The Tate conjecture for K3 surfaces over finite fields

TL;DR

This paper proves Artin's and Tate's conjectures for K3 surfaces over finite fields with characteristic p>3, confirming the conjectures in these cases and extending results to certain symplectic varieties and cubic fourfolds.

Contribution

It establishes the Tate conjecture for K3 surfaces over finite fields of characteristic p>3, advancing the understanding of algebraic cycles in positive characteristic.

Findings

Proves Artin's conjecture for supersingular K3 surfaces in characteristic p>3

Confirms Tate's conjecture for K3 surfaces over finite fields in characteristic p>3

Extends Tate conjecture results to certain holomorphic symplectic varieties and cubic fourfolds

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.