A Néron–Ogg–Shafarevich criterion for K3 surfaces

The naive analogue of the Néron–Ogg–Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified l-adic étale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if (Formula presented.) is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain ‘canonical reduction’ of X. We also prove the corresponding results for p-adic étale cohomology.