomology revealed the Hodge cycles. Here we use more general “quasi-canonical liftings,” and the action of the crystalline Weil group on de Rham cohomology replaces the Galois action on étale cohomology.

Niels Nygaard

Arthur Ogus