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Tate's Conjecture for K3 Surfaces of Finite Height

Niels Nygaard and Arthur Ogus
Annals of Mathematics
Second Series, Vol. 122, No. 3 (Nov., 1985), pp. 461-507
Published by: Annals of Mathematics
DOI: 10.2307/1971327
Stable URL: http://www.jstor.org/stable/1971327
Page Count: 47
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Tate's Conjecture for K3 Surfaces of Finite Height
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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 etale 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 etale cohomology.

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