Misc,
The Tate conjecture for K3 surfaces over finite fields
(2012)cite arxiv:1206.4002Comment: 20 pages, minor changes. Theorem 4 is stated in greater generality, but proofs don't change. Comments still welcome.
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.
