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The Tate conjecture for K3 surfaces over finite fields

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An Erratum to this article was published on 22 April 2015

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.

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Notes

  1. The result of [22] above only works over a smooth base, and \(\widehat{B}[1/p]\) might not be smooth. However, the abelian scheme \(\mathcal{A}\) over \(\widehat{B}\) comes from the universal abelian scheme over \(\mathcal{A}_{g, d', n}\), which is smooth and where Katz’ result applies.

  2. The only difference with Proposition 25 is that \(\overline{Z}_{\varLambda}\) is not normal a priori. However, one can work on the normalization of \(\overline{Z}_{\varLambda}\) and carry on with the proof.

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Acknowledgements

The work on this paper started when I had the chance of discussing the paper [24] with Daniel Huybrechts during the workshop “Advances in hyperkähler and holomorphic symplectic geometry” at the BIRS in Banff.

I am very grateful to D. Huybrechts for the many discussions we had on this subject and his many remarks on early versions of this paper. I would also like to thank the organizers of the workshop for creating a stimulating atmosphere. I thank Davesh Maulik for helpful email correspondence, Olivier Benoist for interesting discussions and Hélène Esnault for helpful remarks on a first draft of this paper. I thank the referee for his careful reading and helpful remarks.

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  1. IRMAR–UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042, Rennes Cedex, France

    François Charles

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Charles, F. The Tate conjecture for K3 surfaces over finite fields. Invent. math. 194, 119–145 (2013). https://doi.org/10.1007/s00222-012-0443-y

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