The Tate conjecture for K3 surfaces over finite fields

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>3. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p>3. 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>3.

20 pages, minor changes. Theorem 4 is stated in greater generality, but proofs don't change. Comments still welcome

References in corpus (2)

• Supersingular K3 surfaces for large primes
• Moduli of abelian varieties and p-divisible groups

Cited by in corpus (26)

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• Effective obstruction to lifting Tate classes from positive characteristic
• On Irreducible Symplectic Varieties of ${K3}^{[n]}$-type in Positive Characteristic
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• Curves and cycles on K3 surfaces
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• On geometric Brauer groups and Tate-Shafarevich groups
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• Cylinder maps of algebraic cycles on cubic hypersurfaces
• Irreducibility of geometric Galois representations and the Tate conjecture for a family of elliptic surfaces
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• The Noether--Lefschetz theorem in arbitrary characteristic
• A variety that cannot be dominated by one that lifts