Twisted derived equivalences and isogenies between K3 surfaces in positive characteristic

We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of isogenous to given surface $X$ terms certain integral sublattices the second rational $\ell$-adic and crystalline cohomology groups $X$. This characteristic analog Huybrechts, extends results author. give applications reduction types surjectivity period morphism. To prove these we describe theory B-fields Mukai lattices characteristic, which may be independent interest. also some on lifting twisted Fourier--Mukai equivalences 0, generalizing Lieblich Olsson.