The Tate conjecture for K3 surfaces over finite fields

Abstract

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 ≥ 5. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p ≥ 5. 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 ≥ 5.

Birational boundedness for holomorphic symplectic varieties, Zarhin's trick for K3 surfaces, and the Tate conjecture

“…Let S n,l,h , S n,h and S n,l be the orthogonal Shimura varieties with spin level n associated to Λ l,h , Λ h and Λ l respectively, see [Cha12,3.6]. Then these three varieties are all defined over Q and we have closed embeddings of S n,l,h into S n,h and S n,l .…”

“…In odd characteristic, it has been proved in [Mau12,Cha12], and independently in [MP13b]. The first of this proofs relies on results of Borcherds on the Picard group of Shimura varieties, while the second one uses construction of canonical models of certain Shimura varieties.…”

2-Adic Integral Canonical Models

“…Of course, the case when the field has characteristic = 2, this is already known by the results of the second author in [17], and also by earlier work by Maulik [21] and Charles [7] (among many others). The new ingredient here is the characteristic 2 case.…”

On the Integral Hodge and Tate Conjectures Over a Number Field

“…The integral Tate conjecture for a smooth projective variety X over the separable closure F of a finitely generated field is the weaker statement that, for k a finitely generated field of definition of X, every element of For example, the proof shows that the integral Hodge conjecture fails for the smooth hypersurface over Q. In this and the later examples, the proof shows more than 'Zariski-dense': the integral Hodge conjecture fails for a positive-density subset of all hypersurfaces over Q of bidegree (3,4), counted by height. The proof also shows that the integral Hodge conjecture fails for a set of hypersurfaces over Q which are dense in the space of all hypersurfaces over R with the classical topology.…”

“…(Note the typographical error in the first example in [16] [4,19,20,23].) We get around the problem by finding a quartic surface with a node in P 3 over F p which has geometric Picard number 1.…”