  We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic
and show that the classical results over the complex numbers all generalize.
The key result is a positive-characteristic version of the Torelli theorem that
uses the derived category in place of the Hodge structure on singular
cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai
kernels to characteristic zero. As a consequence, any Shioda-supersingular K3
surface is uniquely determined up to isomorphism by its derived category of
coherent sheaves. We also study different realizations of Mukai's Hodge
structure in algebraic cohomology theories (etale, crystalline, de Rham) and
use these to prove: 1) the zeta function of a K3 surface is a derived invariant
(discovered independently by Huybrechts); 2) the variational crystalline Hodge
conjecture holds for correspondences arising from Fourier-Mukai kernels on
products of two K3 surfaces.
