The Tate conjecture for K3 surfaces over finite fields

François Charles (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]

François Charles. The Tate conjecture for K3 surfaces over finite fields. Inventiones Mathematicae, 2013, 194 (1), pp.119-145. ⟨10.1007/s00222-012-0443-y⟩. ⟨hal-00709802⟩