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 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 etale cohomology revealed the Hodge cycles. Here we use more general quasi-canonical liftings, and the action of the crystalline Weil group on
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