The Tate conjecture for K3 surfaces over finite fields
Algebraic Geometry
2015-06-05 v2 Number Theory
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.
Cite
@article{arxiv.1206.4002,
title = {The Tate conjecture for K3 surfaces over finite fields},
author = {François Charles},
journal= {arXiv preprint arXiv:1206.4002},
year = {2015}
}
Comments
20 pages, minor changes. Theorem 4 is stated in greater generality, but proofs don't change. Comments still welcome