Inventiones mathematicae

, Volume 194, Issue 1, pp 119–145 | Cite as

The Tate conjecture for K3 surfaces over finite fields

Article

Abstract

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.

Mathematics Subject Classification (2010)

14C22 14C25 14G15 14J28 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.IRMAR–UMR 6625 du CNRSUniversité de Rennes 1Rennes CedexFrance

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