Tate's conjecture for \(K3\) surfaces of finite height (Q1074673)

Let \(X/\mathbb{F}_q\) be a \(K3\) surface over a finite field \(\mathbb{F}_q\), \(q=p^d\) and \(h\) be the height of the formal Brauer group \(^{\wedge} X\). It is proved that Tate's conjecture on algebraic cycles and order of pole of zeta function is true for \(X\) provided that \(h<\infty\) and \(p\geq 5\). The result is deduced from the theorem that if \(V\) is a finite extension of \(W_q\), \(Y/V\) is a quasicanonical lifting of \(X\) then \(X\) satisfies the Tate conjecture. When \(p\geq 13\) the quasicanonical lifting is constructed by standard crystalline deformation theory. For \(p=5,7,11\), this depends on a comparison of the deformations of \(K3\) surfaces, Dieudonné modules of the dual of the enlarged formal Brauer groups and of abstract \(K3\) crystals. The paper contains the theory of quasi-canonical varieties over \(p\)-adic fields and of equicharacteristic deformations of abstract \(F\)-crystals.