Related papers: The Tate conjecture for K3 surfaces over finite fi…

Given a finite field k of characteristic at least 5, we show that the Tate conjecture holds for K3 surfaces defined over the algebraic closure of k if and only if there are only finitely many K3 surfaces over each finite extension of k.

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3…

The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…

Recently N. Levin (Comp. Math. 127 (2001), 1--21) proved the Tate conjecture for ordinary cubic fourfolds over finite fields. In this paper we prove the Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is based on…

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of K3 surfaces over finite fields. We prove every K3 surface of finite height over a finite field admits a…

We investigate boundedness results for families of holomorphic symplectic varieties up to birational equivalence. We prove the analogue of Zarhin's trick by for $K3$ surfaces by constructing big line bundles of low degree on certain moduli…

Arithmetic of K3 surfaces defined over finite fields is investigated. In particular, we show that any K3 surface of finite height over a finite field k of characteristic p > 3 has a quasi-canonical lifting to characteristic 0, and that for…

We show that the classical Kuga-Satake construction gives rise, away from characteristic 2, to an open immersion from the moduli of primitively polarized K3 surfaces (of any fixed degree) to a certain regular integral model for a Shimura…

We prove that the supersingular K3 surface of Artin invariant 1 in characteristic p (where p denotes an arbitrary prime) admits a model over IF_p with Picard number 21.

We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two…

We give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over Z. This gives an analogue for K3 surfaces of Deligne's description of the category of ordinary abelian varieties over a…

A Fano surface of a smooth cubic threefold X in P^4 parametrizes the lines on X. In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2.

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. Andr\'{e} proved this conjecture for…

In 2009, Dolgachev-Keum showed that finite groups of tame symplectic automorphisms of K3 surfaces in positive characteristics are subgroups of the Mathieu group of degree 23. In this paper, we utilize lattice-theoretic methods to…

In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale…

The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a function…

For an abelian variety over a finite field, Clozel (1999) showed that l-homological equivalence coincides with numerical equivalence for infinitely many l, and the author (1999) gave a criterion for the Tate conjecture to follow from Tate's…

We show that the $\ell$-adic Tate conjecture for divisors on smooth proper varieties over finitely generated fields of positive characteristic follows from the $\ell$-adic Tate conjecture for divisors on smooth projective surfaces over…

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over…

We give a short proof that every supersingular K3 surface (except possibly in characteristic $2$ with Artin invariant $\sigma=10$) has an automorphism of Salem degree 22. In particular an infinite subgroup of the automorphism group does not…